2 Dimensional Wave Equation Analytical and Numerical Solution This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI you can find the gui in mathworks file-exchange here https://www.mathworks.com/matlabcentral/fileexchange/55117-2d-wave-equation-simulation-numerical-solution-gui

1. SIMULATION USING MATLAB
2-Dimensional Wave Equation
• Ahmed Hashem
• Ahmed Radwan
• Amr Mousa
• Hazem Hamdy
• Yomna Hamza
BY

2. • Our Goal

3. • Wave Equation (Analytical
Solution)
• Boundary conditions
• Initial Conditions
 Using separation of variables, the analytical
solution for the following equation goes as follows

4.  Dividing both sides by X(x)Y(y)T(t):
• Wave Equation (Analytical
Solution)

5.  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)

6.  Solving for T(t), where
((eigenvalues)
 Our final solution becomes,
• Wave Equation (Analytical
Solution)

7.  The general solution becomes
 We then use the 2 initial conditions to obtain the 2
constants
(double Fourier
series)
• Wave Equation (Analytical
Solution)

8.  Using orthogonality,
• Wave Equation (Analytical
Solution)

9.  The Fourier coefficient becomes
• Wave Equation (Analytical
Solution)

10.  The general solution of the two dimensional wave
equation is then given by the following theorem:
• Wave Equation (Analytical
Solution)

11. • Wave Equation (Analytical
Solution)

12.  Back to the original problem
 Using centred difference in space and time, the
equation becomes
• Wave Equation (Numerical
Solution)

13.  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

14.  Expressing the boundary conditions using our
new notation, we get:
 Starting from m=2, we iterate for every i and j in
our mesh
 Now, we code!
• Wave Equation (Numerical
Solution)

15. NOW, Let’s test the
program

16. References
 Daileda, 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. In Applied Partial Differential
Equations (Fifth ed.). Pearson.