# Preface

This section concerns about two dimensional wave equation.

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Introduction to Linear Algebra with *Mathematica*

## Glossary

# 2D Wave Equations

The wave equation for a function

*u*(

*x*

_{1}, …... ,

*x*

_{n},

*t*) =

*u*(

**x**,

*t*) of

*n*space variables

*x*

_{1}, ... ,

*x*

_{n}and the time

*t*is given by

*c*(having dimensions of speed). The operator □ defined above is known as the

**d'Alembertian**or the d'Alembert operator. The wave equation subject to the initial condisions is known as the initial value problem:

*f*

_{0}(

**x**) and

*f*

_{1}(

**x**) are given (smooth) functions in

*n*-dimensional space ℝ

^{n}. For

*n*= 2, the solution of the initial value problem for wave equation is

*x*

_{1},

*x*

_{2},

*t*) on the initial data consists of the

*solid*disk

*r*≤

*ct*in the (

*y*

_{1}

*y*

_{2})-plane. So disturbances will continue indefinitely, as exhibited by water waves.

**Example: **
We consider vibrations of an elliptical drumhead with vertical displacement \( u = u(x, y,t) \)
governed by the wave equation

*T*and mass density ρ is a constant. We first separate the harmonic time dependence, writing

*v*We now use this equation to convert the Laplacian \( \nabla^2 \) to the elliptical coordinates, where we drop the

*u*-coordinate. This gives

*(q)*is a function of the continuous parameter

*q*in the Mathieu ODEs. It is this parameter dependence that complicates the analysis of Mathieu functions and makes them among the most difficult special functions used in physics. ■

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