Linearized (Airy) Wave Theory

Consider small amplitude waves: (small free surface slope)

Assume amplitude small compared to wavelength, i.e., $\frac{A}{\lambda } < < 1$. Consequently: $\frac{\phi }{{\lambda ^2} \mathord{\left/ {\vphantom {{\lambda ^2} T}} \right. \kern-\nulldelimiterspace} T},\frac{\eta }{\lambda } < < 1$, and we keep only linear terms in $\phi$,$\eta$:

$$\mbox{For example:\ } \left. {\left( \right)} \right\vert _{y... ...x{\tiny {discard}}} + \ldots \mbox{\ \small {Taylor series}}$$

Finally:

$$\begin{array}{l} 1) \nabla ^2\phi = 0, - h < y < 0 \left\{ ... ...ac{\partial \phi }{\partial y} = 0_ ; y = 0 \\ \end{array}$$

Linear (Airy) Waves in constant finite depth $h$ {infinite depth}
   
Given $\phi$:
  
Solution of 2D Periodic Plane Progressive Waves (using separation of variables)
Math ...(solve (1), (2), (3))
Answer:

$$\begin{align}% latex2html id marker 239 \phi & = \frac{gA}{\omega }\sin \left( {... ...cos \left( {kx - \omega t} \right); \left\{ {same} \right\} \notag \end{align}$$

where $A$ is the wave amplitude $= H/2$.

1.

At $t = 0$ (say), $\eta =A\cos kx \to$ periodic in $x$ with wavelength: $\lambda = 2\pi/k [L]$ figure

2.

At $x = 0$ (say), $\eta = A\cos \omega t \to$ periodic in $t$ with period: $T = 2\pi/\omega [T]$ figure

3.

$\eta = A\cos \left[ {k\left( {x - \frac{\omega }{k}t} \right)} \right]_ ;\frac{\omega }{k} = \left[ V \right]$
Following a point with velocity $\frac{\omega }{k}$, i.e. $x_p = \left( {\frac{\omega }{k}} \right)t + const.$ the phase of $\eta$ does not change: $\frac{\omega }{k} = \frac{\lambda }{T} \equiv V_p$ phase velocity.