Section 5.2 Wave parameters
A general travelling wave is written as
\begin{equation}
\eta = a\sin\left[ kx - \omega t \right].\tag{5.2.1}
\end{equation}
An additional phase, e.g. in \(\sin(kx - \omega t - \beta)\) can be included but this can be removed through a shift in space or time. What do the quantities \(k\) and \(\omega\) represent?
We can factor out the \(k\) and obtain
\begin{equation*}
\eta = a\sin \left[k\left( x - \frac{\omega}{k}t\right)\right].
\end{equation*}
The quantity \(k\) is called the wavenumber. If we define it instead to be
\begin{equation}
k \equiv \frac{2\pi}{\lambda},\tag{5.2.2}
\end{equation}
then we see that shifting \(x\) by \(\pm \lambda\) does not change the wave. The quantity \(\lambda\) is called the wavelength.
Let us now turn to the quantity \(\omega/k\text{.}\) If we introduce
\begin{equation}
c \equiv \frac{\omega}{k},\tag{5.2.3}
\end{equation}
then \(c\) is called the phase speed. It corresponds to the speed at which the "phase" of the wave (the crests and troughs) propagate.
At a fixed location in space, what is the time that it takes for the condition at this point to repeat itself? It is the time required for the wave to travel one wavelength, \(\lambda\text{.}\) Hence this is defined as the period of the wave:
\begin{equation}
T \equiv \frac{\lambda}{c}.\tag{5.2.4}
\end{equation}
These concepts can be generalised to wave in a three-dimensional fluid, where we would write instead
\begin{equation}
\eta = a\sin(kx + ly + mz - \omega t) = a \sin(\boldsymbol{K} \cdot \bx - \omega t).\tag{5.2.5}
\end{equation}
In this case, \(\boldsymbol{K} = (k, l, m)\) is the wavenumber vector and has magnitude \(K = \sqrt{k^2 + l^2 + m^2}\text{.}\) The wavelength is then
\begin{equation}
\lambda = \frac{2\pi}{K}.\tag{5.2.6}
\end{equation}
The phase velocity is given by
\begin{equation}
\boldsymbol{c} = \frac{\omega}{K} \frac{\boldsymbol{K}}{K}.\tag{5.2.7}
\end{equation}
