Edge wave

Definition of Edge wave: A gravity wave that propagates along the coast and is trapped near the shoreline by refraction over a seabed with increasing offshore depth, with wave amplitude generally decreasing seaward. This is the common definition for Edge wave, other definitions can be discussed in the article

Characteristics

Edge waves are gravity waves which are trapped in a coastal strip with seaward increasing depth (Huntley, 1976, Huntley et al., 1981). They propagate alongshore, with amplitudes that generally decrease seaward. Edge waves often have infragravity periods. Infragravity waves reflected at the shoreline can become trapped near the coast by refraction over increasing offshore depth and subsequently propagate alongshore as edge waves (Fig. 1). Infragravity waves can be generated in shallow water by nonlinear interaction processes within an irregular wave field, see Infragravity waves. Edge waves can also be excited directly by nonlinear interactions with incident waves or by Greenspan resonance.

Trapping to the coast is due to the greater wave propagation speed in deep water compared to shallow water. However, as the water becomes deeper, the propagation speed becomes less dependent on depth. On a beach profile where the offshore increase in depth becomes small, refraction may no longer be sufficient to trap all the wave energy near the coast. Part of the wave energy may then radiate offshore, giving rise to leaky edge-wave modes. Alongshore bathymetric variations can also scatter trapped wave energy offshore.

Although the largest amplitude of edge waves is likely to be shoreward of the breaker line, the form of edge waves is not directly linked to the breaker line: they may decay in amplitude well within the breaking zone or may extend well beyond it. The cross-shore scale of the seaward decay is related to the alongshore wavelength.

Edge waves occur in different cross-shore modes; mode 0 has no cross-shore node, whereas higher modes have one or more nodes, see the Appendix.

For the fundamental (mode-0) edge wave on a uniformly sloping beach, the amplitude is greatest at the shoreline and decreases exponentially offshore. Higher modes have one or more cross-shore nodes and may exhibit offshore amplitude maxima. However, over a barred beach the amplitude can become large over the bar crest.

Beach cusps along pocket beaches have sometimes been associated with standing edge waves, formed by reflection of progressive edge waves at the bounding headlands (see the article Beach Cusps). Standing edge waves can also result from a resonant instability by which regular shore-normal waves incident on a sloping beach excite a pair of subharmonic edge waves (half the incident frequency) travelling in opposite directions, together forming a standing edge wave (Guza and Davis, 1974). Edge waves have been proposed as a template for rhythmic morphology such as beach cusps, crescentic bars and rip currents (Bowen and Inman, 1969, 1971), but field support is limited; these features are now mostly explained by morphodynamic self-organization (see Rhythmic shoreline features, Beach cusps).

Edge wave dispersion relation(Schäffer and Jonsson, 1992)

An approximate dispersion relation for the fundamental edge-wave mode can be obtained from a wave-ray argument. The argument assumes that the incoming and returning wave components interfere destructively at the offshore turning point, producing a trapped wave pattern. Consider a plane seabed, uniform in the alongshore [math]y[/math]-direction and with a uniform sloping depth [math]h=x \, \tan \beta[/math] in seaward [math]x[/math]-direction. Incoming and reflected waves have to annihilate where the reflected wave is refracting back (at [math]x=x_c[/math]) from seaward directed to onshore directed (see Fig. 2). Destructive interference occurs if the phase [math]k_y L/2[/math] of the incoming wave (wave trough) is equal to the phase [math]2 k_y y_c[/math] of the reflected wave (wave crest). (This condition is similar to the destructive interference of incoming and reflecting waves at the open boundary of a frictionless prismatic canal of length [math]L/4[/math].)

According to Snell's law, the alongshore wavenumber [math]k_y = 2 \pi / L[/math] is independent of [math]x[/math]. We have [math]k_y = k \sin \alpha[/math], where [math]k[/math] is the wave number and [math]\alpha(x)[/math] the wave ray angle. The refraction condition can be written ([math]ds=[/math] length along wave ray, [math]dy=ds \sin \alpha\, , \, ds=dx /\cos \alpha[/math])

[math]\quad y_c = L/4 = \pi/(2k_y) = \int_0^{x_c} \tan \alpha(x) \, dx .\qquad (1)[/math]

An expression of [math]\alpha(x)[/math] can be derived by assuming that the wave propagation speed can be approximated by the shallow-water formula [math]c^2=g \, h = g\, x \, \tan \beta[/math]. We have by definition [math]c=\omega/k=\omega \, \sin \alpha / k_y [/math]. Combining the two equations gives [math]\sin^2 \alpha = (k_y / \omega)^2 g \, x \, \tan \beta[/math]. Because [math]\sin\alpha = 1[/math] at the refraction line [math]x=x_c[/math], we also have [math]1 / x_c = (k_y / \omega)^2 g \, \tan \beta[/math] and [math]\sin^2 \alpha = x / x_c[/math]. From the latter equality follows [math]dx = 2 x_c \sin \alpha \cos \alpha d\alpha[/math], which can be substituted in the integral (1). We then find

[math]y_c = \pi / (2 k_y) = \int_0^{x_c} \tan \alpha \, dx = 2 x_c \int_0^{\pi/2} \sin^2 \alpha \, d \alpha = (\pi/2) x_c . \qquad (2)[/math].

Using the expression of [math]1/x_c[/math] we find the dispersion relation for the 0-mode edge wave: [math]\quad \omega^2 = g \, k_y \, \tan \beta . \qquad (3)[/math]

This is the shallow-water approximation to the exact mode-0 result, Eq. (A10) derived in the Appendix. The two results are equal in the limit of small shoreface slope [math]\beta[/math].

