Chapter VII
Wave flume
89
Bottom boundary condition for horizontal bottom
𝜕𝜑 1
𝜕𝑧
= 0, 𝑎𝑡 𝑧 = −ℎ
Equation VII-12
Kinematic and dynamic free surface boundary condition
𝜕
2 𝜑 1
𝜕𝑡 2 + 𝑔
𝜕𝜑 1
𝜕𝑧
= 0, 𝑎𝑡 𝑧 = 0
Equation VII-13
General wave board boundary condition
𝜕𝜑 1
𝜕𝑥
= (1 +
𝑧
ℎ
)
𝑑𝑋 1
í µí±‘í µí𝑎𝑡 𝑥 = 0
Equation VII-14
Using the method of separation of variables, we can obtain the final solution for the surface
elevation 𝜂(𝑥, 𝑡) in terms of the wave parameters. The solution is given by
𝜂 1 (𝑥, 𝑡) =
𝜎𝐴
𝑔
cosh(𝑘 1 ℎ) cos(𝑘 1 𝑥 − 𝜎𝑡) + sin(𝜎𝑡) ∑
𝜎𝐶 𝑛
𝑔
𝑒
−𝑘 3 𝑛 𝑥
∞
𝑛=1
cos(𝑘 3 𝑛 ℎ)
Equation VII-15
where σ is defined as
𝜎
2 = 𝑔𝑘 1 tanh(𝑘 1 ℎ) = −𝑔𝑘 3 tanh (𝑘 3 ℎ)
Equation VII-16
and 𝐴 is given by:
𝐴 =
2𝜎𝑆 0
𝑘(𝑠𝑖𝑛ℎ 2𝑘ℎ + 2𝑘ℎ)
[sinh 𝑘ℎ
(1 − cosh 𝑘ℎ)
𝑘ℎ
]
Equation VII-17
The second term in Equation VII 15 becomes negligible after the distance x = 3h. This simplifies
the equation to:
𝜂 1 (𝑥, 𝑡) =
𝜎𝐴
𝑔
cosh(𝑘 1 ℎ) cos (𝑘 1 𝑥 − 𝜎𝑡)
Equation VII-18
Therefore, the final expression for 𝜂₁(𝑥, 𝑡) is:
𝜂 1 (𝑥, 𝑡) =
𝐻
2
cos (𝑘 1 𝑥 − 𝜎𝑡)
Equation VII-19