Wavemaker Theories
2.2. Planar Wavemaker in a 2D Channel
Two generic planar wavemaker configurations are shown in Figs. 2.1(a) and 2.1(b).
The fluid motion may be obtained from the negative gradient of a dimensional scalar velocity potential Φ(x, z, t) according to
q(x, z, t) = u(x, z, t)ex+ w(x, z, t)ez=−∇2Φ(x, z, t), (2.2a) where the 2D gradient operator in (2.2a) is given by
∇2(•) = ∂(•)
∂x ex+∂(•)
∂z ez.
Fig. 2.1(a). Definition sketch for a Type I planar wavemaker.
Fig. 2.1(b). Definition sketch for a Type II planar wavemaker.
28 R. T. Hudspeth and R. B. Guenther
The total pressure field P (x, z, t) may be computed from the unsteady Bernoulli equation according to where Q(t) = the Bernoulli constant; and the free surface elevation η(x, t) for zero atmospheric pressure according to
η(x, t) = 1 The scalar velocity potential must be a solution to the Laplace equation
∇22Φ = 0; x ≥ ξ(z, t); −h ≤ z ≤ η(x, t), (2.3a) with the following boundary conditions:
Kinematic Bottom Boundary Condition (KBBC):
∂Φ
∂z = 0; x ≥ ξ(−h, t); z = −h. (2.3b) Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):
∂2Φ Kinematic WaveMaker Boundary Condition (KWMBC):
A Stokes material surface for planar wavemaker is W (x, z, t) = x− ξ(z, t), and the Stokes material derivative gives the KWMBC from
DW Kinematic Radiation Boundary Condition (KRBC):
A KRBC is required as x→ +∞ for uniqueness to insure that propagating waves are only right progressing or that evanescent eigenmodes are bounded. For a temporal dependence proportional to exp±iωt, the KRBC may be expressed by
x→+∞lim
∂
∂x ±iKn
Φ(x, z, t) = 0. (2.3e)
A velocity potential ϕ(x, z) may be defined by the real part of
Φ(x, z, t) = Re{ϕ(x, z) exp −i(ωt + ν)}, (2.4)
Wavemaker Theories 29
where Re{•} means the real part of {•}; and ν = arbitrary phase angle. The linearized WMBVP for kh = O(1) is1
∇22ϕ(x, z) = 0; 0≤ x < +∞; −h ≤ z ≤ 0, (2.5a)
∂ϕ(x, z)
∂z = 0; 0≤ x + ∞; z = −h, (2.5b)
∂ϕ(x, z)
∂z − k0ϕ(x, z) = 0; 0≤ x < +∞; z = 0, (2.5c)
x→+∞lim
∂
∂x −iKn
ϕ(x, z) = 0, (2.5d)
∂ϕ(x, z)
∂x exp−i(ωt + ν) = −∂ξ(z, t)
∂t ; x = 0; −h ≤ z ≤ 0, (2.5e)
η(x, t) = Re
−iω
g ϕ(x, 0) exp −i(ωt + ν)
; x ≥ 0; z = 0, (2.5f)
p(x, z, t) = ρ∂Φ(x, z, t)
∂t ; 0≤ x < +∞; z = 0, (2.5g) where k0= ω2/g.
