Nonlinear wave dispersion relations

A derivation of the linear wave dispersion relation

[math]c = u + \Large\frac{g}{\omega }\normalsize \, \tanh kh \qquad (1)[/math]

is given in the article Shallow-water wave theory. In this formula, [math]c = \lambda / T = \omega / k [/math] is the wave celerity (wave propagation speed), [math]k[/math] is the wave number (related to the wavelength [math]\lambda = 2 \pi / k[/math]), [math]\; \omega[/math] is the radial wave frequency (related to the wave period [math]T = 2 \pi / \omega[/math]), [math]g \approx 9.8 \; ms^{-2}[/math] is the gravitational acceleration, [math]h[/math] is the still water depth and [math]u[/math] is the surface current velocity in the wave propagation direction. In the following we assume that the surface current velocity is nil or very small, [math]u \lt \lt c[/math]. If this condition is not satisfied but the surface current velocity is known, it must be subtracted from the measured wave celerity ([math]c \rightarrow c - u[/math]).

The local water depth [math]h[/math] can be determined by inversion of this formula:

[math]h = \Large\frac{1}{k }\normalsize \, \tanh^{-1} \Big( \Large\frac{\omega c}{g}\normalsize \Big) . \qquad (2)[/math]

A restriction on the use of Eq. (1) are the assumptions underlying linear wave theory. These assumptions are: (i) irrotational wave flow, (ii) the wave amplitude [math]H[/math] is much smaller than the water depth ([math]H/h \lt \lt 1[/math]) and (iii) much smaller than the wavelength ([math]H / \lambda \lt \lt 1[/math]). However, these assumptions are not satisfied in the nearshore zone where waves become skewed and asymmetric and eventually break.

If weak nonlinearity is assumed in the shoaling zone (prior to wave breaking) and if the Ursell number [math]U_r = kH / (kh)^3[/math] is small, nonlinear Stokes theory can be applied. In this case the dispersion relation can be approximated by

[math]c = \Large\frac{g}{\omega }\normalsize \, \sigma \Big( 1 + \Large\frac{9 - 10 \sigma^2+9 \sigma^4}{32 \sigma^4}\normalsize (kH)^2 \Big) + O[(kH)^4] , \qquad \sigma = \tanh kh .\qquad (3)[/math]

Field observations of wave height and wave celerity show that the shallow water linear dispersion relation underestimates the wave speed at wave breaking and inside the surf zone. Measured celerity values can be 20% higher than predicted by the linear dispersion relation or even more. Weak nonlinearity cannot be assumed in the zone where waves are breaking. If the wave after breaking is surfing onshore like a bore, the bore formula for the celerity can be applied (see Tidal bore dynamics, Eq. (1) ),

[math]c = \sqrt{gh} \, \sqrt{(1+\large\frac{H}{h}\normalsize)(1+\large\frac{H}{2h}\normalsize)} . \qquad (4)[/math]

An alternative approach is applying cnoidal wave theory. This gives

[math]c \approx \sqrt{g h} \, \sqrt{1 + \alpha \large\frac{H}{h}\normalsize } , \qquad (5)[/math]

where [math]\alpha[/math] is a function of the Ursell number with value close to 1. Empirical evidence suggests [math]c \approx \sqrt{g h} \, \sqrt{1 + 0.45 \large\frac{H_s}{h}\normalsize } [/math], where [math]H_s[/math] is the significant wave height.

A physics-based approach uses a modified dispersion relation according to the Boussinesq theory that describes the propagation of weakly nonlinear and weakly dispersive waves for Ursell numbers of order unity ([math]O[H/h] \sim O[(kh)^2] \lt \lt 1[/math]). In this theory, nonlinear interactions between resonant triads of frequencies ([math]\omega \, , \, \pm \omega' \, , \, \omega \mp \omega'[/math]) lead to the growth of forced high-frequency components that modify the wave shape in shallow water. The resulting dispersion relation to order [math](kh)^2[/math] is

[math]c(\omega) = \Large\frac{\omega}{k(\omega)}\normalsize = \sqrt{gh} \Big[ 1 + \Large\frac{h \omega^2}{3g} + \frac{h^2 \omega^4}{36g^2} - \frac{1}{h}\normalsize \gamma_{am} \Big]^{-1/2} , \qquad \gamma_{am} = \Large\frac{3}{2 | \hat{\eta}(\omega)|^2}\normalsize \, \int_{-\infty}^{\infty} \Re \big( \hat{\eta}(\omega') \hat{\eta}(\omega - \omega') \hat{\eta}^*(\omega) \big) d \omega' , \qquad (6)[/math]

where [math]\hat{\eta}(\omega) = \hat{\eta}(k, \omega)[/math] is the Fourier transform of the surface elevation [math]\eta (x, t) = \int \int \hat{\eta}(k, \omega) \exp(i(kx-\omega t)) dk d\omega[/math].

This dispersion relation can be used to determine the local water depth from wave data. It requires knowledge of the free surface elevation with high space and time resolution in order to determine [math]\hat{\eta}(\omega)[/math]. Lidars currently offer the most robust and practical solution for collecting such highly-resolved surface elevation data in the field. The depth [math]h[/math] can be determined from Eq. (6) by a least-squares fit to values of [math]c(\omega)[/math] around the peak wave frequency. Using this theory, reasonable agreement was found (within 10%) in laboratory experiments between the real depth profiles in the shoaling and surf zones and depth profiles derived from the measured surface wave pattern using the dispersion relation Eq. (6) .

Symbols

Variable	Description	Variable	Description	Variable	Description
[math]c = \omega / k[/math]	wave celerity	[math]H[/math]	wave height	[math]x[/math]	cross-shore coordinate
[math]g[/math]	gravitational acceleration	[math]t[/math]	time	[math]\eta(x,t)[/math]	wave surface elevation
[math]k = 2 \pi / \lambda[/math]	wave number	[math]T[/math]	wave period	[math]\lambda[/math]	wavelength
[math]h = h(x)[/math]	local still water depth	[math]U_r = kH / (kh)^3[/math]	Ursell number	[math]\omega[/math]	radial wave frequency

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References

The main author of this article is Job Dronkers Please note that others may also have edited the contents of this article. Citation: Job Dronkers (2024): Nonlinear wave dispersion relations. Available from http://www.coastalwiki.org/wiki/Nonlinear_wave_dispersion_relations [accessed on 3-12-2024] For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Dronkers J