7.4. NONLINEAR WAVE GENERATION
377
the inhomogeneous free surface boundary condition. Calculation of the
coefficients (Cn) of the standing wave terms was done numerically by Flick
and Guza, and analytical expressions for coefficients in their solution were
not given in the paper.
Flick and Guza stated that secondary free waves generated by sinusoidal wave board motion have an amplitude comparable to that of the
second-order Stokes wave component, irrespective of wave board geometry or values of kh. Therefore, finite-amplitude waves generated in any
water depth will be seriously contaminated by secondary free waves in a
wave tank with a horizontal bottom. They noted, however, that the secondary free waves generated by a wave board in relatively deep water will
become increasingly unimportant as the waves shoal. This occurs because
the Stokes second-order wave grows much more rapidly than the secondary
free wave as shoaling occurs. Therefore, an effective method of suppressing
the secondary free wave is to generate the waves (using sinusoidal motion)
in deep water, and then allowing the waves to shoal to a shallow depth.
Hudspeth and Sulisz (1986, 1991) formulated a complete second-order
solution for a generic planar wavemaker being forced in a sinusoidal oscillation. The theory holds for wavemakers that are doubly articulated, and it
includes both piston and hinged wave boards of variable draft. Their theory included the first-order standing wave summation terms in evaluating
both the second-order free surface and wave board boundary conditions,
and these terms were shown to be non-negligible for estimating the amplitude of the secondary free wave. Hudspeth and Sulisz also included, for
the first time, a time-independent solution that is needed to satisfy inhomogeneous boundary conditions on the wave board and at the free surface.
The time-independent solution was found to estimate accurately the mean
return flow in a closed wave flume.
Hudspeth and Sulisz’s (1991) formulation is a linear combination of
three time-dependent velocity potentials and two time-independent potentials; and not surprisingly, it is mathematically complex and must be applied numerically. The paper did not discuss the necessary nonsinusoidal
wave board motion that would be needed to suppress the secondary free
wave.
Synolakis (1989) derived a second-order approximate solution for the
instantaneous hydrodynamic force (per unit width) on a piston-type wave
board generating long waves. The theory was developed for the case where
water is only on one side of the wave board, and the approximations used in
the theoretical development limit application to cases when h/L << 0.15,
which is more restrictive than Madsen’s (1971) approximate second-order
wavemaker theory.
In dimensional form, the instantaneous force per unit width on a piston
377
the inhomogeneous free surface boundary condition. Calculation of the
coefficients (Cn) of the standing wave terms was done numerically by Flick
and Guza, and analytical expressions for coefficients in their solution were
not given in the paper.
Flick and Guza stated that secondary free waves generated by sinusoidal wave board motion have an amplitude comparable to that of the
second-order Stokes wave component, irrespective of wave board geometry or values of kh. Therefore, finite-amplitude waves generated in any
water depth will be seriously contaminated by secondary free waves in a
wave tank with a horizontal bottom. They noted, however, that the secondary free waves generated by a wave board in relatively deep water will
become increasingly unimportant as the waves shoal. This occurs because
the Stokes second-order wave grows much more rapidly than the secondary
free wave as shoaling occurs. Therefore, an effective method of suppressing
the secondary free wave is to generate the waves (using sinusoidal motion)
in deep water, and then allowing the waves to shoal to a shallow depth.
Hudspeth and Sulisz (1986, 1991) formulated a complete second-order
solution for a generic planar wavemaker being forced in a sinusoidal oscillation. The theory holds for wavemakers that are doubly articulated, and it
includes both piston and hinged wave boards of variable draft. Their theory included the first-order standing wave summation terms in evaluating
both the second-order free surface and wave board boundary conditions,
and these terms were shown to be non-negligible for estimating the amplitude of the secondary free wave. Hudspeth and Sulisz also included, for
the first time, a time-independent solution that is needed to satisfy inhomogeneous boundary conditions on the wave board and at the free surface.
The time-independent solution was found to estimate accurately the mean
return flow in a closed wave flume.
Hudspeth and Sulisz’s (1991) formulation is a linear combination of
three time-dependent velocity potentials and two time-independent potentials; and not surprisingly, it is mathematically complex and must be applied numerically. The paper did not discuss the necessary nonsinusoidal
wave board motion that would be needed to suppress the secondary free
wave.
Synolakis (1989) derived a second-order approximate solution for the
instantaneous hydrodynamic force (per unit width) on a piston-type wave
board generating long waves. The theory was developed for the case where
water is only on one side of the wave board, and the approximations used in
the theoretical development limit application to cases when h/L << 0.15,
which is more restrictive than Madsen’s (1971) approximate second-order
wavemaker theory.
In dimensional form, the instantaneous force per unit width on a piston