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-u = L / T = \omega / k [/math] is the wave celerity (wave propagation speed), [math]k[/math] is the wave number related to the wavelength [math]L = 2 \pi / k[/math], [math]\; \omega[/math] is the intrinsic angular 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] for shallow and intermediate waters 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]
The applicability of Eq. (1) is restricted by the assumptions underlying linear wave theory. These assumptions are: (i) irrotational wave flow, (ii) small depth variation over the wavelength and (iii) limits on [math]H/L[/math] and [math]h/L[/math] as indicated in Fig. 1. According to the expression (1), for large values of [math]kh[/math] the wave celerity hardly depends on depth. In this case Eq. (2) has no practical solution.
For waves of moderate steepness and sufficiently small Ursell number (see Fig. 1), second-order Stokes theory provides a nonlinear correction to the linear dispersion relation. In this case the dispersion relation can be approximated by[2]
[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[3] or even more[4]. Weak nonlinearity cannot be assumed in the zone where waves are breaking. After breaking, waves may propagate shoreward as turbulent bores. Their propagation speed can then be approximated by the hydraulic bore formula, where [math]H[/math] represents the difference in water depth across the bore (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]
For larger Ursell numbers, cnoidal wave theory may be applied. This gives[3]
[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. It reaches a value close to 1 in the solitary-wave limit. Evidence from a field experiment[5] 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 derived from Boussinesq theory, which accounts for nonlinear and dispersive effects in shallow-water waves. This approach is particularly relevant for Ursell numbers of order 25 or more, where nonlinear effects can substantially modify the wave dispersion relation. 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[6][7]
[math]c(\omega) = \dfrac{\omega}{k(\omega)} = \sqrt{gh} \Big[ 1 + \dfrac{h \omega^2}{3g} - \dfrac{\gamma_{am}}{h} \Big]^{-1/2} , \qquad \gamma_{am} = \dfrac{3}{2 | \hat{\eta}(\omega)|^2} \, \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, x)[/math] is the Fourier transform of the surface elevation [math]\eta (x, t) = \int_{-\infty}^{\infty} \hat{\eta}(\omega, x) \exp(-i\omega t) d\omega[/math].
For irregular waves, [math]c(\omega)[/math] represents the frequency-dependent phase velocity of the wave field, which may differ from the propagation speed of individual wave crests or breaking fronts. The 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[7]. 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) [7].
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 / L[/math] | wave number | [math]T[/math] | wave period | [math]L[/math] | wavelength |
| [math]h = h(x)[/math] | local still water depth | [math]Ur = H L^2/ h^3[/math] | Ursell number | [math]\omega[/math] | intrinsic angular wave frequency |
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References
- ↑ Zhao, K., Wang, Y. and Liu, P.L-F. 2024. A guide for selecting periodic water wave theories - Le Méhauté (1976)’s graph revisited. Coastal Engineering 188, 104432
- ↑ Whitham, G.B. 1974. Linear and nonlinear waves. Wiley-Interscience
- ↑ 3.0 3.1 Thornton, E.B. and Guza, R.T. 1982. Energy saturation and phase speeds measured on a natural beach. J. Geophys. Res. 87: 9499-9508
- ↑ Suhayda, J.N. and Pettigrew, N.R. 1977. Observations of wave height and wave celerity in the surf zone. J. Geophys. Res. 82: 1419-1424
- ↑ Holland, K.T. 2001. Application of the linear dispersion relation with respect to depth inversion and remotely sensed imagery. IEEE Trans. on Geos. and Rem. Sens. 39: 2060-2071
- ↑ Herbers, T.H.C., Elgar, S., Sarap, N.A. and Guza, R.T. 2002. Nonlinear dispersion of surface gravity waves in shallow water. Journal of Physical Oceanography 32: 1181–1193
- ↑ 7.0 7.1 7.2 Martins, K., Bonneton, P., de Viron, O., Turner, I.L., Harley, M.D. and Splinter, K. 2023. New Perspectives for Nonlinear Depth-Inversion of the Nearshore Using Boussinesq Theory. Geophysical Research Letters 50(2), e2022GL100498
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