3-1 3.4.2. wavenumber
a. We can easily calculate the phase speed with a given wavenumber and a given water depth as discussed previously.
b. In practical problems, however, you are frequently asked to calculate the wavenumber (or wavelength, L) with a given angular frequency (or wave period, T) and a given water depth. As we can see, a direct solving of the dispersion relation is not a piece of cake. You should use a numerical technique such as the Newton-Raphson method.
c. The attached computer program provides very accurate wavenumbers (accurate probably up to the fifteenth after decimal point). The program calculates not only a wavenumber for propagating mode but also wavenumbers for evanescent modes which will be discussed later.
3.4.3. Newton-Raphson method
For reference, the basic procedure of the Newton-Raphson method is described as follows:
a. Suppose that we have a function f(x) and that we are looking for the root of f(x) = 0 . If we have a guess x0, then we can expand the function at the root, x1 about x0, that is
f(x1) ≈f(x0) + (x1-x0)fx(x0) + (x1-x0)2fxx(x0)/2! + ⋯
b. If |x0-x1| is very small, that is two solutions are close enough, the higher order terms can be neglected. Then, we have
f(x1) ≈f(x0) + (x1-x0)fx(x0) = 0
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in which x1 is assumed to be an exact solution of f(x) . It is remarked that if |x1-x0| is not small, you can not use the Taylor series expansion.
c. Rewriting the above equation yields
x1=x0- f(x0) fx(x0)
in which x1 is a more accurate solution thanx0. The procedure can be repeated until a convergent criterion is satisfied.
d. In general, the following convergent criterion is used:
|xn+ 1- xn| < ε with ε being a tiny number.
e. The general procedure of the Newton-Raphson method can be written as:
xn+ 1=xn- f(xn) fx(xn)
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Table. Calculated wavenumbers ( ω = (πg)1/2, h= 0.1m)
