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## Homework Statement

The wave equation for ψ=ψ(t,x,y) is given by

##\frac{\partial ^2 \phi}{\partial t^2} - \frac{\partial ^2 \phi}{\partial x^2} - \frac{\partial ^2 \phi}{\partial y^2}##

Use separation of variables to separate the equation into 3 ODEs for T, X and Y. Use the separation constants

##-k_{x}^{2}X## and ##-k_{y}^{2}Y##

Do not introduce any more separation constants for T.

## Homework Equations

## The Attempt at a Solution

I'm fairly sure I know how to start.

Ansatz ψ(t,x,y) = T(t)X(x)Y(y). Sub the derivatives of this into the the wave equation:

##XY\frac{\partial ^2 T}{\partial t^2} - TY\frac{\partial ^2 X}{\partial x^2} - TX\frac{\partial ^2 Y}{\partial y^2}##=0

Then divide by TXY:

##\frac{1}{T}\frac{\partial ^2 T}{\partial t^2} - \frac{1}{X}\frac{\partial ^2 X}{\partial x^2} - \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}##=0

Rearrange:

##\frac{1}{T}\frac{\partial ^2 T}{\partial t^2} = \frac{1}{X}\frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}##

Which is only possible if LHS = RHS = constant, I think? But then I would get

##\frac{1}{X}\frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}## = constant, and how do I separate that further? Why would I need more than one constant of separation?

Because if

##\frac{1}{X}\frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}## = constant

Then surely each individual term must also be a constant, and I can just write

##\frac{1}{X}\frac{\partial ^2 X}{\partial x^2}## = ##-k_{x}^{2}X##

##\frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}## = ##-k_{y}^{2}X##

And why wouldn't I introduce a new constant for T? What do I write instead,

##\frac{1}{T}\frac{\partial ^2 T}{\partial t^2}## = constant?