https://www.coastalwiki.org/w/index.php?title=Breaker_index&feed=atom&action=history
Breaker index - Revision history
2024-03-29T10:19:16Z
Revision history for this page on the wiki
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https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=80490&oldid=prev
Dronkers J at 15:09, 13 October 2023
2023-10-13T15:09:29Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:09, 13 October 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D \approx  Q_b \rho g \Large\frac{H^3}{4hT}\normalsize ,  \qquad (A3)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D \approx  Q_b \rho g \Large\frac{H^3}{4hT}\normalsize ,  \qquad (A3)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>T</math> is the peak wave period, <math>H</math> the wave <del class="diffchange diffchange-inline">height and </del><math><del class="diffchange diffchange-inline">Q_b</del></math> <del class="diffchange diffchange-inline">the fraction of broken waves in the surf zone. </del>Baldock et al. (1998<ref name=B98/>) and Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>) derived an analytical expression for <math>Q_b</math> by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]]). The following expression for the wave energy dissipation is found:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <ins class="diffchange diffchange-inline"><math>Q_b</math> the fraction of broken waves in the surf zone, </ins><math>T</math> is the peak wave period, <ins class="diffchange diffchange-inline">and </ins><math>H</math> the <ins class="diffchange diffchange-inline">height of an individual </ins>wave<ins class="diffchange diffchange-inline">. </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[For a derivation, see [[Tidal bore dynamics]] Eq. (5) with:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><math>\quad D = \Large\frac{Q \Delta E}{L}\normalsize, \quad D_1=h, \; D_2 - D_1=H, \quad v_2 = - \sqrt{gh}\sqrt{\Large\frac{H+2h}{2 (H+h)}\normalsize} \approx - \sqrt{gh}= -\Large\frac{L}{T}\normalsize \, , \;</math> where </ins><math><ins class="diffchange diffchange-inline">L=</ins></math> <ins class="diffchange diffchange-inline">wavelength ]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Baldock et al. (1998<ref name=B98/>) and Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>) derived an analytical expression for <math>Q_b</math> by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]]). The following expression for the wave energy dissipation is found:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td></tr>
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Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=80290&oldid=prev
Dronkers J at 15:22, 30 June 2023
2023-06-30T15:22:58Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:22, 30 June 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:WaveBreakingTypes.jpg|thumb|center|800px|Fig. 1. Schematic of wave evolution towards breaking, for swell waves (left panel) and sea waves (right panel).]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:WaveBreakingTypes.jpg|thumb|center|800px|Fig. 1. Schematic of wave evolution towards breaking, for swell waves (left panel) and sea waves (right panel).]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The process of wave breaking starts on the shoaling zone where incident waves become progressively skewed by interaction with the seabed: the wave crest becomes sharper and higher, while the wave trough widens and flattens (Fig. 1). Incident waves with short wavelengths (locally generated 'sea') are steeper than incident waves with longer wavelengths (remotely generated swell); the former category of waves will break earlier, thus at greater depth <math>h_b</math> than the latter category<ref name=A/><ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>, implying a smaller value of  <math>\gamma_b</math> <ref>Xu, J., Liu, S., Li, J. and Jia, W. 2021. Experimental study of wave height, crest, and trough distributions of directional irregular waves on a slope. Ocean Engineering 242: 110136</ref>. These short sea waves are also less skewed before breaking; they break when the crest becomes unstable and flows down the front face of the wave, producing spilling breaker bores that surf towards the shoreline (Fig. 2). This breaking mode is also favoured on gently sloping shorefaces when the [[surf similarity parameter]] <math>\xi=m/\sqrt{H_0/L_0}</math> (ratio of shoreface slope and square root of offshore wave steepness) has values typically smaller than about 0.4. Incident waves with longer wavelengths will break later, thus at smaller depth <math>h_b</math>. While shoaling, they are strongly skewed and break when the crest curls over the front face and falls into the base of the wave, producing a so-called plunging breaker<ref name=CL>Camenen, B. and Larson, M. 2007. Predictive Formulas for Breaker Depth Index and Breaker Type. Journal of Coastal Research 23: 1028–1041</ref> (Fig. 3). This breaking mode is also favoured on steep sloping shorefaces when the [[surf similarity parameter]] has values typically larger than 0.4.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The process of wave breaking starts on the shoaling zone where incident waves become progressively skewed by interaction with the seabed: the wave crest becomes sharper and higher, while the wave trough widens and flattens (Fig. 1). Incident waves with short wavelengths (locally generated 'sea') are steeper than incident waves with longer wavelengths (remotely generated swell); the former category of waves will break earlier, thus at greater depth <math>h_b</math> than the latter category<ref name=A/><ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>, implying a smaller value of  <math>\gamma_b</math> <ref>Xu, J., Liu, S., Li, J. and Jia, W. 2021. Experimental study of wave height, crest, and trough distributions of directional irregular waves on a slope. Ocean Engineering 242: 110136</ref>. These short sea waves are also less skewed before breaking; they break when the crest becomes unstable and flows down the front face of the wave, producing spilling breaker bores that surf towards the shoreline (Fig. 2). This breaking mode is also favoured on gently sloping shorefaces when the [[surf similarity parameter]] <math>\xi=m/\sqrt{H_0/L_0}</math> (ratio of shoreface slope and square root of offshore wave steepness) has values typically smaller than about 0.4. Incident waves with longer wavelengths will break later, thus at smaller depth <math>h_b</math>. While shoaling, they are strongly skewed and break when the crest curls over the front face and falls into the base of the wave, producing a so-called plunging breaker<ref name=CL>Camenen, B. and Larson, M. 2007. Predictive Formulas for Breaker Depth Index and Breaker Type. Journal of Coastal Research 23: 1028–1041</ref> (Fig. 3). This breaking mode is also favoured on steep sloping shorefaces when the [[surf similarity parameter]] has values typically larger than 0.4<ins class="diffchange diffchange-inline">. Direct measurements give an approx. 20% larger breaker index for plunging waves compared to spilling waves<ref>Weishar, L. L. and Byrne, R. J. 1978. Field study of breaking wave characteristics. In Procs. Coastal Eng. Conf. 1978, ASCE, pp. 487–506</ref><ref name=C21> Carini, R. J., Chickadel, C. C. and Jessup, A. T. 2021. Surf zone waves at the onset of breaking: 2. Predicting breaking and breaker type. Journal of Geophysical Research: Oceans 126, e2020JC016935</ref></ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:BreakerIndex.jpg|thumb|right|300px|Fig. 4. The area covered by experimental data for the breaker index as a function of the surf similarity parameter. The density of the experimental data is represented by varying degrees of redness. The black line is the Battjes formula Eq. 4.  