430
CHAPTER 7. LABORATORY WAVE GENERATION
represents a free-standing wavemaker within a basin, and this boundary
condition was given as
= 0 for y < — B/2 and y > B/2.
Dalrymple and Greenberg (1985) presented a first-order solution covering an infinitely long directional wavemaker, and they gave the more
complicated solutions for finite length and point wavemakers placed in a
wall or free standing in a basin. In this chapter we will examine only the
solution for an infinitely long wavemaker. For details of the other cases, see
Dalrymple and Greenberg (1985).
The solution for the first-order velocity potential corresponding to the
generation of an oblique regular wave by an infinitely long wavemaker is
found using the separation of variables technique. This was illustrated
for the two-dimensional problem in the section entitled First-Order Wave
Generation. Skipping the details, the resulting velocity potential is (Dean
and Dalrymple 1984)
0i(i, y, z, t) = J4cosh[A:1(/i + z)] sin ( (kf — A2) x + Xy — at I +
oo
. _______
+ cos(Aj/ — at)
Cn e v 3n + A * cos[fc3n(z + à)]
n=l
(7.281)
where ky is the wavenumber of the progressive wave (the first term in
Eqn. 7.281) found from the equation a2 = gki tanh(fciTi), and k^n are the
wavenumbers of the standing waves (summation terms in Eqn. 7.281) found
from the equation a2 — —gk3 tan^/i).
Equation 7.281 is a slightly extended version of Eqn. 7.46 derived for
the two-dimensional wave generation problem. Because of the radical sign,
the solution requires that fa > A. Physically, this corresponds to the requirement that no waves can be generated at an angle greater than 90° to
the x-axis.
The wavenumber k\ in the direction of wave propagation is the hypotenuse of a right triangle with the wavenumber A aligned with the y-axis
and the wavenumber \/k2 — A2 parallel to the x-axis (see Figure 7.11). If
we define 9 as the angle between the x-axis and fa, then
A = fa sin 9
and
\Jk2 — X2 — ki cos 9
(7.282)
The next step in the first-order directional wavemaker problem is to
evaluate the wave board boundary condition given by Eqn. 7.277. Substituting fa from Eqn 7.281 and dXoJdt from Eqn. 7.280 and performing the
algebra yields
CHAPTER 7. LABORATORY WAVE GENERATION
represents a free-standing wavemaker within a basin, and this boundary
condition was given as
= 0 for y < — B/2 and y > B/2.
Dalrymple and Greenberg (1985) presented a first-order solution covering an infinitely long directional wavemaker, and they gave the more
complicated solutions for finite length and point wavemakers placed in a
wall or free standing in a basin. In this chapter we will examine only the
solution for an infinitely long wavemaker. For details of the other cases, see
Dalrymple and Greenberg (1985).
The solution for the first-order velocity potential corresponding to the
generation of an oblique regular wave by an infinitely long wavemaker is
found using the separation of variables technique. This was illustrated
for the two-dimensional problem in the section entitled First-Order Wave
Generation. Skipping the details, the resulting velocity potential is (Dean
and Dalrymple 1984)
0i(i, y, z, t) = J4cosh[A:1(/i + z)] sin ( (kf — A2) x + Xy — at I +
oo
. _______
+ cos(Aj/ — at)
Cn e v 3n + A * cos[fc3n(z + à)]
n=l
(7.281)
where ky is the wavenumber of the progressive wave (the first term in
Eqn. 7.281) found from the equation a2 = gki tanh(fciTi), and k^n are the
wavenumbers of the standing waves (summation terms in Eqn. 7.281) found
from the equation a2 — —gk3 tan^/i).
Equation 7.281 is a slightly extended version of Eqn. 7.46 derived for
the two-dimensional wave generation problem. Because of the radical sign,
the solution requires that fa > A. Physically, this corresponds to the requirement that no waves can be generated at an angle greater than 90° to
the x-axis.
The wavenumber k\ in the direction of wave propagation is the hypotenuse of a right triangle with the wavenumber A aligned with the y-axis
and the wavenumber \/k2 — A2 parallel to the x-axis (see Figure 7.11). If
we define 9 as the angle between the x-axis and fa, then
A = fa sin 9
and
\Jk2 — X2 — ki cos 9
(7.282)
The next step in the first-order directional wavemaker problem is to
evaluate the wave board boundary condition given by Eqn. 7.277. Substituting fa from Eqn 7.281 and dXoJdt from Eqn. 7.280 and performing the
algebra yields