Seperation of variables in the 2 dimensional wave equation

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[SOLVED] separation of variables in the 2 dimensional wave equation

I'd like to apologize right away for the terrible formatting. I was trying to make it pretty and easy to read but I guess I'm just not used the system yet and I had one problem after another. As you'll see at one point the formatting pretty much went out the window. I hope you can still figure out what I'm trying to say. Sorry!

Homework Statement

Solve the wave equation in 2 dimensions by separation of variables.
($$\delta$$$$^{2}$$u)/($$\delta$$t$$^{2}$$)=4(($$\delta$$$$^{2}$$u)/($$\delta$$x$$^{2}$$)+($$\delta$$$$^{2}$$u)/($$\delta$$y$$^{2}$$))
a=$$\pi$$/2 , b=$$\pi$$
V(0,y,t)=V(a,y,t)=0
V(x,0,t)=V(x,b,t)=0
V(x,y,0)=0
($$\delta$$V)/($$\delta$$t)(x,y,0)=g(x,y)=($$\pi$$/2-x)($$\pi$$-y)
Show that
V(x,y,t)=Sigma(n=1,$$\infty$$)Sigma(m=1,$$\infty$$)(((sin(2sqrt(lambda sub(nm))t))/(nm*sqrt(lambda sub(nm)))*sin(2nx)sin(m)y)
where
lambda sub(nm)=4n^2+m^2

Homework Equations

All I know is it's going to be semi-related to the general solution of the wave equation:
($$\delta$$$$^{2}$$u)/($$\delta$$t$$^{2}$$)=(k^2)*($$\delta$$$$^{2}$$u)/($$\delta$$x$$^{2}$$)

The Attempt at a Solution

I know that k^2 is going to be equal to the 4. Other then that I'm lost. If possible, well explained steps would be best so I can figure out what I'm doing.

Thanks for your help!

 

[Chronothread]

[SOLVED] separation of variables in the 2 dimensional wave equation

I'd like to apologize right away for the terrible formatting. I was trying to make it pretty and easy to read but I guess I'm just not used the system yet and I had one problem after another. As you'll see at one point the formatting pretty much went out the window. I hope you can still figure out what I'm trying to say. Sorry!

Homework Statement

Solve the wave equation in 2 dimensions by separation of variables.
($$\delta$$$$^{2}$$u)/($$\delta$$t$$^{2}$$)=4(($$\delta$$$$^{2}$$u)/($$\delta$$x$$^{2}$$)+($$\delta$$$$^{2}$$u)/($$\delta$$y$$^{2}$$))
a=$$\pi$$/2 , b=$$\pi$$
V(0,y,t)=V(a,y,t)=0
V(x,0,t)=V(x,b,t)=0
V(x,y,0)=0
($$\delta$$V)/($$\delta$$t)(x,y,0)=g(x,y)=($$\pi$$/2-x)($$\pi$$-y)
Show that
V(x,y,t)=Sigma(n=1,$$\infty$$)Sigma(m=1,$$\infty$$)(((sin(2sqrt(lambda sub(nm))t))/(nm*sqrt(lambda sub(nm)))*sin(2nx)sin(m)y)
where
lambda sub(nm)=4n^2+m^2

I would have left the relevant equations and attempted solutions parts, but for some reason I can't get it to post with that and I didn't have very much to say in those anyway. If possible, well explained steps would be best because I'm kind of lost, and then I can figure out what I'm doing.

Thanks for your help!

 

[Chronothread]

Dang it, it was posting but it came up with an error. Um... I don't know how to delete this right off hand, but give me a minute I'll see if I can figure it out.

 

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It doesn't seem I can delete. If an admin reads this please delete this and one of the other copies.

 

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I'll set this one and one other to solved to try to remove the amount of people opening it.

 

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Sorry for the copies.

 

[Chronothread]

Please delete this one, setting it to solved to try to help remove additional readers of the copy.