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Revision as of 14:20, 29 June 2019
This article describes the physical processes responsible for tidal wave deformation in shallow coastal inlet systems – tidal lagoons and estuaries. A qualitative discussion is given of the mutual interaction between tidal asymmetry generation and morphological development of these systems, which is generally referred to by the term selforganizing morphodynamics. This article is largely based on the book Dynamics of Coastal Systems ^{[1]}.
Contents
 1 Tidal wave deformation in shallow water
 2 Tidal wave deformation in the absence of friction
 3 Tidal wave deformation in the presence of friction and intertidal areas
 4 Morphology of shallow tidal basins with small river inflow
 5 Tidal wave deformation in a converging channel
 6 Morphology of estuaries with tidal rivers
 7 Related articles
 8 References
 9 Appendix
Tidal wave deformation in shallow water
Tides result from the response of the ocean water bodies to the attractive gravitational forces by sun and moon. Tidal motion in the oceans can be described by a limited number of sinusoidal components, because earth's rotation and the relative movements of sun and moon have a cyclical character, see the article Ocean and shelf tides. The semidiurnal lunar tide (M2) is usually the dominant component, in which case the ocean tide can be described fairly accurately with a single sine function.
However, it is well known that this simple description is not valid for shallow coastal areas. The tidal wave is distorted as it propagates from the ocean to coastal areas where the tidal amplitude is no longer negligibly small compared to the average water depth. Negligibly small means: a tidal amplitude that is no greater than a few percent of the water depth. In some coastal areas, particularly in estuaries and tidal lagoons, the tidal amplitude is just a few times smaller than the average water depth or even of similar magnitude.
In shallow coastal areas, the distortion of the tide can be so strong that the durations of rising tide and falling tide become very different and that a large difference arises between the peak flow velocities of flood and ebb. Often the duration of rising tide is much shorter than the duration of falling tide. This is illustrated in Fig. 1 for the Hooghly estuary. In the most extreme case, the duration of tidal rise becomes so short that a hydraulic jump develops at the front of the tidal wave. The front of the tidal wave appears as a propagating wall of water, a socalled tidal bore.
Tidal wave deformation in the absence of friction
The distortion of the tide described above is related to a difference in propagation speed of the highwater crest of the tidal wave and the lowwater trough. This difference already occurs when the water depth below the wave crest is slightly greater than the water depth below the wave trough. This can be easily demonstrated for a tidal wave that propagates in [math]x[/math]direction with time [math]t[/math] in deep water with little loss of friction. The tidal equations (momentum balance and mass balance) can in this case be simplified to
[math]\Large\frac{\partial u}{\partial t}\normalsize + u \Large \frac{\partial u}{\partial x}\normalsize + g \Large \frac{\partial \zeta}{\partial x}\normalsize =0 , \quad \quad (1)[/math]
[math]\Large \frac{\partial \zeta}{\partial t}\normalsize + h \Large \frac{\partial u}{\partial x}\normalsize + \Large \frac{\partial u \zeta }{\partial x}\normalsize =0 , \quad \quad (2)[/math]
where [math]\zeta[/math] is the tidal wave surface elevation, [math]u[/math] is the depthaveraged tidal velocity, [math]h[/math] is the mean water depth and [math]g[/math] the gravitational acceleration. The equations are linear except for the second term in Eq. (1) and the last term in Eq. (2). These nonlinear terms are much smaller in the present case than the linear terms.
The equations can therefore be solved to a first approximation by substituting the solution [math]\zeta^{(1)}[/math] of the linear equations into the nonlinear terms. The solution of the resulting linearized equations introduces a small M4 tidal component [math]\zeta^{(2)}[/math] with twice the M2tidal frequency. This M4 component modifies the duration of rising water and falling water, because the propagation speed [math]c^+[/math] of the highwater wave crest is higher than the propagation speed [math]c^[/math] of the lowwater wave trough:
[math]c^{\pm}=(1 \pm \Large \frac{3a}{2h}\normalsize) \; \sqrt{gh} , \quad \quad(3)[/math]
where [math]a[/math] is the amplitude of the M2 tide. The resulting tidal distortion is illustrated in Fig. 2; a derivation is given in the appendix. This equation suggests that after some time the highwater wave crest overtakes the lowwater wave trough. However, this can only happen when the amplitude of the M4 is of the same order of magnitude as the M2 tide, which violates the assumption underlying the linearization of the tidal equations.
