2-Dimensional Wave Equation

SIMULATION USING MATLAB2-Dimensional Wave EquationAhmed Hashem Ahmed RadwanAmr Mousa Hazem Hamdy Yomna HamzaBY

Our Goal

Wave Equation (Analytical Solution) Boundary conditions

Initial Conditions

Using separation of variables, the analytical solution for the following equation goes as follows

Dividing both sides by X(x)Y(y)T(t):

Wave Equation (Analytical Solution)

The boundary conditions become:

Using the conditions and solving for X(x) and Y(y), we get the non-trivial solutions (eigenfunctions)

with eigenvalues

Wave Equation (Analytical Solution)

Solving for T(t), where

((eigenvalues)Our final solution becomes,

Wave Equation (Analytical Solution)

The general solution becomes

We then use the 2 initial conditions to obtain the 2 constants

(double Fourier series)

Wave Equation (Analytical Solution)

Using orthogonality,

Wave Equation (Analytical Solution)

The Fourier coefficient becomes

Wave Equation (Analytical Solution)

The general solution of the two dimensional wave equation is then given by the following theorem:

Wave Equation (Analytical Solution)

Wave Equation (Analytical Solution)

Back to the original problem

Using centred difference in space and time, the equation becomes

Wave Equation (Numerical Solution)

Isolating the term that marches in time, we get

Wave Equation (Numerical Solution)

Stability condition (where c2 =1) :By optimizing the problem : Cmax was found to equal 1

Expressing the boundary conditions using our new notation, we get:

Starting from m=2, we iterate for every i and j in our meshNow, we code!

Wave Equation (Numerical Solution)

NOW, Lets test the program

ReferencesDaileda, R. (2012, March 1).The two dimensional wave equation. Lecture presented at Partial Differential Equations in Trinity University, San Antonio, Texas.Haberman, R. (n.d.). Finite Difference Numerical Methods for Partial Differential Equations. InApplied Partial Differential Equations(Fifth ed.). Pearson.