USSC3002 Oscillations and Waves Lecture 7 Boundary Values

1. USSC3002 Oscillations and Waves Lecture 7 Boundary Values Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749 1

2. BOUNDARY CONDITIONS We consider a solution u(x,t) of the homogeneous wave equation in the region given and Theorem 1 If x-ct is constant on BC and AD and x+ct is constant on BA and CD then u(A) + u(C) = u(B) + u(D) Proof By D’Alembert u(x,t) = f(x-ct)+g(x+ct) … Question 1. Complete the proof. 2

3. BOUNDARY CONDITIONS Since the values of u in the domain are determined by the initial data on theorem 1 implies that u is determined at points P outside this domain by domain determined by initial data if P,Q,R,S are vertices of a quadrilateral with sides as in theorem 1. Question 2. Derive the coordinates of the points Q, R, S above from the coordinates of point P. 3

4. REFLECTION METHOD Initial conditions boundary conditions and the general solution imply so D’Alembert’s formula  Definitions Fixed Boundary Conditions: specify Free Boundary Conditions: specify 4

5. WAVE EQUATION ON AN INTERVAL General solution of the wave equation with boundary conditions is The coefficients are determined by initial conditions 5

6. WAVE EQUATION ON A RECTANGLE General solution of the wave equation with boundary conditions on the boundary is Question 3. How is r related to a and b ? Question 4. How are the coefficients determined from the initial conditions ? 6

7. TUTORIAL 7 • Do problem 2 on page 30 in Coulson and Jeffrey. 2. Do problem 14 on page 32 in C & J. 3. Do problem 17 on page 33 in C & J. 4. Do problem 18 on page 34 in C & J. 5. Learn about Bessel functions and derive the general solution of the wave equation with fixed boundary conditions on a circular membrane. 7