Characteristics of a Plane Linear Progressive Wave

$$k = \frac{2\pi }{\lambda },\ \omega = \frac{2\pi }{T},\ H = 2A$$

Define $U \equiv \omega A$.
Linear Solution:

$$\phi = \frac{Ag}{\omega }\frac{\cosh k\left( {y + h} \right)... ...ega t} \right) ; \eta = A\cos \left( {kx - \omega t} \right)$$

with

$$\omega ^{2} = gk \tanh kh$$

Velocity Field:

$\begin{array}{l} u = \frac{\partial \phi }{\partial x} = \frac{Agk}{\omega }\... ...{y + h} \right)}{\sinh kh}\cos \left( {kx - \omega t} \right) \\ \end{array}$

$\begin{array}{l} v = \frac{\partial \phi }{\partial y} = \frac{Agk}{\omega }\... ...{y + h} \right)}{\sinh kh}\sin \left( {kx - \omega t} \right) \\ \end{array}$

On y = 0:

$\begin{array}{l} u = U_o = A\omega \frac{1}{\tanh kh}\cos \left( {kx - \omega... ... 1 & \mbox{\small {shallow water}} \\ \end{array}} \right. \\ \end{array}$

$\begin{array}{l} v = v_o = A\omega \sin \left( {kx - \omega t} \right) = \fra... ...h} & \mbox{\small {shallow water}} \\ \end{array}} \right. \\ \end{array}$

Finite Depth Deep Water

Pressure Field: dynamic pressure $p_d = - \rho \frac{\partial \phi }{\partial t};p = p_d - \rho gy$

Finite depth	Deep water
$\begin{array}{l} p_d = \rho gA\frac{\cosh k\left( {y + h} \right)}{\cosh kh}\... ...t} = \rho g\frac{\cosh k\left( {y + h} \right)}{\cosh kh}\eta \\ \end{array}$ $\frac{p_d }{p_{d_o}}$ same picture as $\frac{u}{u_o } ; \frac{p_d(- h)}{p_{d_o} } = \frac{1}{\cosh kh}$ shallow water: $p_{d}=\rho g\eta$ (no decay) $p = \rho g(\eta - y) \leftarrow$ ``hydrostatic'' approximation	$\begin{array}{l} p_d = \rho ge^{ky}\eta \\ p = \rho g\left[ {\eta e^{ky} -... ...\\ \frac{p_d \left( { - h} \right)}{p_{d _o} } = e^{ - ky} \\ \end{array}$

Particle Orbit/ Velocity (Lagrangian).


Let $x_{p}(t), y_{p}(t)$ be the position of the particle P, then $x_p( t) = \bar {x} + x'( t) ; y_p( t) = \bar {y} + y'( t)$ where $( {\bar {x};\bar {y}})$ is the mean position of P.

$$\begin{align}v_p & \approx v\left( {\bar {x},\bar {y},t} \right) \left( {\bar {... ...ght) y' + \ldots }_{\mbox{\small {ignore - linear theory}}} \notag \end{align}$$

$$\begin{align}x_p & = \bar {x} + \int {dt\ u\left( {\bar {x},\bar {y},t} \right)}... ... 0, y' = A\cos \left( {k\bar {x} - \omega t} \right) = \eta \notag \end{align}$$

$$\frac{{x'}^2}{a^2} + \frac{{y'}^2}{b^2} = 1 \mbox{\ or\ } \fr... ...ght)}{a^2} + \frac{\left( {y_p - \bar {y}} \right)}{b^2} = 1$$

where $a = A\frac{\cosh k\left( {\bar {y} + h} \right)}{\sinh kh}; b = A\frac{\sinh k\left( {\bar {y} + h} \right)}{\sinh kh},$ i.e. the particle orbits form closed ellipses with horizontal and vertical axes $a$ and $b$.

Summary of Plane Progressive Wave Characteristics

	Deep water/ short waves $kh > \pi$ (say)	Shallow water/ long waves $kh << 1$
$\begin{array}{l} \frac{\cosh k\left( {y + h} \right)}{\cosh kh} = f_1 \left( y \right)\sim \\ \mbox{e.g.} p_d \\ \end{array}$	$e^{ky}$	1
$\begin{array}{l} \frac{\cosh k\left( {y + h} \right)}{\sinh kh} = f_2 \left( y \right)\sim \\ \mbox{e.g.} u,a \\ \end{array}$	$e^{ky}$	$\frac{1}{kh}$
$\begin{array}{l} \frac{\sinh k\left( {y + h} \right)}{\sinh kh} = f_3 \left( y \right)\sim \\ \mbox{e.g.\ }v,b \\ \end{array}$	$e^{ky}$	$1 + \frac{y}{h}$

$C\left( x \right) = \cos \left( {kx - \omega t} \right)$	$S\left( x \right) = \sin \left( {kx - \omega t} \right)$
$\frac{\eta }{A} = C\left( x \right)$
$\frac{u}{A\omega } = C\left( x \right)f_2 \left( y \right)$	$\frac{v}{A\omega } = S\left( x \right)f_3 \left( y \right)$
$\frac{p_d }{\rho gA} = C\left( x \right)f_1 \left( y \right)$
$\frac{{y}'}{A} = C\left( x \right)f_3 \left( y \right)$	$\frac{{x}'}{A} = - S\left( x \right)f_2 \left( y \right)$
$\frac{a}{A} = f_2 \left( y \right)$	$\frac{b}{A}=f_3 \left( y \right)$