Because the boundary conditions defined by (2.5b)–(2.5e) are prescribed on boundaries with constant values of the independent variables x and z, a solution by the method of separation of (independent) variables may be computed.1The instan-taneous wavemaker displacement ξ(z, t) from its mean position x = 0 is assumed to be strictly periodic in time with period T = 2π/ω, and may be expressed by
ξ(z/h, t) = Re
i
S
(∆/h)
χ(z/h) exp −i(ωt + ν)
=
S
(∆/h)
χ(z/h) sin(ωt + ν). (2.6)
The specified shape function χ(z/h) for the Type I wavemaker shown in Fig. 2.1(a) is valid for either a double-articulated piston or hinged wavemaker of variable draft and is given by the following dimensionless equation for a straight line2:
χ(z/h) = [α(z/h) + β][U(z/h + 1 − d/h) − U(z/h + b/h)], (2.7a) where α, β = dimensionless constants; U (•) = the Heaviside step function with two boundary conditions given by
[S/(∆/h)]χ(z/h =−1 + d/h + ∆b/h + ∆/h) = S, (2.7b) [S/(∆/h)]χ(z/h =−1 + d/h + ∆b/h) = Sb, (2.7c)
30 R. T. Hudspeth and R. B. Guenther
that may be solved simultaneously for the dimensionless coefficients α, β to obtain α = (1 − Sb/S); β = ∆/h + α(1 − d/h − ∆b/h − ∆/h). (2.7d,e) The coefficients α and β for the specified shape function χ(z/h) in (2.7a) may be obtained for the Type II wavemaker shown in Fig. 2.1(b) by substituting
S = ¯S + ˆS; Sb= ¯S; ∆ = h− b − d into the following boundary conditions2 in (2.7b) and (2.7c):
( ¯S + ˆS) 1− b/h − d/h
χ(z/h = −b/h) = ¯S + ˆS, (2.8a)
( ¯S + ˆS) 1− b/h − d/h
χ(z/h = −1 + d/h) = ¯S, (2.8b)
that may be solved simultaneously for the constant coefficients α, β to obtain α = Sˆ
S + ˆ¯ S; β = 1 − d h −
S¯ S + ˆ¯ S
b
h. (2.8c,d)
2.2.1. Eigenfunction solution to the WMBVP
Because all of the boundary conditions defined by (2.5b)–(2.5e) are now prescribed for constant values of the independent variables (x, z) and the dimensionless parameter kh = O(1), a solution by separation of independent variables1 is suggested according to
ϕ(x, z) = X(x) • Z(z). (2.9)
The eigenseries solution may be written compactly as1,3,4 Φ(x, z, t; Kn) =
n=1
Cncosh Kn(z + h) exp +i(Knx − ωt + ν), (2.10a)
where Kn= k for n = 1 and Kn = +iκn for n≥ 2 provided that
koh − kh tanh kh = koh + κnh tan κnh = 0; n > 2. (2.10b) The eigenseries (2.10a) may be separated into a propagating Φp(x, z, t; k) and evanescent eigenmodes Φe(x, z, t; κn) or “local ” wave components3 according to
Φ(x, z, t; Kn) = Φp(x, z, t; k) + Φe(x, z, t; κn)
=
C1cosh k(z + h) +
n=2
Cncos κn(z + h)
exp +i(Knx − ωt + ν).
(2.10c)
Wavemaker Theories 31
The wave number k = 2π/λ where λ = wavelength. Because the numerical value of kh must be computed from an eigenvalue problem in the vertical z coordinate, equiv-alence of the eigenvalue k to the wave number 2π/λ requires a pseudo-horizontal boundary condition of periodicity given by k = 2π/λ and ϕ(x + λ, z) = ϕ(x, z). It is computationally efficient to normalize the eigenseries in (2.10a) according to
Ψn(Kn, z/h) = cosh Knh(1 + z/h)
Nn ; n = 1, 2, . . . , (2.11a) where the nondimensional normalizing constant Nn is
Nn2= The eigenseries in (2.10a) may be written as an orthonormal eigenseries by
Φ(x, z, t; Kn) = ∞ n=1
CnΨn(Kn, z/h) exp i(Knx − ωt − ν), (2.12)
where the orthonormal eigenfunction Ψn(•,•) is dimensionless.
2.2.2. Evaluation of Cn by WM vertical displacement χ(z/h)
The following dimensionless coefficient computed from (2.5e) will replace integral calculus with algebraic substitution for the coefficients Cn in the eigenseries (2.12):
In(α, β, b, d, Kn) = that is dimensionless when α and β are given by (2.7d) and (2.7e) or (2.8c) and (2.8d). The coefficients Cn may be computed algebraically by (2.13) from the KWMBC (2.5e) to obtain
Cn= iSωh
Kn∆In(α, β, b, d, Kn), (2.14) and the orthonormal eigenseries (2.12) is given by
Φ(x, z, t; Kn)
32 R. T. Hudspeth and R. B. Guenther
2.2.3. Decay distance of evanescent eigenmodes n ≥ 2
Numerical solutions and experimental measurements of ocean and coastal designs require that the KRBC (2.5d) be applied far enough away so that only the prop-agating eigenmode for n = 1 in (2.12) is measurable. The evanescent eigenseries in (2.12) for n ≥ 2 will decay spatially at least as fast as the smallest evanescent eigenvalue κ2. This eigenvalue must be κ2h > (n − 3/2)π = π/2. If the smallest value for κ2h > π/2, then κ2 > π/2h and ϕ(x, z) ∝ exp −(πx/2h). For the values of the evanescent eigenseries to be less than 1% of their values at the wavemaker, ϕ(x, z) ∝ exp −(πxd/2h) = 0.01 and πxd/(2h) = 4.6 ≈ 3π/2, and the minimum decay distance is xd ≥ 3h.
