Edge wave
Definition of Edge waves:
Obliquely incident waves trapped near the shore by refraction and reflection over seaward-increasing water depth.
This is the common definition for Edge waves, other definitions can be discussed in the article
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Characteristics
Edge waves are waves which are trapped in a coastal strip with seaward increasing depth. The wavelength is comparable with or smaller than the width of the sloping coastal strip. These longer period waves (generally called infragravity waves) do not break on the shore, but are (partially) reflected. The reflected wave is refracted back toward the coast and then travel alongshore as an edge wave (Fig. 1). Infragravity waves can be generated in shallow water by nonlinear interaction processes within short wave groups, see Infragravity waves. They can also be generated by other causes, see Seiche.
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. Thus, if reflected infragravity waves are not sufficiently refracted back toward the coast before reaching deeper water, they can escape seaward as leaky waves. Alongshore variations in bathymetry 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.
Over a beach that slopes continuously seawards, the maximum amplitude of an edge wave occurs at the shoreline, with an exponential seaward decay in case of a constant cross-shore slope. 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[1].
Edge wave dispersion relation[2]
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. An approximate dispersion relation for the mode-0 edge wave can be obtained from a simple wave-ray argument. This condition states that 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]\quad y_c = L/4 = \pi/(2k_y) = \int_0^{x_c} \cot \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} \cot \alpha \, dx = 2 x_c \int_0^{\pi/2} \cos^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 result, Eq. (A12) derived in the Appendix. The two results are equal in the limit of small shoreface slope [math]\beta[/math].
Appendix: Mathematical derivation
Here we derive expressions for full linear potential flow on a plane sloping seabed (Fig.3). 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 generally 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.
Seabed friction is ignored. 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: [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[2] [math]\quad \omega^2 = g k_y \sin ((2n+1) \beta) . \qquad (A11)[/math]
For a plane beach, for the higher 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]
with [math](2n+1) \beta \lt \pi/2[/math].
See also
References
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