Chapter VII
Wave flume
92
𝜎
2 = 𝑔𝑘 tanh(𝑘ℎ)
Equation VII-26
For a fixed period and a constant water depth, there is only one 𝑘 that satisfies Equation VII-26;
This value is determined by employing an iterative process (Table VII-2) to solve the equation.
Table VII-2 Iterative process for solving the dispersion equation.
For each period of range do.
1. Calculate the angular frequency 𝜎 for the period.
2. Choose an initial value for k.
3. Calculate the angular frequency 𝜎 ̅ corresponding to the value of k.
4. Evaluate the difference 𝜎
2 − 𝜎 ̅
2
.
5. If the difference is greater than the desired tolerance, increment k by STEP and go to
step3.
6. Else, save the value of k as the wave number of the considered period and go to the next
period (go to step1).
The STEP is evaluated by 𝑆𝑇𝐸𝑃 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ×
𝑑𝑘
𝑑(𝜎 2 )
|
𝑘=𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘
. It’s clear from the
Figure VII-3 that 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑘,
𝑑𝑘
𝑑(𝜎 2 )
= 0.1
Figure VII-3 Plot of k as a function of σ^2.
To visualize the different limits of actuator stroke, the equations (Equation VII-22, Equation
VII-23, Equation VII-24 and Equation VII-25) are plotted after applying the transformation
𝑆 𝑐 =
ℎ 𝑐
ℎ
× í µí±† 0 .
From the graph illustrated in Figure VII-4, it is evident that the maximum extent of actuator
strokes, represented as 𝑆 𝑐 𝑚𝑎𝑥, is initially constrained by the wave break limit 𝑆 𝑐,𝑤𝑏 𝑚𝑎𝑥 until
𝑇 = 1.23 𝑠. Afterward, its limitation shifts to the physical capacity of the actuator.