Figure adapted from Ostendorf and Madsen (1979<ref name=OM>Ostendorf, D. and Madsen, O.  1979. An Analysis of Longshore Current and Associated Sediment Transport in the Surf Zone. Boston: Massachusetts Institute of Technology, Department of Civil Engineering Technical Report 241, 169p.</ref>).]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:BreakerIndex.jpg|thumb|right|300px|Fig. 4. The area covered by experimental data for the breaker index as a function of the surf similarity parameter. The density of the experimental data is represented by varying degrees of redness. The black line is the Battjes formula Eq. 4.  Figure adapted from Ostendorf and Madsen (1979<ref name=OM>Ostendorf, D. and Madsen, O.  1979. An Analysis of Longshore Current and Associated Sediment Transport in the Surf Zone. Boston: Massachusetts Institute of Technology, Department of Civil Engineering Technical Report 241, 169p.</ref>).]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A theoretical limit for <math>\gamma_b</math> was established by Miche (1944<ref>Miche, R. 1944. Mouvements ondulatoires de l’océan pour une eau profonde constante et décroissante. Annales des Ponts et Chaussées 114: 369-406</ref>), who analysed the wave motion over a horizontal seabed to determine the maximum crest steepness of the incident wave before collapse (occurring when the surface fluid velocity equals the wave crest celerity). <del class="diffchange diffchange-inline">The result is</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A theoretical limit for <math>\gamma_b</math> was established by Miche (1944<ref>Miche, R. 1944. Mouvements ondulatoires de l’océan pour une eau profonde constante et décroissante. Annales des Ponts et Chaussées 114: 369-406</ref>), who analysed the wave motion over a horizontal seabed to determine the maximum crest steepness of the incident wave before collapse (occurring when the surface fluid velocity equals the wave crest celerity). <ins class="diffchange diffchange-inline">Using a second-order theory of small-amplitude wave propagation, Miche found</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_b < \Large\frac{0.88}{k_b \, h_b}\normalsize \tanh(k_b\, h_b) , \quad k_b=2 \pi / L_b , \qquad (2)</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math><ins class="diffchange diffchange-inline">\Large\frac{H_b}{L_b}\normalsize < 0.14 \, \tanh(k_b \, h_b) \quad</math> or <math>\quad </ins>\gamma_b < \Large\frac{0.88}{k_b \, h_b}\normalsize \tanh(k_b\, h_b) , \quad k_b=2 \pi / L_b , \qquad (2)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>L_b</math> is the wavelength at the breakpoint. <del class="diffchange diffchange-inline">Miche used a second-order theory </del>of <del class="diffchange diffchange-inline">small-amplitude wave propagation</del>. <del class="diffchange diffchange-inline">Detailed </del>numerical modeling <del class="diffchange diffchange-inline">shows that wave crests may already break with a steepness value smaller than Miche's limiting criterion</del><ref>Derakhti, M., Kirby, J. T., Banner, M. L., Grilli, S. T. and Thomson, J. 2020. A unified breaking onset criterion for surface gravity water waves in arbitrary depth. Journal of Geophysical Research: Oceans 125: 1–28</ref>.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>L_b</math> is the wavelength at the breakpoint. <ins class="diffchange diffchange-inline">Field observations<ref name=C21> Carini, R. J., Chickadel, C. C. and Jessup, A. T. 2021. Surf zone waves at the onset </ins>of <ins class="diffchange diffchange-inline">breaking: 2. Predicting breaking and breaker type</ins>. <ins class="diffchange diffchange-inline">Journal of Geophysical Research: Oceans 126, e2020JC016935</ref> and detailed </ins>numerical modeling<ref>Derakhti, M., Kirby, J. T., Banner, M. L., Grilli, S. T. and Thomson, J. 2020. A unified breaking onset criterion for surface gravity water waves in arbitrary depth. Journal of Geophysical Research: Oceans 125: 1–28</ref> <ins class="diffchange diffchange-inline">indicate that wave crests already break with a steepness value <math>H_b/L_b</math> smaller than Miche's limiting criterion</ins>.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Experiments by Battjes (1974<ref name=B>Battjes, J.A. 1974. Surf similarity. Proceedings 14th International Conference on Coastal Engineering, pp. 466–480</ref>) suggested that the dependence of <math>\gamma_b</math> on wave steepness and bed slope could be represented by the [[surf similarity parameter]] <math>\xi</math> at the breakpoint,</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Experiments by Battjes (1974<ref name=B>Battjes, J.A. 1974. Surf similarity. Proceedings 14th International Conference on Coastal Engineering, pp. 466–480</ref>) suggested that the dependence of <math>\gamma_b</math> on wave steepness and bed slope could be represented by the [[surf similarity parameter]] <math>\xi</math> at the breakpoint,</div></td></tr>
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Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=80158&oldid=prev
Dronkers J at 15:04, 16 April 2023
2023-04-16T15:04:09Z
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<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:04, 16 April 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l44" >Line 44:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_b = 1.06 + 0.14 \ln \xi_b . \qquad (3)</math>  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_b = 1.06 + 0.14 \ln \xi_b . \qquad (3)</math>  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Another frequently used formula was established by Goda (2010<ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>)<del class="diffchange diffchange-inline">,</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Another frequently used formula was established by Goda (2010<ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{b}  = 0.17 \Large\frac{L_0}{h_{b}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{b}}{L_0}\normalsize (1+11m^{4/3})\big)\Big].\qquad (4)</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{b}  = 0.17 \Large\frac{L_0}{h_{b}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{b}}{L_0}\normalsize (1+11m^{4/3})\big)\Big]. \qquad (4)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This formula requires an iterative solution because of the dependence on <math>h_b</math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This formula requires an iterative solution because of the dependence on <math>h_b</math<ins class="diffchange diffchange-inline">>. Goda's formula overestimates the breaker index for <math>h_b/L_0 > 0.1</math> <ref>Kim, M., Lee, S. and Hong, J-W. 2022. Empirical estimation of the breaker index using a stereo camera system. Ocean Engineering 265, 112522</ref</ins>>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Irregular waves==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Irregular waves==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The formulas (2-<del class="diffchange diffchange-inline">5</del>) were derived for regular monochromatic waves. However, incident wave trains in nature usually have an irregular, random character. An irregular wave field can be characterized by a wave spectrum with root-mean-square wave height <math>H_{rms}</math> (see [[Statistical description of wave parameters]]). Since individual waves in a wave train have different breakpoints in the surf zone, the breaker index concept cannot be applied as such for <math>H_{rms}</math>. This problem can be addressed in two ways. The first way is to consider the breaking of individual waves in an incident wave train, represented by a wave-by-wave breaker index. The second way is to consider the location in the surf zone where <math>H_{rms}</math> starts