A method for the numerical solution of steadily progressing periodic waves on irrotational flow over a horizontal bed is presented. No analytical approximations are made. A finite Fourier series, similar to Dean's stream function series, is used to give a set of nonlinear equations which can be solved using Newton's method. Application to laboratory and field situations is emphasized throughout. When compared with known results for wave speed, results from the method agree closely. Results for fluid velocities are compared with experiment and agreement found to be good, unlike results from analytical theories for high waves.The problem of shoaling waves can conveniently be studied using the present method because of its validity for all wavelengths except the solitary wave limit, using the conventional first-order approximation that on a sloping bottom the waves at any depth act as if the bed were horizontal. Wave period, energy flux and mass flux are conserved. Comparisons with experimental results show good agreement.

A Fourier approximation method for steady water waves

The numerical model developed previously for coastal structures is slightly modified and applied to predict the wave transformation in the surf and swash zones on gentle slopes as well as the wave reflection and swash oscillation on relatively steep beaches. The numerical model is one-dimensional in the cross-shore direction and is based on the finite amplitude, shallow water equations, including the effect of bottom friction, which are solved in the time domain for the incident wave train specified as input at the seaward boundary of the computation located outside the breakpoint. The slight modification is related to the effect of the time-averaged current on the seaward boundary condition and improves the agreement between the computed and measured mean water levels on gentle slopes. The modified numerical model is compared with available small-scale test data for monochromatic waves spilling on gentle slopes as well as for monochromatic waves plunging and surging on a relatively steep slope. Additional comparisons are made with small-scale tests conducted using transient monochromatic and grouped waves on a 1:8 smooth slope with and without an idealized nearshore bar at the toe of the 1:8 slope. As a whole, the numerical model is shown to be capable of predicting both time-varying and time-averaged hydrodynamic quantities in the surf and swash zones on gentle as well as steep slopes.

Wave transformation and swash oscillation on gentle and steep slopes

The Kakutani equation is extended to include the effect of the variable width of channel, which is important in such a practical problem as refraction.Assuming that the wave profile is that of cnoidal waves and that this profile is maintained throughout the travel, a law of shoaling is derived.A comparison with experimental data shows that the present theory describes very well the shoaling for gHT2/d2>30, while for gHT2/d2<30 the theory of linear surface waves gives good agreement.For 30< gHT2/d2<50, is Hd2/7=const. and for 50< gHT2/d2, it is Hd3/2[≠ gHT2-2≠3]=const. in case of variable depth only.Due to the lack of experimental data, the theoretical result for variable width could not be examined for the present.

Nonlinear Long Waves in a Channel of Variable Section

Monochromatic wave overtopping over the crest of an impermeable coastal structure located on a sloping beach is predicted numerically by expanding the numerical model developed previously for predicting wave run-up on such a structure located on the horizontal seabed The expanded numerical model predicts the temporal variations of the velocity and depth of the flow over the crest of the structure from which the average overtopping rate per unit width is computed The model accounts for the effect of wave shoaling on the sloping beach in front of the structure located in relatively shallow water The computed average overtopping rates are shown to be in agreement with available small-scale test data for which smooth impermeable structures were fronted by a 1:10 slope The numerical model also predicts the decrease of wave reflection due to the increase of wave overtopping However, more detailed measurements will be required to further calibrate and evaluate the numerical model which may be extended to examine the armor stability of overtopped breakwaters

Wave overtopping on coastal structures

Measurements of wave elevation and orbital velocity in the shoaling, breaking, and bore regime of single-frequency laboratory waves show that third-order Stokes theory, when energy flux is conserved, predicts the wave height change and harmonic growth in the regime where the Ursell number Ur = (H/ h)/(kh)2 is 0(1) or less. Shoreward of the Stokes region and up to the breakpoint, harmonic amplitudes are well described by the cnoidal theory. It is shown theoretically that a smooth transition regime exists between Stokes and cnoidal regions for waves which eventually break by plunging. The wave profile asymmetry about the vertical plane observed in near-breaking waves and bores is due to slow changes of phase of the harmonics relative to the primary wave as the wave train shoals. By contrast, only asymmetry about the horizontal plane is possible in the Stokes and cnoidal wave theories, since these classical solutions allow no relative phase shifts between harmonics. Velocity measurements made with hot-film anemometers show that ‘unorganized’ fluctuations at the bottom under breaking waves are of the order of half the rms amplitude of the wave-induced ‘organized’ flow. The correlation between surface elevation and bottom velocity under breakers and bores suggests that turbulence contributes more strongly to the unorganized flow at the bottom under plunging than under spilling waves.

Elevation and velocity measurements of laboratory shoaling waves

An equation is derived which governs the propagation of a cnoidal wave train over a gently sloping bottom. The equation is solved numerically, the solution being tabulated in terms of fH (Eq. 47) as a function of Ei = (Etr/pg) 1/3/gT2 and hi = h/gT2. Results are compared with sinusoidal wave theory. Two numerical examples are included.

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O. Brink-Kjaer

Papers

Shoaling of cnoidal waves