# Category:Simple Linear Waves

## Introduction

The principle topic of this wiki is linear water waves, however other simpler linear wave systems are discussed in some detail, especially as they relate to water wave problems.

## Waves on a variable density string / waves on variable depth shallow water

The equation is

$\rho(x)\partial_t^2 \zeta = \partial_x \left(h(x) \partial_x \zeta \right).$

subject to the initial conditions

$\zeta_{t=0} = \zeta_0(x)\,\,\,{\rm and}\,\,\, \partial_t\zeta_{t=0} = \partial_t\zeta_0(x)$

where $\zeta$ is the displacement, $\rho$ is the string density and $h(x)$ is the variable depth (note that we are unifying the variable density string and the wave equation in variable depth because the mathematical treatment is identical).

The problem on waves on a string of variable density is closely related to water wave equation and is discussed in detail in Variable Depth Shallow Water Wave Equation.

## Waves on a Variable Beam

There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler Beam theory (other beam theories include the Timoshenko Beam theory and Reddy-Bickford Beam theory where shear deformation of higher order is considered). For a Bernoulli-Euler Beam, the equation of motion is given by the following

$\partial_x^2\left(\beta(x)\partial_x^2 \zeta\right) + \gamma(x) \partial_t^2 \zeta = p$

where $\beta(x)$ is the non dimensionalised flexural rigidity, and $\gamma$ is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that $h$ is the thickness of the plate, $p$ is the pressure and $\zeta$ is the plate vertical displacement) .

The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).

$\partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0$

at the edges of the plate.

The problem is subject to the initial conditions

$\zeta(x,0)=f(x) \,\!$
$\partial_t \zeta(x,0)=g(x)$

The solution for this is discuss in Waves on a Variable Beam

## Pages in category "Simple Linear Waves"

The following 3 pages are in this category, out of 3 total.