to decay (the intersection of the shoaling zone where <math>H_{rms}</math> increases and the surf zone where <math>H_{rms}</math> decreases<ref>Kamphuis, J. W. 1991. Incipient wave breaking. Coastal Engineering 15: 185–203</ref>); this location is called the incipient breakpoint, indicated by the subscript <math>ib</math>. The wave-by-wave breaker index is similar the breaker index for regular waves<ref name=X>Xu, J., Liu, S., Li, J. and Jia, W. 2020. Experimental study of breaker index of normal and oblique incident unidirectional and multidirectional irregular waves on slope. Ocean Engineering 213, 107792</ref>. The incipient breaker index <math>\gamma_{ib}</math> is significantly smaller than the breaker index for regular waves, but the dependence on wave steepness and bed slope is similar. The empirical formulas for regular waves can be applied for irregular waves if the value of <math>\gamma_b</math> is reduced by about 50% <ref name=OM/><ref name=X/>. According to Goda (2010<ref name=G10/>), at incipient breaking (subscript <math>ib</math>):</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The formulas (2-<ins class="diffchange diffchange-inline">4</ins>) were derived for regular monochromatic waves. However, incident wave trains in nature usually have an irregular, random character. An irregular wave field can be characterized by a wave spectrum with root-mean-square wave height <math>H_{rms}</math> (see [[Statistical description of wave parameters]]). Since individual waves in a wave train have different breakpoints in the surf zone, the breaker index concept cannot be applied as such for <math>H_{rms}</math>. This problem can be addressed in two ways. The first way is to consider the breaking of individual waves in an incident wave train, represented by a wave-by-wave breaker index. The second way is to consider the location in the surf zone where <math>H_{rms}</math> starts to decay (the intersection of the shoaling zone where <math>H_{rms}</math> increases and the surf zone where <math>H_{rms}</math> decreases<ref>Kamphuis, J. W. 1991. Incipient wave breaking. Coastal Engineering 15: 185–203</ref>); this location is called the incipient breakpoint, indicated by the subscript <math>ib</math>. The wave-by-wave breaker index is similar the breaker index for regular waves<ref name=X>Xu, J., Liu, S., Li, J. and Jia, W. 2020. Experimental study of breaker index of normal and oblique incident unidirectional and multidirectional irregular waves on slope. Ocean Engineering 213, 107792</ref>. The incipient breaker index <math>\gamma_{ib}</math> is significantly smaller than the breaker index for regular waves, but the dependence on wave steepness and bed slope is similar. The empirical formulas for regular waves can be applied for irregular waves if the value of <math>\gamma_b</math> is reduced by about 50% <ref name=OM/><ref name=X/>. According to Goda (2010<ref name=G10/>), at incipient breaking (subscript <math>ib</math>):</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{ib} \equiv \Large\frac{H_{rms,ib}}{h_{ib}}\normalsize = 0.084 \Large\frac{L_0}{h_{ib}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{ib}}{L_0}\normalsize (1+11m^{4/3})\big)\Big] . \qquad (5)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{ib} \equiv \Large\frac{H_{rms,ib}}{h_{ib}}\normalsize = 0.084 \Large\frac{L_0}{h_{ib}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{ib}}{L_0}\normalsize (1+11m^{4/3})\big)\Big] . \qquad (5)</math></div></td></tr>
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Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=80157&oldid=prev
Dronkers J at 10:11, 16 April 2023
2023-04-16T10:11:06Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 10:11, 16 April 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The process of wave breaking plays a crucial role in the morphodynamics of the shoreface and the beach, such as accretion or erosion trends and the development and dynamics of nearshore bars and rip currents. The breaker index concept has been introduced to estimate the location on the shoreface where wave breaking occurs based on empirical formulas, without the need for a detailed simulation of the breaker process.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The process of wave breaking plays a crucial role in the morphodynamics of the shoreface and the beach, such as accretion or erosion trends and the development and dynamics of nearshore bars and rip currents. The breaker index concept has been introduced to estimate the location on the shoreface where wave breaking occurs based on empirical formulas, without the need for a detailed simulation of the breaker process.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_b < \Large\frac{0.88}{k_b \, h_b}\normalsize \tanh(k_b\, h_b) , \quad k_b=2 \pi / L_b , \qquad (2)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_b < \Large\frac{0.88}{k_b \, h_b}\normalsize \tanh(k_b\, h_b) , \quad k_b=2 \pi / L_b , \qquad (2)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>L_b</math> is the wavelength at the breakpoint. <del class="diffchange diffchange-inline">This relation was empirically refined by Ostendorf and Madsen (1979</del><ref <del class="diffchange diffchange-inline">name=OM</del>><del class="diffchange diffchange-inline">Ostendorf</del>, <del class="diffchange diffchange-inline">D</del>. <del class="diffchange diffchange-inline">and Madsen</del>, <del class="diffchange diffchange-inline">O</del>. <del class="diffchange diffchange-inline"> 1979. An Analysis of Longshore Current and Associated Sediment Transport in the Surf Zone</del>. <del class="diffchange diffchange-inline">Boston: Massachusetts Institute of Technology</del>, <del class="diffchange diffchange-inline">Department of Civil Engineering Technical Report 241</del>, <del class="diffchange diffchange-inline">169p</del>.<del class="diffchange diffchange-inline"></ref>) who proposed the formula</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>L_b</math> is the wavelength at the breakpoint. <ins class="diffchange diffchange-inline">Miche used a second-order theory of small-amplitude wave propagation. Detailed numerical modeling shows that wave crests may already break with a steepness value smaller than Miche's limiting criterion</ins><ref><ins class="diffchange diffchange-inline">Derakhti</ins>, <ins class="diffchange diffchange-inline">M</ins>., <ins class="diffchange diffchange-inline">Kirby, J</ins>. <ins class="diffchange diffchange-inline">T</ins>., <ins class="diffchange diffchange-inline">Banner</ins>, <ins class="diffchange diffchange-inline">M</ins>. <ins class="diffchange diffchange-inline">L</ins>., <ins class="diffchange diffchange-inline">Grilli</ins>, <ins class="diffchange diffchange-inline">S. T. and Thomson</ins>, <ins class="diffchange diffchange-inline">J</ins>. <ins class="diffchange diffchange-inline">2020</ins>. <ins class="diffchange diffchange-inline">A unified breaking onset criterion for surface gravity water waves in arbitrary depth</ins>. <ins class="diffchange diffchange-inline">Journal of Geophysical Research: Oceans 125: 1–28</ins></<ins class="diffchange diffchange-inline">ref</ins>><ins class="diffchange diffchange-inline">. </ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><math>\gamma_b =  \Large\frac{0</del>.<del class="diffchange diffchange-inline">88}{k_b h_b}\normalsize \tanh(p \</del>, <del class="diffchange diffchange-inline">k_b \</del>, <del class="diffchange diffchange-inline">h_b) </del>, <del class="diffchange diffchange-inline">\quad p = 0</del>.<del class="diffchange diffchange-inline">8 + 5 \; min(m, 0</del>.