Note: Use the prime notation for derivatives, so the derivative of [math] is written as [math]. Do NOT use [math]. The variable [math] is typed lambda, [math] is typed as mu.

Solve the two-dimensional wave equation with clamped edges and the given initial conditions:

[math]
[math] [math] [math] [math]

Part 1 Separating the variables
We try for a separable solution [math], plugging [math] into the PDE for [math] we get

[math]


Therefore [math] =
Divide both sides by [math] leads to the separated equations:
[math] [math]


For now we will work with the middle of this equation, the equation in Y and T. This equation also separates:
[math] = [math]

We now have three separated differential equations on the three variables:
ODE in X: [math] (use m for the multiple of [math])
ODE in Y: [math] (use n for the multiple of [math])
ODE in T: [math]

Part 2 The Sturm-Liouville Problems
From the boundary conditions on the plate we see that
X(0) =
X(b) =
Y(0) =
Y(c) =


That makes the differential equations in X and Y Sturm-Liouville problems. We can use them to find [math] and [math], and then X,Y, T:
X(x) = (use m as the multiple of [math])
Y(y) = (use n as the multiple of [math])
T(t) = (use Amn as the coefficient on cos, Bmn as the coefficient on sin)

Part 3 The series solution
[math]

We now need to find the coefficients [math] and [math]. Using the first initial condition:

[math] [math]
Therefore
[math]  
----------
 
[math]
[math]
 
[math]
[math]
 [math]   dydx


Now we use the second initial condition:
[math] [math]
Therefore
[math]  [math]
-----------------------------------
 
[math]
[math]
 
[math]
[math]
 [math]   dydx

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