# 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 [math]\displaystyle{ \zeta }[/math] is the displacement, [math]\displaystyle{ \rho }[/math] is the string density and [math]\displaystyle{ h(x) }[/math] 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 [math]\displaystyle{ \beta(x) }[/math] is the non dimensionalised flexural rigidity, and [math]\displaystyle{ \gamma }[/math] is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that [math]\displaystyle{ h }[/math] is the thickness of the plate, [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \zeta }[/math] 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

- [math]\displaystyle{ \zeta(x,0)=f(x) \,\! }[/math]
- [math]\displaystyle{ \partial_t \zeta(x,0)=g(x) }[/math]

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.