The approximate validity of the result (3) is restricted to the zone where M4 is much smaller than M2. In spite of the restricted validity of Eq. (3) to large water depths and small tidal amplitude, the following general rule remains valid: the difference in propagation speed between high water and low water increases when the ratio [math]a/h[/math] increases. Numerical simulations with fully nonlinear mathematical models show that tidal distortion, with decreasing tidal rise period, occurs when the value of the parameter [math]a/h[/math] remains sufficiently large when the tidal wave propagates over a shallow seabed ^{[3]}. In this case, the tidal wave transforms into a tidal bore when the propagation distance over the shallow seabed is sufficiently large.
Tidal wave deformation in the presence of friction and intertidal areas
Tidal asymmetry develops during upchannel propagation into a shallow tidal basin. Pronounced tidal asymmetry (possibly leading to tidal bore formation) only occurs if during propagation a sufficiently large ratio [math]a/h[/math] is maintained. In many cases this condition is not met, because the tidal amplitude decreases during propagation. The two main reasons for decrease of the tidal amplitude are: (1) tidal wave damping by friction and (2) lateral spreading of the flood tidal wave. These two nonlinear processes also influence the propagation of the highwater wave crest and the lowwater wave trough. This can be demonstrated in the same way as previously, assuming that the nonlinear terms in the equations are much smaller than the linear terms.
We consider a straight prismatic tidal channel with mean depth [math]h[/math] , channel width [math]b_C[/math] and lateral flood storage zones of width [math]b_S= b_{S0} \; ( 1 + \zeta / a)[/math], where [math]\zeta / a[/math] is the ratio of tidal surface elevation and tidal amplitude, see Fig. 3 . When frictional effects are much stronger than inertial effects, the onedimensional crosssectionaveraged tidal equations for small values of [math]a/h[/math] can be simplified to^{[4]}
[math]g \Large \frac{\partial \zeta}{\partial x}\normalsize + r \Large \frac{u}{h+\zeta}\normalsize \approx g\Large \frac{\partial \zeta}{\partial x}\normalsize + r \Large \frac{u}{h}\normalsize – r \Large \frac{u \zeta}{h^2}\normalsize =0 , \quad \quad (4)[/math]
[math](b_C+ b_S) \Large \frac{\partial \zeta}{\partial t} \normalsize + \Large \frac{\partial (u b_C(h+\zeta)) }{\partial x} \normalsize = (b_C+ b_{S0}) \Large \frac{\partial \zeta}{\partial t} \normalsize + b_{S0} \Large \frac{\zeta}{a} \frac{\partial \zeta}{\partial t} \normalsize + h b_C \Large \frac{\partial u}{\partial x}\normalsize + b_C \Large \frac{\partial (\zeta u)}{\partial x}\normalsize =0 . \quad \quad (5)[/math]
The friction term in Eq. (4) has been linearized; [math]r[/math] is the associated friction coefficient (dimension [m/s]) that relates momentum dissipation at the channel bed to the depthaveraged current velocity. Its value typically ranges between 0.001  0.003 m/s ^{[1]}. The precise value of the friction coefficient for estuarine flow is generally not well known as it is influenced by many factors such as density stratification (lower friction), small bed forms (higher friction) and fluid mud layers (lower friction). The friction coefficient may even be different for ebb and flood because of differences in salinity stratification. Apart from salinity stratification effects, there is no significant influence of river flow on bed friction in most estuaries. The linearization of the friction term is often a minor approximation compared to the uncertainty in the value of the friction coefficient. Neglect of the inertial term [math]\partial u / \partial t[/math] in the momentum equation is a valid approximation if [math]r / h \omega \lt \lt 1[/math]. In shallow estuaries, [math]h \le 5 m[/math], the friction term is on average a few times larger than the inertial term.
