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Rectangular Drum Problem #9

Just #9, please.

Solve the rectangular drum problem

u_xx + u_yy = (c^-2)u_tt 0 < x < a 0 < y < b t > 0
u(x,y,t) = 0, u(a,y,t) = 0 0 < y < b t > 0
u(x,0,t) = 0, u(x,b,t) = 0 0 < x < a t> 0
u(x,y,0) = xy and (u_t)(x,y,0) = 0.

Find the solution explicitly in the case
u(x,y,0) = xy and (u_t)(x,y,0) = 0.

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Solution Preview

First we assume that:
u(x,y,t)=X(x)Y(y)T(t), so if we plug this into the equation, we will get:

X"YT+Y"XT=(1/c^2)T"XY or
X"/X+Y"/Y=(1/c^2)T"/T or
X"/X=-Y"/Y+(1/c^2)T"/T
We let the above expression be -k1^2 in that case we will have:
X"/X=-k1^2
X(x)=Acos(k1x)+Bsin(k1x)
Ok, because u(0,y,t)=X(0)Y(y)T(t)=0 ---> ...

Solution Summary

In this solution, a rectangular drum problem is solved. A Word copy of the solution is also attached.

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