<del class="diffchange diffchange-inline">1)  </del>. <del class="diffchange diffchange-inline">\qquad (3)</del></<del class="diffchange diffchange-inline">math</del>></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Experiments by Battjes (1974<ref name=B>Battjes, J.A. 1974. Surf similarity. Proceedings 14th International Conference on Coastal Engineering, pp. 466–480</ref>) suggested that the dependence of <math>\gamma_b</math> on wave steepness and bed slope could be represented by the [[surf similarity parameter]] <math>\xi</math> at the breakpoint,</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Experiments by Battjes (1974<ref name=B>Battjes, J.A. 1974. Surf similarity. Proceedings 14th International Conference on Coastal Engineering, pp. 466–480</ref>) suggested that the dependence of <math>\gamma_b</math> on wave steepness and bed slope could be represented by the [[surf similarity parameter]] <math>\xi</math> at the breakpoint,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_b = 1.06 + 0.14 \ln \xi_b . \qquad (<del class="diffchange diffchange-inline">4</del>)</math>  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_b = 1.06 + 0.14 \ln \xi_b . \qquad (<ins class="diffchange diffchange-inline">3</ins>)</math>  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Another frequently used formula was established by Goda (2010<ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>),</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Another frequently used formula was established by Goda (2010<ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>),</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{b}  = 0.17 \Large\frac{L_0}{h_{b}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{b}}{L_0}\normalsize (1+11m^{4/3})\big)\Big].\qquad (<del class="diffchange diffchange-inline">5</del>)</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{b}  = 0.17 \Large\frac{L_0}{h_{b}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{b}}{L_0}\normalsize (1+11m^{4/3})\big)\Big].\qquad (<ins class="diffchange diffchange-inline">4</ins>)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">The formulas (3) and (5) both require </del>an iterative solution because of the dependence on <math>h_b</math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">This formula requires </ins>an iterative solution because of the dependence on <math>h_b</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l62" >Line 62:</td>
<td colspan="2" class="diff-lineno">Line 58:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The formulas (2-5) were derived for regular monochromatic waves. However, incident wave trains in nature usually have an irregular, random character. An irregular wave field can be characterized by a wave spectrum with root-mean-square wave height <math>H_{rms}</math> (see [[Statistical description of wave parameters]]). Since individual waves in a wave train have different breakpoints in the surf zone, the breaker index concept cannot be applied as such for <math>H_{rms}</math>. This problem can be addressed in two ways. The first way is to consider the breaking of individual waves in an incident wave train, represented by a wave-by-wave breaker index. The second way is to consider the location in the surf zone where <math>H_{rms}</math> starts to decay (the intersection of the shoaling zone where <math>H_{rms}</math> increases and the surf zone where <math>H_{rms}</math> decreases<ref>Kamphuis, J. W. 1991. Incipient wave breaking. Coastal Engineering 15: 185–203</ref>); this location is called the incipient breakpoint, indicated by the subscript <math>ib</math>. The wave-by-wave breaker index is similar the breaker index for regular waves<ref name=X>Xu, J., Liu, S., Li, J. and Jia, W. 2020. Experimental study of breaker index of normal and oblique incident unidirectional and multidirectional irregular waves on slope. Ocean Engineering 213, 107792</ref>. The incipient breaker index <math>\gamma_{ib}</math> is significantly smaller than the breaker index for regular waves, but the dependence on wave steepness and bed slope is similar. The empirical formulas for regular waves can be applied for irregular waves if the value of <math>\gamma_b</math> is reduced by about 50% <ref name=OM/><ref name=X/>. According to Goda (2010<ref name=G10/>), at incipient breaking (subscript <math>ib</math>):</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The formulas (2-5) were derived for regular monochromatic waves. However, incident wave trains in nature usually have an irregular, random character. An irregular wave field can be characterized by a wave spectrum with root-mean-square wave height <math>H_{rms}</math> (see [[Statistical description of wave parameters]]). Since individual waves in a wave train have different breakpoints in the surf zone, the breaker index concept cannot be applied as such for <math>H_{rms}</math>. This problem can be addressed in two ways. The first way is to consider the breaking of individual waves in an incident wave train, represented by a wave-by-wave breaker index. The second way is to consider the location in the surf zone where <math>H_{rms}</math> starts to decay (the intersection of the shoaling zone where <math>H_{rms}</math> increases and the surf zone where <math>H_{rms}</math> decreases<ref>Kamphuis, J. W. 1991. Incipient wave breaking. Coastal Engineering 15: 185–203</ref>); this location is called the incipient breakpoint, indicated by the subscript <math>ib</math>. The wave-by-wave breaker index is similar the breaker index for regular waves<ref name=X>Xu, J., Liu, S., Li, J. and Jia, W. 2020. Experimental study of breaker index of normal and oblique incident unidirectional and multidirectional irregular waves on slope. Ocean Engineering 213, 107792</ref>. The incipient breaker index <math>\gamma_{ib}</math> is significantly smaller than the breaker index for regular waves, but the dependence on wave steepness and bed slope is similar. The empirical formulas for regular waves can be applied for irregular waves if the value of <math>\gamma_b</math> is reduced by about 50% <ref name=OM/><ref name=X/>. According to Goda (2010<ref name=G10/>), at incipient breaking (subscript <math>ib</math>):</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{ib} \equiv \Large\frac{H_{rms,ib}}{h_{ib}}\normalsize = 0.084 \Large\frac{L_0}{h_{ib}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{ib}}{L_0}\normalsize (1+11m^{4/3})\big)\Big]</math><del class="diffchange diffchange-inline">.</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma_{ib} \equiv \Large\frac{H_{rms,ib}}{h_{ib}}\normalsize = 0.084 \Large\frac{L_0}{h_{ib}}\normalsize \Big[1-\exp\big(-\Large\frac{3 \pi}{2}\frac{h_{ib}}{L_0}\normalsize (1+11m^{4/3})\big)\Big] <ins class="diffchange diffchange-inline">. \qquad (5)</ins></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l73" >Line 73:</td>