The nonlinear terms in the tidal equations are much smaller than the linear terms if [math]b_S \lt \lt b_C[/math] and [math]a/h \lt \lt 1[/math]. In this case a first order solution [math]\zeta^{(1)}[/math] can be found by substituting the solution of the linear equations in the nonlinear terms. The nonlinear terms then generate a small M4 tidal component [math]\zeta^{(2)}[/math], which affects the duration of tidal rise and tidal fall. This is because the propagation speed [math]c^+[/math] of the highwater wave crest differs from the propagation speed [math]c^[/math] of the lowwater wave trough (see the appendix):
[math]c^{\pm} \approx [1 \pm (2  \sqrt{2}) \Large\frac{a}{h}\normalsize \mp \Large\frac{b_{S0}}{2b_C}\normalsize ] \; \sqrt{gh} \; \sqrt{ \Large\frac{2 \omega h}{r} \frac{b_C}{b_C +b_{S0}}\normalsize} , \quad \quad (6) [/math]
where [math]\omega[/math] is the M2 radial frequency. This expression shows that the tidal propagation speed is reduced for increasing values of the nondimensional friction coefficient [math]r / (h \omega)[/math]. As in the previous example, the highwater (HW) propagation speed [math]c^+[/math] increases and the lowwater (LW) propagation speed decreases with increasing values of [math]a/h[/math]. The propagation of the LW trough of the tidal wave is more delayed than the propagation of the HW crest, due to stronger friction at smaller water depths. The period during which the tide rises is thus shortened with respect to the duration of falling tide. However, the opposite occurs for increasing values of the relative flood storage width [math]b_{S0}/b_C[/math]. The HW propagation speed [math]c^+[/math] decreases relative to the LW propagation and the period during which the tide rises is thus lengthened with respect to the duration of falling tide. This can be understood by noting that the lateral expansion of the tidal flood wave over the intertidal area reduces its propagation speed and thus delays the time of HW in the upperchannel zone. Hence, shallowness of the channel (large [math]a/h[/math]) and extension of the intertidal area (large [math]b_{S0}/b_C[/math]) have counteracting effects on tidal wave distortion.
Morphology of shallow tidal basins with small river inflow
In the foregoing it was shown that the tidal wave that enters a shallow prismatic channel is distorted due to the opposite effects of friction and intertidal areas on the upchannel propagation of HW on the one hand and LW on the other. In short tidal basins, these effects are partially offset by the reflected tidal wave at the landward basin boundary. However, in the case of strong friction, the reflected tidal wave is much smaller than the incoming tidal wave in a large part of the basin. In this part of the tidal basin the tidal velocity [math]u[/math] is mainly determined by the water surface slope [math]\partial \zeta /\partial x[/math], according to Eq. 4. A short period of tidal rise compared to the period of tidal fall implies steeper water surface slopes during flood than during ebb. Hence, maximum flood velocities are higher than maximum ebb velocities in the absence of significant river inflow.