2.2.4. Transfer function for wave amplitude from wavemaker stroke
The average rate of work or power done by a wavemaker of width B is1
˙W τ =Pτ = B
τ+1
τ
h
0
−1
p(x, z, τ)u(x, z, τ)d(z/h)dτ, (2.16a)
where the temporal averaging operator is defined by
•τ =
τ+1
τ
(•)dτ, (2.16b)
and
˙W τ =Pτ =
ρω3S2Bh4
∆22kh
I12(α, β, b, d, k), (2.16c)
so that all of the average power from a wavemaker is transferred to only the propagating eigenmode. The average energy flux in a linear wave is given by1
˙Eτ =
ρgBA2 2
CG, (2.16d)
where the group velocity CG is given by1 CG= C
2
1 + 2kh sinh 2kh
. (2.16e)
Equating (2.16c) to (2.16d) gives the following transfer function for a planar wavemaker:
A S =
koh kh
Ψ1(k, 0)I1(α, β, b, d, k). (2.16f)
Wavemaker Theories 33
2.2.5. Hydrodynamic pressure loads (added mass and radiation damping)
The wave loads on a planar wavemaker may be estimated by integrating the total pressure over the wetted surface of the wavemaker, i.e.,
F M
=
0S
P
n
r× n
dS, (2.17a,b)
where the outward pointing unit normal n points from the wavemaker into the fluid, and the pseudo-unit normal n for the rotational modes is given by
n= r× n = (z + h − d)nxey = nyey. (2.17c) Force. For the Type I piston wavemaker of total width B, the horizontal component of the pressure force on the fluid side only may be computed from the real part of
F1(t) = Re
iρωBh
n=1
Cn
−b/h
−1+d/h
Ψn(Kn, z/h)d(z/h) exp −i(ωt + ν)
=−F1cos(ωt + ν− α1), (2.18a)
where the static component of the pressure force on the fluid side only is Fs=−ρgBh2
2 [1− 2(d/h) + (d/h)2− (b/h)2]. (2.18b) The hydrodynamic component of F1(t) may be separated linearly into a propagating and an evanescent component that are related to the piston wavemaker translational velocity and acceleration, respectively, from the real part of
F1(t) =−Re{[λ11(Sω) + µ11(−iSω2)] exp−i(ωt + ν)}
=−µ11(−Sω2sin(ωt + ν))− λ11(Sω cos(ωt + ν)) (2.18c)
= Re{−µ11X¨1(t)− λ11X˙1(t)}, (2.18d) where the added mass coefficient µ11 may be computed from the evanescent eigenmodes only, and the radiation damping coefficient λ11may be computed from the propagating eigenmode only. The average power may be computed from
−F1X˙1t= λ11(Sω)2
2 . (2.19a)
Equating (2.19a) to (2.16d) yields λ11=
A1
S1
2ρBh
kohCG, (2.19b)
34 R. T. Hudspeth and R. B. Guenther
that relates the radiation damping coefficient to the square of the ratio of the radiated wave amplitude to the amplitude of the wavemaker displacement.
Moment. For the Type I wavemaker of width B, the dynamic pressure moment on one side only of the wavemaker may be computed from the real part of
M5(t) = Re
and the static component of the pressure moment on the fluid side only is Ms= ρgBh3 The pressure moment M5(t) in (2.20a) may be separated linearly into a propagating and an evanescent component that are related to the rotational velocity and accel-eration from the real part of1
M5(t) =−Re
Havelock5 applied Fourier integrals to develop a theory for surface gravity waves forced by circular wavemakers in water of both infinite and finite depth. The fluid motion may be obtained from the negative gradient of a scalar velocity potential Φ(r, θ, z, t) according to
q(r, θ, z, t) =−∇Φ(r, θ, z, t), (2.22a)