<td colspan="2" class="diff-lineno">Line 69:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>in which for both terms explicit expressions as function <math>H(x)</math> have to be substituted. The wave energy dissipation depends on several factors, including the proportion of broken waves in the surf zone. If all waves are broken, the surf zone is said to be saturated, but for an irregular (random) wave field this is generally not the case. Explicit expressions for the terms in Eq. (6) have been proposed by Battjes and Janssen (1978<ref name=BJ>Battjes, J.A. and Janssen, J.P.F.M. 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Int. Conf. on Coastal Engineering. ASCE, New York, pp. 570–587</ref>) and Baldock et al. (1998<ref name=B98>Baldock, T.E., Holmes, P., Bunker, S. and Van Weert, P. 1998. Crossshore hydrodynamics within an unsaturated surf zone. Coast. Eng. 34: 173–196</ref>), but they rely on several assumptions (see the Appendix).</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>in which for both terms explicit expressions as function <math>H(x)</math> have to be substituted. The wave energy dissipation depends on several factors, including the proportion of broken waves in the surf zone. If all waves are broken, the surf zone is said to be saturated, but for an irregular (random) wave field this is generally not the case. Explicit expressions for the terms in Eq. (6) have been proposed by Battjes and Janssen (1978<ref name=BJ>Battjes, J.A. and Janssen, J.P.F.M. 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Int. Conf. on Coastal Engineering. ASCE, New York, pp. 570–587</ref>) and Baldock et al. (1998<ref name=B98>Baldock, T.E., Holmes, P., Bunker, S. and Van Weert, P. 1998. Crossshore hydrodynamics within an unsaturated surf zone. Coast. Eng. 34: 173–196</ref>), but they rely on several assumptions (see the Appendix).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Many laboratory and field investigations have therefore been conducted to establish empirical relationships for the dependence of wave height on water depth in the surf zone,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Many laboratory and field investigations have therefore been conducted to establish empirical relationships for the dependence of <ins class="diffchange diffchange-inline">the </ins>wave height <ins class="diffchange diffchange-inline"><math>H(x)</math> </ins>on <ins class="diffchange diffchange-inline">the local water depth <math>h(x)</math> in the surf zone. The simplest relationship is a constant ratio of wave height and </ins>water depth in the surf zone,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>H(x)<del class="diffchange diffchange-inline">/</del>h(x) <del class="diffchange diffchange-inline">= \gamma . </del>\<del class="diffchange diffchange-inline">qquad (7)</math>. </del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math><ins class="diffchange diffchange-inline">\Large\frac{</ins>H(x)<ins class="diffchange diffchange-inline">}{</ins>h(x)<ins class="diffchange diffchange-inline">}</ins>\<ins class="diffchange diffchange-inline">normalsize </ins>= \gamma_b  . \qquad (<ins class="diffchange diffchange-inline">7</ins>)</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">    </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">The simplest relationship is a constant ratio of wave height and water depth in the surf zone,</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><math>\gamma (x) </del>= \gamma_b  . \qquad (<del class="diffchange diffchange-inline">8</del>)</math></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Flume experiments suggest that this relationship is satisfied to some extent for rather steep shoreface profiles (<math>m \approx 1/30</math>), but not for more gently sloping shorefaces<ref>Dally, W. R., Dean, R. G. and Dalrymple, R. A. 1985. A model for breaker decay on beaches. Proc. 19th Int. Conf. on Coastal Engineering, Vol. 1, pp. 82–98</ref>. For more gently sloping profiles the ratio <math>H/h</math> displays a notable increase at small water depths<ref>Raubenheimer, B., Guza, R. T. and Elgar, S. 1996. Wave transformation across the inner surf zone. J. Geophys. Res. 101(C11): 25,589–25,597</ref><ref name=G10/><ref name=X/><ref>Power, H. E., Hughes, M. G., Aagaard, T. and Baldock, T. E. 2010. Nearshore wave height variation in unsaturated surf. J. Geophys. Res. 115, C08030</ref>. This may be due to wave regeneration after breaking, which commonly occurs in the troughs between [[nearshore sandbars]]. These regenerated waves propagate shoreward with less energy loss than broken wave bores. For an irregular wavefield, the increase in <math>H_s/h</math> may also be due to a large percentage of waves that have not yet broken past the incipient breakpoint; the proportion of these shoaling waves (still increasing in height) gradually decreases as the wave train travels towards the coast<ref name=G10/><ref name=X/>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Flume experiments suggest that this relationship is satisfied to some extent for rather steep shoreface profiles (<math>m \approx 1/30</math>), but not for more gently sloping shorefaces<ref>Dally, W. R., Dean, R. G. and Dalrymple, R. A. 1985. A model for breaker decay on beaches. Proc. 19th Int. Conf. on Coastal Engineering, Vol. 1, pp. 82–98</ref>. For more gently sloping profiles the ratio <math>H/h</math> displays a notable increase at small water depths<ref>Raubenheimer, B., Guza, R. T. and Elgar, S. 1996. Wave transformation across the inner surf zone. J. Geophys. Res. 101(C11): 25,589–25,597</ref><ref name=G10/><ref name=X/><ref>Power, H. E., Hughes, M. G., Aagaard, T. and Baldock, T. E. 2010. Nearshore wave height variation in unsaturated surf. J. Geophys. Res. 115, C08030</ref>. This may be due to wave regeneration after breaking, which commonly occurs in the troughs between [[nearshore sandbars]]. These regenerated waves propagate shoreward with less energy loss than broken wave bores. For an irregular wavefield, the increase in <math>H_s/h</math> may also be due to a large percentage of waves that have not yet broken past the incipient breakpoint; the proportion of these shoaling waves (still increasing in height) gradually decreases as the wave train travels towards the coast<ref name=G10/><ref name=X/>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The <del class="diffchange diffchange-inline">use of Eq. (7) </del>can be justified if the surf zone depth decreases monotonically towards the shoreline, with an approximately constant slope <math>m</math>. In other situations there is no reason to assume <del class="diffchange diffchange-inline">that the wave height is related to the local depth</del><ref name=BJ/>. For example, for a wave propagating across a surf zone with [[nearshore sandbars]], the wave height at the trough between sandbars depends on the crest height of the previous sandbar.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <ins class="diffchange diffchange-inline">assumption that the wave height depends on the local depth </ins>can be justified if the surf zone depth decreases monotonically towards the shoreline, with an approximately constant slope <math>m</math>. In other situations there is no reason to assume <ins class="diffchange diffchange-inline">such a dependence</ins><ref name=BJ/>. For example, for a wave propagating across a surf zone with [[nearshore sandbars]], the wave height at the trough between sandbars depends on the crest height of the previous sandbar.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where the approximation holds for gently sloping surf zone profiles (<math>m<0.01</math>) and where <math>erf</math> is the error function. The expression (A4) represents wave dissipation by breaking waves. Dissipation by other processes such as energy dissipation due to bottom friction and energy <del class="diffchange diffchange-inline">transferring </del>to <del class="diffchange diffchange-inline">long </del>waves <del class="diffchange diffchange-inline">also </del>need to be considered in specific situations.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where the approximation holds for gently sloping surf zone profiles (<math>m<0.01</math>) and where <math>erf</math> is the error function. The expression (A4) represents wave dissipation by breaking waves. Dissipation by other processes such as energy dissipation due to bottom friction and energy <ins class="diffchange diffchange-inline">transfer </ins>to <ins class="diffchange diffchange-inline">[[infragravity </ins>waves<ins class="diffchange diffchange-inline">]] are usually small but </ins>need to be considered in specific situations.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An expression of the breaker wave height in the surf zone <math>H_b(x)</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An expression of the breaker wave height in the surf zone <math>H_b(x)</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td></tr>