Because the transported sediment load increases more than linearly with the current velocity (see the articles Sand transport and Sediment transport formulas for the coastal environment), sediment fluxes during flood tide are higher than sediment fluxes during ebb tide. Flooddominant tidal asymmetry thus produces a net import of sediment into the basin. Sediment infill could possibly go on until no tidal basin is left. This may have happened in the past to some tidal basins, but still many tidal basins without river inflow still survive. The reason is that flood dominance is neutralized by several processes. One of these processes is wave action, which can suspend large amounts of sediment in the HW period that are subsequently transported out of the basin by ebb currents. However, tidal basins do not depend only on wave action for their survival. One reason is the socalled Stokes transport, the water outflow compensating for the net influx due to greater mean water depth during flood than during ebb. Another reason is the reduction of tidal asymmetry due to the presence of intertidal areas as discussed in the previous section. During the development of intertidal areas by flooddominant sediment transport, tidal asymmetry is weakened until the average sediment transport by flood currents has become comparable to the average transport by ebb currents. In order to neutralize flood dominance with increasing relative tidal amplitude [math]a/h[/math], the counteracting effect of intertidal areas should also increase. Fig. 4 shows that this is indeed the case for natural tidal basins with small (or without) river inflow: tidal basins with larger relative tidal amplitude have larger intertidal areas. One may thus conclude that selforganizing processes can produce a natural equilibrium morphology for tidal basins in a sedimentary environment without geological constraints^{[1]}^{[5]} (see also the article Morphology of estuaries).
Tidal wave deformation in a converging channel
The influence of friction on tidal propagation increases with decreasing depth. The LW propagation is slowed down more strongly than the HW propagation, which results in a larger tidal asymmetry. However, the tidal amplitude is decreased by frictional damping. During the past century many estuaries have been deepened for navigational purposes and intertidal areas have been reclaimed. The effect of these interventions on tidal propagation is illustrated in Fig. 5 for the Seine estuary and tidal river system. The tide propagates now much faster into the estuary and the tidal amplitude is much larger, especially in the upstream river. The tide propagation speed has increased more for the low waters than for the high waters, although the propagation of the high waters also benefits of the reduction of the intertidal areas. In the past a high tidal bore developed each spring tide in the downstream river. After the interventions it takes a much larger distance before the HW wave crest overtakes the LW wave trough. A small tidal bore now develops far upstream and only for very high tidal coefficients^{[6]}.
As discussed before, the expansion of the tidal flood wave over large intertidal areas decreases its height and propagation speed. The opposite occurs when the tide propagates into a tidal channel that becomes progressively narrower in upchannel direction, see Fig. 6. Instead of expanding laterally, the tidal wave is contracted when propagating. In the hypothetical case of no friction, conservation of the tidal energy flux along the channel requires upchannel amplification of the tidal amplitude. Many estuaries with significant river inflow have an upstream converging channel. Intertidal areas are rather small, partly as a result of natural sedimentation but often also as a result of human reclamation. The channel depth along the thalweg is fairly uniform^{[8]}, but shoals may be present in the mouth zone. The uniformity of the depth can also be due to dredging works for navigation purposes. In the following we consider an idealized estuary with exponentially converging width [math]b=b_0 \; e^{x/L_b} [/math] and uniform depth [math]h[/math]. It should be borne in mind that although many estuaries have an upstream converging width, the assumption of exponential width convergence and uniform depth is for most estuaries a very rough approximation. Often only a limited part of the estuary can be represented in this way.
