Choosing between wave equation solutions with given boundary conditions

[gomes.]

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Im a bit stuck at part (e) and (f)

The 4 basic solutions are:
sinkx*coskct
sinkx*sinkct
coskx*coskct
coskx*sinkct

I got the 4 boundary conditions as:
y(0,t)=0
y(L,t)=0
y(x,0)=sin (2pi/l)x
dy/dt at t=0, =0

For part (e), I could eliminate coskx*coskct and coskx*sinkct from y(0,t)=0.

How do I choose between the other 2 solutions?

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for part (f)
How do I do it? Thanks.

 

[omoplata]

The last two of your boundary conditions are wrong.

If they are corrected solutions for parts (e) and (f) can be found. Even if they are not corrected a solution can be found for part (e), but not (f).

After the boundary conditions are corrected:

part (e): Compare the boundary condition for y(x,0) with the remaining two equations. Only one can have finite solutions for all possible values of x.

part (f): Apply the boundary condition y(L,t)=0 to the singled out solution and find a condition for k. Then use the boundary condition for dy/dt at t=0 to find a value for c. If you have k and c, you have the frequencies.

Tell me if you have trouble.

PS: I didn't check your 4 basic solutions and assumed them to be correct.