</table>
Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=79961&oldid=prev
Dronkers J at 21:24, 9 December 2022
2022-12-09T21:24:24Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:24, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l99" >Line 99:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>g</math> is the gravitational acceleration, <math>\rho</math> the seawater density, <math>\overline \theta</math> the mean incident wave direction and <math>c_g</math> the wave group velocity, which in shallow water is about equal to the wave propagation speed <math>c \approx \sqrt{gh}</math>.  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>g</math> is the gravitational acceleration, <math>\rho</math> the seawater density, <math>\overline \theta</math> the mean incident wave direction and <math>c_g</math> the wave group velocity, which in shallow water is about equal to the wave propagation speed <math>c \approx \sqrt{gh}</math>.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>An expression for the energy dissipation <math>D</math> by breaking waves was derived by Battjes and Janssen (1978<ref name=BJ/>) based on analogy with the power dissipated by a bore</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>An expression for the energy dissipation <math>D</math> by breaking waves was derived by Battjes and Janssen (1978<ref name=BJ/>) based on analogy with the power dissipated by a bore<ins class="diffchange diffchange-inline">,</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D \approx  Q_b \rho g \Large\frac{H^3}{4hT}\normalsize ,  \qquad (A3)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D \approx  Q_b \rho g \Large\frac{H^3}{4hT}\normalsize ,  \qquad (A3)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>T</math> is the peak wave period, <math>H</math> the wave height and <math>Q_b</math> the fraction of broken waves in the surf zone. <del class="diffchange diffchange-inline">Alsina and </del>Baldock (<del class="diffchange diffchange-inline">2007</del><ref<del class="diffchange diffchange-inline">>Alsina, J.M. and Baldock, T.E. 2007. Improved representation of breaking wave energy dissipation in parametric wave transformation models. Coast. Eng. 54: 765–769<</del>/<del class="diffchange diffchange-inline">ref</del>>) and Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>) derived an analytical expression for <math>Q_b</math> by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]]). The following expression for the wave energy dissipation <del class="diffchange diffchange-inline"><math>D</math> </del>is found:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>T</math> is the peak wave period, <math>H</math> the wave height and <math>Q_b</math> the fraction of broken waves in the surf zone. Baldock <ins class="diffchange diffchange-inline">et al. </ins>(<ins class="diffchange diffchange-inline">1998</ins><ref <ins class="diffchange diffchange-inline">name=B98</ins>/>) and Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>) derived an analytical expression for <math>Q_b</math> by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]]). The following expression for the wave energy dissipation is found:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where the approximation<del class="diffchange diffchange-inline"><ref name=B98/> </del>holds for gently sloping surf zone profiles (<math>m<0.01</math>) and where <math>erf</math> is the error function. The expression (A4) represents wave dissipation by breaking waves. Dissipation by other processes such as energy dissipation due to bottom friction and energy transferring to long waves also need to be considered in specific situations.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where the approximation holds for gently sloping surf zone profiles (<math>m<0.01</math>) and where <math>erf</math> is the error function. The expression (A4) represents wave dissipation by breaking waves. Dissipation by other processes such as energy dissipation due to bottom friction and energy transferring to long waves also need to be considered in specific situations.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An expression of the breaker wave height in the surf zone <math>H_b(x)</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>An expression of the breaker wave height in the surf zone <math>H_b(x)</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td></tr>
</table>
Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=79960&oldid=prev
Dronkers J at 18:57, 9 December 2022
2022-12-09T18:57:44Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:57, 9 December 2022</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>g</math> is the gravitational acceleration, <math>\rho</math> the seawater density, <math>\overline \theta</math> the mean incident wave direction and <math>c_g</math> the wave group velocity, which in shallow water is about equal to the wave propagation speed <math>c \approx \sqrt{gh}</math>.  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>g</math> is the gravitational acceleration, <math>\rho</math> the seawater density, <math>\overline \theta</math> the mean incident wave direction and <math>c_g</math> the wave group velocity, which in shallow water is about equal to the wave propagation speed <math>c \approx \sqrt{gh}</math>.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>An expression for the <del class="diffchange diffchange-inline">wave </del>energy dissipation <math>D</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on analogy with the power dissipated by a bore</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>An expression for the energy dissipation <math>D</math> <ins class="diffchange diffchange-inline">by breaking waves </ins>was derived by Battjes and Janssen (1978<ref name=BJ/>) based on analogy with the power dissipated by a bore</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D \approx  Q_b \rho g \Large\frac{H^3}{4hT}\normalsize ,  \qquad (A3)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D \approx  Q_b \rho g \Large\frac{H^3}{4hT}\normalsize ,  \qquad (A3)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>T</math> is the peak wave period, <math>H</math> the wave height and <math>Q_b</math> the fraction of broken waves. Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>)<del class="diffchange diffchange-inline">, </del>by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]])<del class="diffchange diffchange-inline">, derived </del>for <math>D</math> <del class="diffchange diffchange-inline">the approximate expression</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>T</math> is the peak wave period, <math>H</math> the wave height and <math>Q_b</math> the fraction of broken waves <ins class="diffchange diffchange-inline">in the surf zone. Alsina and Baldock (2007<ref>Alsina, J.M. and Baldock, T.E. 2007. Improved representation of breaking wave energy dissipation in parametric wave transformation models</ins>. <ins class="diffchange diffchange-inline">Coast. Eng. 54: 765–769</ref>) and </ins>Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>) <ins class="diffchange diffchange-inline">derived an analytical expression for <math>Q_b</math> </ins>by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]])<ins class="diffchange diffchange-inline">. The following expression </ins>for <ins class="diffchange diffchange-inline">the wave energy dissipation </ins><math>D</math> <ins class="diffchange diffchange-inline">is found:</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] \approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where the approximation<ref name=B98/> holds for gently sloping surf zone profiles (<math>m<0.01</math>) and where <math>erf</math> is the error function. An expression of the breaker wave height in the surf zone <math>H_b(x)</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where the approximation<ref name=B98/> holds for gently sloping surf zone profiles (<math>m<0.01</math>) and where <math>erf</math> is the error function. <ins class="diffchange diffchange-inline">The expression (A4) represents wave dissipation by breaking waves. Dissipation by other processes such as energy dissipation due to bottom friction and energy transferring to long waves also need to be considered in specific situations.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>An expression of the breaker wave height in the surf zone <math>H_b(x)</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>H_b=\Large\frac{0.88}{k}\normalsize \tanh(\large\frac{\gamma}{0.88}\normalsize kh) , \qquad (A5)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>H_b=\Large\frac{0.88}{k}\normalsize \tanh(\large\frac{\gamma}{0.88}\normalsize kh) , \qquad (A5)</math></div></td></tr>