The linearized tidal equations (crosssectionally averaged) for this idealized estuary read:
[math]\Large\frac{\partial u}{\partial t}\normalsize + g \Large \frac{\partial \zeta}{\partial x}\normalsize + r \Large \frac{u}{h}\normalsize = 0 , \quad \quad (7)[/math]
[math]b \Large \frac{\partial \zeta}{\partial t}\normalsize + \Large \frac{\partial (bhu)}{\partial x}\normalsize = b [ \Large \frac{\partial \zeta}{\partial t}\normalsize + h \Large \frac{\partial u}{\partial x}\normalsize – u \Large \frac{h}{L_b} \normalsize ] = 0 . \quad \quad (8)[/math]
Solving these linear equations (only tide, no river discharge) yields
[math]\zeta = a e^{\mu x} \; cos(kx\omega t) \quad [/math] with [math]\quad 2 L_b \; \mu = \normalsize 1 + \Large[\normalsize 1 – (\Large \frac{1}{2} \normalsize + 2 K_0^2) +\large[\normalsize (\Large \frac{1}{2}\normalsize + 2 K_0^2 )^2+ 4 (K_c^2  K_0^2) \large]^{\large 1/2} \Large]^{\large 1/2} \normalsize, \quad \quad (9)[/math]
where [math]K_0 = \Large \frac{\omega L_b}{\sqrt{gh}}\normalsize , \; K_c = \Large \frac{r \omega L_b^2}{g h^2} \normalsize [/math]. The damping factor [math]\mu[/math] is positive for large friction and large convergence length ([math]K_c \gt K_0[/math]). However, for small friction and small convergence length ([math]K_c \lt K_0[/math]) the damping factor is negative: the tide is amplified when propagating upchannel. Even in the case of strong friction, the tide is only slightly damped or even amplified if the convergence length [math]L_b[/math] is sufficiently small. In cases where tidal damping dominates over the effect of channel convergence (large [math]L_b[/math]), the relative tidal amplitude decreases along the estuary; tidal asymmetry then becomes less relevant for upstream sediment transport. As noted before, the friction factor [math]r[/math] can vary greatly between estuaries because of salinity stratification and the type of bed sediments (coarse or muddy).
The solution of the Eqs. (7) and (8) also yields an expression for the wave propagation velocity [math]c[/math]:
[math]c = \Large \frac{\omega}{k}\normalsize = 2 \omega L_b \Large[\normalsize  1 + (\Large \frac{1}{2} \normalsize + 2 K_0^2) +\large[\normalsize (\Large \frac{1}{2}\normalsize + 2 K_0^2 )^2+ 4 [K_c^2  K_0^2] \large]^{\large 1/2} \Large]^{\large 1/2}\normalsize . \quad \quad (10)[/math]
The expressions (9) and (10) show that tidal wave propagation depends on only two parameters, [math] K_0[/math] and [math] K_c [/math]. The linear equations (7) and (8) do not describe tidal wave deformation; for this, nonlinear terms have to be included ([math]u \partial u /\partial x[/math] and time varying water depth [math]h+\zeta[/math] in Eq. (7) and [math]b \zeta \partial u / \partial x[/math] in Eq. (8)). For weak nonlinearity these equations can be solved analytically^{[9]}; the solution involves the additional small parameter [math]a/h[/math]. Tidal wave deformation in an exponentially convergent estuary with uniform depth is entirely determined by three parameters: [math] K_0, K_c, a/h[/math] ^{[10]}^{[11]}. If intertidal areas [math]b_S[/math] and mean river discharge [math]Q_R[/math] are included, two additional parameters [math]b_{S0}/b_C[/math] and [math]Q_R/(a b_C L_b \omega)[/math] are needed to characterize tidal wave deformation.
A large positive difference between the HW and LW propagation speeds requires a large relative tidal amplitude [math]a/h[/math]. This is illustrated in Fig. 7 for the Charente estuary, by comparing the curves for tidal elevation and current velocity for springtide (large [math]a/h[/math]) and neap tide (small [math]a/h[/math]). During springtide the tidal rise is much steeper than for neap tide. The maximum flood current velocity is larger than the maximum ebb tidal velocity for springtide, while the opposite holds for neap tide. In Fig. 8 the relative difference between HW and LW propagation speeds [math]\Delta c / c = 2(c^+ c^)/(c^++c^)[/math] are compared for estuaries with different relative tidal amplitude [math]a/h[/math]. The figure shows a positive correlation between [math]\Delta c / c [/math] and [math]a/h[/math]. Although [math]a/h[/math] is the most important parameter, other factors also influence the relation between [math]a/h[/math] and [math]\Delta c / c[/math], such as [math]K_0[/math] and [math]K_c[/math] (representing depth [math]h[/math], convergence length [math]L_b[/math] and friction parameter [math]r[/math]), the relative intertidal area [math]b_{S0}/b_C[/math] and the mean river discharge [math]Q_R[/math]. The dependence of [math]\Delta c / c [/math] on [math]a/h[/math] is therefore different for each estuary.