Fig. 1. Principle of wave trapping to the shore by wave refraction and reflection. The wave crest is shown in red at successive times (from left to right), black arrows indicate the wave propagation speed and the dotted line displays the wave ray. The wave propagation speed is larger in deep water (dark blue) than in shallow water (light blue). Incident waves reflected at the shoreline without substantial dissipation therefore cannot escape to deep water but travel along the shore. Such waves are called edge waves.

Fig. 2. Wave ray of incoming and reflected wave crest (red) and wave ray of incoming wave trough (black line). The edge wave is trapped in a coastal strip of width [math]x_c[/math].

Appendix: Mathematical derivation

Here we provide expressions for full linear potential flow on a plane sloping seabed (Fig.3), according to the derivation by Ursell, 1952. For simplicity, the bathymetry is assumed uniform in the alongshore direction. The flow is assumed inviscid and irrotational (thin boundary layer), while non-hydrostatic pressure is retained. Edge waves are mostly considered to be infragravity waves (intermediate between short breaking waves and long tidal waves); dissipation and seabed friction are neglected. The equations are linearized for the sake of simplicity.

The continuity equation: [math]\quad \Large\frac{\partial u }{\partial x}\normalsize + \Large\frac{\partial v}{\partial y}\normalsize + \Large\frac{\partial w}{\partial z}\normalsize =0 . \qquad (A1) [/math]

The vertical momentum equation: [math]\quad \Large\frac{\partial w}{\partial t}\normalsize + \Large\frac{1}{\rho}\frac{\partial p}{\partial z}\normalsize = -g . \qquad (A2)[/math]

The condition at the sloping seabed ([math]z = -h(x)[/math]) : [math] \quad w = - u \Large\frac{dh}{dx}\normalsize . \qquad (A3) [/math]

Symbols (Fig. 3): [math]x=[/math] cross-shore coordinate (seaward >0), [math]y=[/math] longshore coordinate, [math]z=[/math] vertical coordinate (upward >0), [math] u, v, w[/math] are the velocities in the [math]x, y, z[/math] directions, [math]h(x)=[/math] mean depth, [math]dh/dx=\tan \beta=[/math] seabed slope, [math]\eta=[/math] wave elevation, [math]p=[/math] pressure, [math]g=[/math] gravitational acceleration, [math]\rho=[/math] seawater density.

The wave motion is irrotational, and can thus be described by a velocity potential [math]\phi(x,y,z,t)[/math] ,

[math]u = \Large\frac{\partial \phi}{\partial x}\normalsize , \quad v = \Large\frac{\partial \phi}{\partial y}\normalsize , \quad w = \Large\frac{\partial \phi}{\partial z}\normalsize . \qquad (A4)[/math]

The wave equations can then be written as

Continuity equation: [math]\quad \Large\frac{\partial^2 \phi}{\partial x^2}\normalsize + \Large\frac{\partial^2 \phi}{\partial y^2}\normalsize +\Large\frac{\partial^2 \phi}{\partial z^2}\normalsize =0 . \qquad (A5)[/math]

Vertical momentum equation: [math]\quad \Large\frac{\partial^2 \phi}{\partial z \partial t}\normalsize + \Large\frac{1}{\rho}\frac{\partial p}{\partial z}\normalsize = -g . \qquad (A6) [/math]

Seabed condition at [math]z = −h(x) = −x \tan \beta[/math]: [math]\quad \Large\frac{\partial \phi}{\partial z}\normalsize = -\tan \beta \Large\frac{\partial \phi}{\partial x}\normalsize . \qquad (A7)[/math]

The pressure is eliminated by integrating Eq. (A6) to the surface, [math]\quad \Large\frac{\partial \phi}{\partial t}\normalsize = - g \eta \quad [/math] and differentiation near the surface [math]z=0[/math]:

[math]\quad \Large\frac{\partial^2 \phi}{\partial t^2}\normalsize = - g \dfrac{\partial \eta}{\partial t} = - g w = - g \Large\frac{\partial \phi}{\partial z}\normalsize \, . \qquad (A8) [/math]

The solution of the linear equations Eqs. (A5, A7, A8) which is harmonic in the alongshore direction, represents an edge wave propagating along the coast with period [math]T = 2 \pi / \omega [/math] and wave number [math]k_y[/math],

[math]\phi(x,y,z,t) = e^{i \omega t + i k_y y} f(x,z) , \qquad (A9)[/math]

where [math]f(x,z)[/math] is a superposition of exponential functions. The simplest function (mode 0 edge wave or Stokes edge wave) is

[math]f_0(x,z) = a_0 e^{k_y(- x \cos \beta + z \sin \beta)} , \quad [/math] with corresponding dispersion relation [math]\quad \omega^2 = g k_y \sin \beta . \qquad (A10)[/math]

This expression demonstrates the exponential offshore decay, which has an e-folding scale [math]\dfrac{1}{k_y \cos \beta}[/math] related to the alongshore wavelength [math]\dfrac{2 \pi}{k_y}[/math].

The dispersion relation of higher edge wave modes [math]f_n(x,z) [/math] is given by [math]\quad \omega^2 = g k_y \sin ((2n+1) \beta) . \qquad (A11)[/math]

For a plane beach, for the linear edge-wave modes the corresponding alongshore wavelengths are

[math]\lambda_y = \Large\frac{2 \pi}{k_y}\normalsize = \Large\frac{g T^2}{2 \pi}\normalsize \sin ((2n+1) \beta) \, . \qquad (A12)[/math]

Trapped modes exist only for [math](2n+1) \beta \lt \pi/2[/math]

See also

• Infragravity waves
• Seiche
• Surf beat
• Beach Cusps

References

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