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. 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 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).

We begin with