Morphology of estuaries with tidal rivers
In a converging (funnelshaped) estuary with strong friction the tidal velocity [math]u[/math] is mainly determined by the water surface slope [math]\partial \zeta /\partial x[/math], according to Eq. 4. A short period of tidal rise compared to the period of tidal fall implies steeper water surface slopes during flood than during ebb, with maximum flow rates that are higher during flood than during ebb, as illustrated in Fig. 7. Therefore, sediment fluxes during flood tide are higher than sediment fluxes during ebb tide, resulting in a net import of sediment into the estuary. As shown before, flood dominance increases with increasing relative tidal amplitude [math]a/h[/math]. The time span over which flood dominance develops also plays a role. For converging estuaries the relevant nondimensional time duration indicator is mainly determined by the ratio of convergence length and tide propagation speed [math]L_b \omega/ c[/math]. The value of the nondimensional parameter [math]a L_b \omega / (a c)[/math] can be considered an indication for the strength of flood dominance.
Infill of estuaries is limited by sediment export through river flow, although dredging may also play a role. The influence of river flow on sediment export can be represented by the nondimensional parameter [math]Q_R/Q_{tide}[/math], where [math]Q_R[/math] is the mean river discharge and [math]Q_{tide}[/math] the maximum tidal discharge in the midestuarine zone ([math]x \approx L_b/2[/math]). For estuaries in morphological equilibrium, sediment import due to tidal asymmetry (flood dominance) should be approximately balanced by export due to river flow. Comparing different estuaries one may thus expect a positive correlation between the parameters [math]a L_b \omega / (a c)[/math] and [math]Q_R/Q_{tide}[/math] ^{[13]}. As shown in Fig. 9, such a positive correlation exists, although the spread in the data is large. This spread can be due to many other factors, which influence sediment import and export in different ways. Possible important factors are^{[1]}:
 dredging,
 import/export by wave activity,
 import by estuarine circulation,
 sediment recirculation in ebb/floodchannel cells,
 sediment import/export related to settling and erosion time lags,
 fluvial sediment supply,
 type of sediment.
Related articles
 Morphology of estuaries
 Ocean and shelf tides
 Tidal bore dynamics
 Tidal motion in shelf seas
 Estuarine circulation
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} Dronkers, J. 2017. Dynamics of Coastal Systems. World Scientific Publ. Co, Singapore, 740 pp.
 ↑ Banerjee, A.P., Dutta, S. and Majumdar, A. 2015. Quest for the determination of environmental flow assessment for hilsa fish of the Hooghly estuary by hydraulic rating method. ARPN Journal of Engineering and Applied Sciences 10: 78857899
 ↑ Filippini, A.G., Arpaia, L., Bonneton, P. and Ricchiuto, M. 2019. Modeling analysis of tidal bore formation in convergent estuaries. European Journal of Mechanics  B/Fluids 73: 5568
 ↑ LeBlond, P. 1978. On tidal propagation in shallow rivers, J. Geophys. Res., 83: 4717–4721
 ↑ Dronkers, J. 1998. Morphodynamics of the Dutch Delta. In: Physics of estuaries and coastal seas. Ed.: J.Dronkers and M.B.A.M. Scheffers, Balkema, Rotterdam: 297304
 ↑ Bonneton, N., Bonneton, P., Parisot, JP., Sottolichio, A. and Detandt G. 2012. Tidal bore and Mascaret  example of Garonne and Seine Rivers. Comptes Rendus Geosciences, 344, 508515
