7.4. NONLINEAR WAVE GENERATION
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wave nonlinearities was considered minor relative to other laboratory and
scale effects influencing the experimental result. However, there are numerous instances when an experiment requires precise waves conforming to a
higher-order mathematical wave theory, either to validate new analytical
theories or to develop correspondence between theory and more complex
phenomena such as sediment transport. The following sections discuss, in
varying levels of detail, laboratory generation of nonlinear waves of permanent form in two-dimensional wave tanks.
7.4.1 Second-Order Stokes Waves
Attempts to generate regular waves in a wave flume using a sinusoidallyvarying wave board motion, as derived for first-order wavemaker theory,
also produce unwanted free secondary waves which move at a speed that
is slightly slower than the primary wave. The combination of the primary
and secondary waves results in a combined wave form that varies both
spatially and temporally as illustrated in Figure 7.9. This modification
to the primary wave becomes more pronounced as wave steepness (H/L)
increases and/or relative depth (h/L) decreases, and experiments requiring
uniform waves may be compromised if the wave distortion becomes severe.
Fontanet (1961) developed a complete second-order theory describing
the waves produced by a sinusoidally-moving plane wavemaker. The development was in Lagrangian coordinates rendering it cumbersome to use.
Madsen (1970, 1971) developed an approximate second-order wavemaker theory for generating relatively long (h/L < 0(0.1)) second-order
Stokes waves of permanent form. Previously, the problem of secondary
waves produced by a sinusoidally-varying piston-type wave board had been
studied using numerical simulations. In presenting the approximate secondorder theory, Madsen made the observation:
To obtain general information about a certain problem, a relatively simple theoretical model that leads to analytical results is
often superior to a numerical model. (Madsen 1971).
Madsen’s approximate wavemaker theory provides an useful analytical
solution for the wave board motion necessary to suppress the unwanted free
second harmonic waves for relatively long second-order Stokes waves. The
theory is presented below, and it has been extended to include the case of
a Wave board hinged a distance I below the bottom as per Flick and Guza
(1980).
367
wave nonlinearities was considered minor relative to other laboratory and
scale effects influencing the experimental result. However, there are numerous instances when an experiment requires precise waves conforming to a
higher-order mathematical wave theory, either to validate new analytical
theories or to develop correspondence between theory and more complex
phenomena such as sediment transport. The following sections discuss, in
varying levels of detail, laboratory generation of nonlinear waves of permanent form in two-dimensional wave tanks.
7.4.1 Second-Order Stokes Waves
Attempts to generate regular waves in a wave flume using a sinusoidallyvarying wave board motion, as derived for first-order wavemaker theory,
also produce unwanted free secondary waves which move at a speed that
is slightly slower than the primary wave. The combination of the primary
and secondary waves results in a combined wave form that varies both
spatially and temporally as illustrated in Figure 7.9. This modification
to the primary wave becomes more pronounced as wave steepness (H/L)
increases and/or relative depth (h/L) decreases, and experiments requiring
uniform waves may be compromised if the wave distortion becomes severe.
Fontanet (1961) developed a complete second-order theory describing
the waves produced by a sinusoidally-moving plane wavemaker. The development was in Lagrangian coordinates rendering it cumbersome to use.
Madsen (1970, 1971) developed an approximate second-order wavemaker theory for generating relatively long (h/L < 0(0.1)) second-order
Stokes waves of permanent form. Previously, the problem of secondary
waves produced by a sinusoidally-varying piston-type wave board had been
studied using numerical simulations. In presenting the approximate secondorder theory, Madsen made the observation:
To obtain general information about a certain problem, a relatively simple theoretical model that leads to analytical results is
often superior to a numerical model. (Madsen 1971).
Madsen’s approximate wavemaker theory provides an useful analytical
solution for the wave board motion necessary to suppress the unwanted free
second harmonic waves for relatively long second-order Stokes waves. The
theory is presented below, and it has been extended to include the case of
a Wave board hinged a distance I below the bottom as per Flick and Guza
(1980).