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Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=79959&oldid=prev
Dronkers J at 15:49, 9 December 2022
2022-12-09T15:49:25Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:49, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l114" >Line 114:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Different empirical relationships have been established for the breaker index <math>\gamma</math> defined by equation (A5), by solving Eq. (A1) and comparing the results with data from laboratory experiments and field surveys. Battjes and Stive (1985<ref>Battjes, J. A. and</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Different empirical relationships have been established for the breaker index <math>\gamma</math> defined by equation (A5), by solving Eq. (A1) and comparing the results with data from laboratory experiments and field surveys. Battjes and Stive (1985<ref>Battjes, J. A. and</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Stive , M. J. F . 1985. Calibration and verification of a dissipation model for random breaking waves. Journal of Geophysical Research: Oceans 90: 9159-9167</ref>) found a dependence of <math>\gamma</math> on the offshore wave steepness <math>S_0=H_{rms,0}/L_0</math>, whereas Ruessink et al. (2003<ref>Ruessink, B.G., Walstra, D.J.R. and Southgate, H.N. 2003. Calibration and verification of a parametric wave model on barred beaches. Coastal Eng. 48: 139–149</ref>) found that the breaker index for beaches with nearshore sandbars depends on <math>kh</math>. Zhang et al. (2021<ref>Zhang, C., Li, Y., Cai, Y., Shi, J., Zheng, J., Cai, F. and Qi, H. 2021. Parameterization of nearshore wave breaker index. Coast Eng. 168, 103914</ref>) found that <math>\gamma</math> is an increasing function of <math>kh</math> for large values of the offshore wave steepness <math>S_0</math>, and a decreasing function of <math>kh</math> for smaller offshore wave steepness, according to the empirical formula</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Stive , M. J. F . 1985. Calibration and verification of a dissipation model for random breaking waves. Journal of Geophysical Research: Oceans 90: 9159-9167</ref>) found a dependence of <math>\gamma</math> on the offshore wave steepness <math>S_0=H_{rms,0}/L_0</math>, whereas Ruessink et al. (2003<ref>Ruessink, B.G., Walstra, D.J.R. and Southgate, H.N. 2003. Calibration and verification of a parametric wave model on barred beaches. Coastal Eng. 48: 139–149</ref>) found that the breaker index for beaches with nearshore sandbars depends on <math>kh</math>. <ins class="diffchange diffchange-inline">Analyzing a large set of field data, </ins>Zhang et al. (2021<ref>Zhang, C., Li, Y., Cai, Y., Shi, J., Zheng, J., Cai, F. and Qi, H. 2021. Parameterization of nearshore wave breaker index. Coast Eng. 168, 103914</ref>) found that <math>\gamma</math> is an increasing function of <math>kh</math> for large values of the offshore wave steepness <math>S_0</math>, and a decreasing function of <math>kh</math> for smaller offshore wave steepness, according to the empirical formula<ins class="diffchange diffchange-inline">, valid for <math>0<S_0<0.05</math> and <math>0.3<kh<1.2</math>:</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma = (237 \, S_0^2-34.8 \, S_0 +1.46) \, \exp \big(1.96 \, kh \, \ln(38.64 \, S_0)\big) . \qquad (A6)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma = (237 \, S_0^2-34.8 \, S_0 +1.46) \, \exp \big(1.96 \, kh \, \ln(38.64 \, S_0)\big) . \qquad (A6)</math></div></td></tr>
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Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=79937&oldid=prev
Dronkers J at 10:13, 30 October 2022
2022-10-30T10:13:16Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 10:13, 30 October 2022</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>==Appendix==</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>==Appendix <ins class="diffchange diffchange-inline">Surf zone decay irregular waves</ins>==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wave energy dissipation in the surf zone can be described by Eq. (6),</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wave energy dissipation in the surf zone can be described by Eq. (6),</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>T</math> is the peak wave period, <math>H</math> the wave height and <math>Q_b</math> the fraction of broken waves. Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>), by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]]), derived for <math>D</math> the approximate expression</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>T</math> is the peak wave period, <math>H</math> the wave height and <math>Q_b</math> the fraction of broken waves. Janssen and Battjes (2007<ref>Janssen, T.T. and Battjes, J.A. 2007. A note on wave energy dissipation over steep beaches. Coastal Engineering 54: 711–716</ref>), by assuming a Rayleigh distribution of the incident waves (see [[Statistical description of wave parameters]]), derived for <math>D</math> the approximate expression</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] , \quad R=\Large\frac{H_b}{H_{rms}}\normalsize <del class="diffchange diffchange-inline">  </del>, \qquad (A4)</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>D =  \rho g  \Large\frac{H_{rms}^3}{4hT}\normalsize \; \Big[(R^3 + \frac{3}{2} R) \exp(-R^2) \, + \, \Large\frac{3 \sqrt{\pi}}{4}\normalsize \, (1-erf(R)) \Big] <ins class="diffchange diffchange-inline">\approx  \rho g \, \Large\frac{H_{rms}^2}{4T}\normalsize \, (1+R^2) \, \exp(-R^2) </ins>, \quad R=\Large\frac{H_b}{H_{rms}}\normalsize, \qquad (A4)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>erf</math> is the error function. An expression of the breaker wave height <math>H_b</math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">where the approximation<ref name=B98/> holds for gently sloping surf zone profiles (<math>m<0.01</math>) and </ins>where <math>erf</math> is the error function. An expression of the breaker wave height <ins class="diffchange diffchange-inline">in the surf zone </ins><math>H_b<ins class="diffchange diffchange-inline">(x)</ins></math> was derived by Battjes and Janssen (1978<ref name=BJ/>) based on Miche's criterion,  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>H_b=\Large\frac{0.88}{k}\normalsize \tanh(\large\frac{\gamma}{0.88}\normalsize kh) , \qquad (A5)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>H_b=\Large\frac{0.88}{k}\normalsize \tanh(\large\frac{\gamma}{0.88}\normalsize kh) , \qquad (A5)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>k=2 \pi / (cT)</math> is the wave number <del class="diffchange diffchange-inline">and </del><math><del class="diffchange diffchange-inline">\gamma</del></math> <del class="diffchange diffchange-inline">is a </del>breaker <del class="diffchange diffchange-inline">index (</del><math>H_b \approx \gamma h</math> <del class="diffchange diffchange-inline">in shallow water</del>).  