 ↑ Comoy, M. 1881. Etude pratique sur les marées fluviales. GauthiersVillars, Paris
 ↑ Savenije, H.H.G. 2012. Salinity and Tides in Alluvial Estuaries, second ed., Salinity and Tides in Alluvial Estuaries, second ed., www.salinityandtides.com
 ↑ Friedrichs C.T. and Aubrey, D.G. 1994. Tidal propagation in strongly convergent channels. J.Geophys.Res. 99: 33213336
 ↑ Bonneton, P., Filippini, A.G., Arpaia, L., Bonneton, N. and Ricchiuto, M. 2016. Conditions for tidal bore formation in convergent alluvial estuaries, Estuar. Coast. Shelf Sci. 172: 121127
 ↑ Lanzoni, S. and Seminara, G. 1998. On tide propagation in convergent estuaries, J. Geophys. Res. 103: 30793–30812
 ↑ Toublanc, F., Brenon, I., Coulombier, T. and LeMoine, O. 2015. Fortnightly tidal asymmetry inversions and perspectives on sediment dynamics in a macrotidal estuary (Charente, France). Continental Shelf Res. 94: 42–54
 ↑ ^{13.0} ^{13.1} Dronkers, J. 2017. Convergence of estuarine channels. Continental Shelf Res. 144: 120–133
Appendix
The solution of the first order linear equations (1) and (2) is
[math]\zeta^{(1)}=a \cos \theta , \; u^{(1)} = \Large\frac{c_0}{h}\normalsize \zeta^{(1)} , [/math]
where [math]\theta= k_0 x\omega t[/math], [math]\omega[/math] is the M2 radial frequency, the wave number [math]k_0 = \Large \frac{\omega}{gh} \normalsize[/math] and [math]c_0=\omega / k[/math].
After substitution in the nonlinear terms we can solve the second order approximation:
[math]\zeta = \zeta^{(1)} + \zeta^{(2)} , \quad \zeta^{(2)} = \Large \frac{3 a^2}{4h}\normalsize kx \sin 2 \theta [/math].
The location of the wave crest at time [math]t[/math] is called [math]x^+(t) [/math]. At this location the surface slope is zero:
[math] \Large \frac{\partial \zeta}{\partial x}\normalsize (x^+(t),t) = a k_0 \sin \theta^+ + \Large \frac{3 a^2}{4h}\normalsize k_0 [ \sin 2 \theta^+ + 2kx \cos 2 \theta^+ ] \quad \quad[/math] , with [math]\theta^+=k_0 x^+(t)  \omega t [/math].
Because [math]a/h[/math] is small the wave crest is at a location where [math]\theta^+[/math] is small ([math]\theta^+ \lt \lt 1[/math]). An approximate expression for the location of the wave crest is then given by
[math]x^+(t) \approx (1  \Large \frac{3a}{2h}\normalsize)^{1} c_0 t [/math].
The propagation speed of the HW wave crest (Eq. 3) follows from [math]c^+(t)=dx^+/dt[/math].
The solution of the first order linear equations (4) and (5) is
[math]\zeta^{(1)} = \Large\frac{1}{2}\normalsize a e^{i (\kappa x  \omega t)} + c.c. [/math], where [math]c.c. [/math] is the complex conjugate, [math]\kappa = k+i \mu , \; k = \mu = \sqrt { \Large \frac{\omega r}{2 g h^2}\normalsize }[/math].
After substitution in the nonlinear terms the second order linear equations can be solved, yielding [math]\zeta = \zeta^{(1)} + \zeta^{(2)} , \quad \zeta^{(2)} \approx (\Large \frac{b_{S0}}{8 b_C}  \frac{a}{2h}\normalsize) e^{2i \omega t} (e^{2i \kappa x} e^{\sqrt{2} i \kappa x} ) + \Large \frac{a}{4h}\normalsize (1 – e^{2 \mu x}) + c.c. [/math].
In the same way as before the location of the wave crest is derived from the condition
[math] \Large \frac{\partial \zeta}{\partial x} \normalsize (x^+(t),t) =0[/math] for [math] k x^+  \omega t\lt \lt 1[/math].
The propagation speed of the HW wave crest (Eq. 6) follows from [math]c^+(t)=dx^+/dt[/math].
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