Question

Problem 1. Consider the wave equation ∂ 2 ∂t2 u = ∇2u with c 2 = 1 on a rectangle 0 < x < 2, 0 < y < 1 with u = 0 on the boundary (fixed boundary condition). Find two independent eigenfunctions um1,n1 (x, y, t) and um2,n2 (x, y, t) with either m1 6= m2 or n1 6= n2 (or both) which have the same eigenvalue (frequency).

Problem 1. Consider the wave equation a2 u= at2 v4 with ? = 1 on a rectangle () < < 2,0<y<1 with u = 0 on the boundary (fixed boundary condition) Find two independent eigenfunctions Um2,n] (I, y,t) and Um2,n2 (1, y,t) with either mı ? m, or ni 7 nz (or both) which have the same eigenvalue (frequency).

Answer 1

Given that

soli Griven that consider the wave equation 24 247 O2222 3+* 04 de at U10,4+)-4109.774) 0 409,00z4(,1, H20 Assuming the Separaciones of variable. ucaly, t) - X(1) 4 (7) T(+). from egn ① then, we geb. T" + 3 T х --ll, and y T from ega ③, then a t - * T" + - MITO Now, eqn , theen *(0) 2X2) 20, 400) 2461) 20 since T(+) 20 for all to unigt) so but we for non-livial Solution for u Cooking ulnly,t) then, we get the following stoum- lisuile eigenvaluee problem for xew),yy): you + Xx20, X10) = X(2) 20 qoc

Y"+sely 0,4(0)24110 - . the eigen values and eigenfunction of a & © equations to (1.xm(1) - Sin (12), m=1,2,3.. all m (na) Ymly csin (ny),12,1,2,3... Now can (21 > T" + ( + n)at. athen solution is Tag(t) = Amin of me ton i A Bann sind. nat ... The product solution is Um, n baig,t) a Xm() 4 mly) Trom, olt) = mm52 Jan [hy] from compromet a Bormio e do'n Theen two independent eigen function to 0 - 200 4,64%t )=sin() (992 sm(14am cos(TT+e, som ( (ny,t) = sin(Ta) sonbry> [A2, cos (art)+ sin lut)] Hence proved +nTt elai Bril Hous