# 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

subject to the initial conditions

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

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).

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.