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>k=2 \pi / (cT)</math> is the wave number<ins class="diffchange diffchange-inline">. In shallow water (</ins><math><ins class="diffchange diffchange-inline">kh << 1</ins></math><ins class="diffchange diffchange-inline">), the </ins>breaker <ins class="diffchange diffchange-inline">wave height </ins><math>H_b \approx \gamma h</math><ins class="diffchange diffchange-inline">. The breaker index <math>\gamma</math> should not be confused with <math>\gamma_b</math> (Eq. 1</ins>).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Different empirical relationships have been established for the breaker index <math>\gamma</math> defined by equation (A5), by solving Eq. (A1) and comparing the results with data from laboratory experiments and field surveys. Battjes and Stive (1985<ref>Battjes, J. A. and</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Different empirical relationships have been established for the breaker index <math>\gamma</math> defined by equation (A5), by solving Eq. (A1) and comparing the results with data from laboratory experiments and field surveys. Battjes and Stive (1985<ref>Battjes, J. A. and</div></td></tr>
</table>
Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=79936&oldid=prev
Dronkers J at 21:14, 29 October 2022
2022-10-29T21:14:57Z
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<a href="https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=79936&oldid=79934">Show changes</a>
Dronkers J
https://www.coastalwiki.org/w/index.php?title=Breaker_index&diff=79934&oldid=prev
Dronkers J at 15:54, 28 October 2022
2022-10-28T15:54:33Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:54, 28 October 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16" >Line 16:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:WaveBreakingTypes.jpg|thumb|center|800px|Fig. 1. Schematic of wave evolution towards breaking, for swell waves (left panel) and sea waves (right panel).]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:WaveBreakingTypes.jpg|thumb|center|800px|Fig. 1. Schematic of wave evolution towards breaking, for swell waves (left panel) and sea waves (right panel).]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The process of wave breaking starts on the shoaling zone where incident waves become progressively skewed by interaction with the seabed: the wave crest becomes sharper and higher, while the wave trough widens and flattens (Fig. 1). Incident waves with short wavelengths (locally generated 'sea') are steeper than incident waves with longer wavelengths (remotely generated swell); the former category of waves will break earlier, thus at greater depth <math>h_b</math> than the latter category<ref name=A/><ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>, implying a <del class="diffchange diffchange-inline">larger </del>value of  <math>\gamma_b</math>. These short sea waves are also less skewed before breaking; they break when the crest becomes unstable and flows down the front face of the wave, producing spilling breaker bores that surf towards the shoreline (Fig. 2). This breaking mode is also favoured on gently sloping shorefaces when the [[surf similarity parameter]] <math>\xi=m/\sqrt{H_0/L_0}</math> has values typically smaller than about 0.4. Incident waves with longer wavelengths will break later, thus at smaller depth <math>h_b</math>. While shoaling, they are strongly skewed and break when the crest curls over the front face and falls into the base of the wave, producing a so-called plunging breaker<ref name=CL>Camenen, B. and Larson, M. 2007. Predictive Formulas for Breaker Depth Index and Breaker Type. Journal of Coastal Research 23: 1028–1041</ref> (Fig. 3). This breaking mode is also favoured on steep sloping shorefaces when the [[surf similarity parameter]] has values typically larger than 0.4.  </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The process of wave breaking starts on the shoaling zone where incident waves become progressively skewed by interaction with the seabed: the wave crest becomes sharper and higher, while the wave trough widens and flattens (Fig. 1). Incident waves with short wavelengths (locally generated 'sea') are steeper than incident waves with longer wavelengths (remotely generated swell); the former category of waves will break earlier, thus at greater depth <math>h_b</math> than the latter category<ref name=A/><ref name=G10>Goda, Y. 2010. Reanalysis of regular and random breaking wave statistics. Coastal Engineering Journal 52: 71-106</ref>, implying a <ins class="diffchange diffchange-inline">smaller </ins>value of  <math>\gamma_b</math<ins class="diffchange diffchange-inline">> <ref>Xu, J., Liu, S., Li, J. and Jia, W. 2021. Experimental study of wave height, crest, and trough distributions of directional irregular waves on a slope. Ocean Engineering 242: 110136</ref</ins>>. These short sea waves are also less skewed before breaking; they break when the crest becomes unstable and flows down the front face of the wave, producing spilling breaker bores that surf towards the shoreline (Fig. 2). This breaking mode is also favoured on gently sloping shorefaces when the [[surf similarity parameter]] <math>\xi=m/\sqrt{H_0/L_0}</math> has values typically smaller than about 0.4. Incident waves with longer wavelengths will break later, thus at smaller depth <math>h_b</math>. While shoaling, they are strongly skewed and break when the crest curls over the front face and falls into the base of the wave, producing a so-called plunging breaker<ref name=CL>Camenen, B. and Larson, M. 2007. Predictive Formulas for Breaker Depth Index and Breaker Type. Journal of Coastal Research 23: 1028–1041</ref> (Fig. 3). This breaking mode is also favoured on steep sloping shorefaces when the [[surf similarity parameter]] has values typically larger than 0.4.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l112" >Line 112:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma \approx  0.29+0.76 \, kh  . \qquad (A6)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\gamma \approx  0.29+0.76 \, kh  . \qquad (A6)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Zhang et al. (2021<ref>Zhang, C., Li, Y., Cai, Y., Shi, J., Zheng, J., Cai, F. and Qi, H. 2021. Parameterization of nearshore wave breaker index. Coast Eng. 168, 103914</ref>) and Myrhaug et al. (2022<ref>Myrhaug, D., Wang, H. and Holmedal, L.E. 2022. Discussion/comments of “Parameterization of nearshore wave breaker index” by Chi Zhang, Yuan Li, Yu Cai, Jian Shi, Jinhai Zheng, Feng Cai and Hong Shuai. Coastal Engineering 173, 104042</ref>) provided evidence that the dependence of the breaker index <math>\gamma</math> on <math>kh</math> is in fact a function of the offshore wave steepness.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Zhang et al. (2021<ref>Zhang, C., Li, Y., Cai, Y., Shi, J., Zheng, J., Cai, F. and Qi, H. 2021. Parameterization of nearshore wave breaker index. Coast Eng. 168, 103914</ref>) and Myrhaug et al. (2022<ref>Myrhaug, D., Wang, H. and Holmedal, L.E. 2022. Discussion/comments of “Parameterization of nearshore wave breaker index” by Chi Zhang, Yuan Li, Yu Cai, Jian Shi, Jinhai Zheng, Feng Cai and Hong Shuai. Coastal Engineering 173, 104042</ref>) provided evidence that the dependence of the breaker index <math>\gamma</math> on <math>kh</math> is in fact a function of the offshore wave steepness.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>    </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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Dronkers J