Chapter VII
Wave flume
88
𝜕𝜂
í µí¼•í µí+
𝜕𝜑
𝜕𝑥
𝜕𝜂
𝜕𝑥
−
𝜕𝜑
𝜕𝑧
= 0, 𝑎𝑡 𝑧 = 𝜂
Equation VII-4
𝜕𝜑
í µí¼•í µí+
1
2
[ (
𝜕𝜑
𝜕𝑥
)
2
+ (
𝜕𝜑
𝜕𝑧
)
2
] + 𝑔𝜂 = 0, 𝑎𝑡 𝑧 = 𝜂
Equation VII-5
𝜕𝜑
𝜕𝑥
= (1 +
𝑧
ℎ
)
𝜕𝑥 0 (𝑡)
í µí¼•í µí,
𝑎𝑡 𝑥 = 𝑋(𝑧, 𝑡)
Equation VII-6
The last condition requires that the angle 𝜃 be very small, 𝜃 ≤ 10
°
, and that the wavemaker be
flat, solid, and impermeable.
To solve the problem mentioned above, the perturbation method is employed. This method is a
mathematical technique used to approximate solutions to problems by expanding the unknown
functions in a series of terms based on a small parameter 𝜀. In this case, we are interested in the
generation of first-order waves, so the unknown functions are expressed as
𝜑 = ∑ 𝜀
𝑛 𝜑 𝑛 = 𝜀𝜑 1 + 𝑂(𝜀
2 )
∞
𝑛=1
Equation VII-7
𝜂 = ∑ 𝜀
𝑛 𝜂 𝑛 = 𝜀𝜂 1 + 𝑂(𝜀
2 )
∞
𝑛=1
Equation VII-8
𝜃 = ∑ 𝜀
𝑛 𝜃 𝑛 = 𝜀𝜃 1 + 𝑂(𝜀
2 )
∞
𝑛=1
Equation VII-9
𝑋 0 = ∑ 𝜀
𝑛 𝑋 𝑛 = 𝜀𝑋 1 + 𝑂(𝜀
2 )
∞
𝑛=1
Equation VII-10
In summary, the wavemaker problem with the first-order approximation can be described by the
following equations
Laplace equation
𝜕
2 𝜑 1
𝜕𝑥 2 +
𝜕
2 𝜑 1
𝜕𝑧 2 = 0
Equation VII-11