PDE: wrong initial conditions given for the wave equation?

I think, in part d) of the question the initial conditions are given should be other way around u_t(x,0)=0 and vice versa. Do you agree? With the current initial conditions I am stuck. However, with the initial conditions given I can find d'Alembert's solution but that’s not the question is asking.

(edited 8 months ago)

EB1E0FF3-1D48-49B1-BBD0-052D575F51DA.jpg360.4KB

6BD918F0-48AA-49DC-9557-B9580947E870.jpeg349.7KB

Reply 1

RichE

Original post by Shaonx

There's no reason to think that. The string starts from equilibrium having been struck. Further the V in u_t suggests velocity, so think things are correct as they are.

Reply 2

Shaonx

Original post by RichE

There's no reason to think that. The string starts from equilibrium having been struck. Further the V in u_t suggests velocity, so think things are correct as they are.

I applied the first initial condition and it got rid of one of the constant. But with the 2nd condition i am not sure what to do. Any suggestions?

Reply 3

RichE

Original post by Shaonx

I applied the first initial condition and it got rid of one of the constant. But with the 2nd condition i am not sure what to do. Any suggestions?

Isn't the rest just Fourier analysis? Sorry off to bed now.

Reply 4

RDKGames

Study Forum Helper

Original post by Shaonx

I applied the first initial condition and it got rid of one of the constant. But with the 2nd condition i am not sure what to do. Any suggestions?

As @RichE says, you need Fourier analysis knowledge to figure that out.

See below. If you have found Fourier Series before then this should feel familiar.

Reply 5

Shaonx

Original post by RDKGames

As @RichE says, you need Fourier analysis knowledge to figure that out.

See below. If you have found Fourier Series before then this should feel familiar.

Please see my attempt below and tell me how to proceed from there onwards.

Reply 6

RichE

Have you met Fourier series? If so, RDK gave you the integrals to work out. If not then you shouldn't be attempting this question.

Reply 7

Shaonx

Original post by RichE

Have you met Fourier series? If so, RDK gave you the integrals to work out. If not then you shouldn't be attempting this question.

Yes I know the Fourier series, but after I put the initial condition in I get the sine Fourier series, but my initial condition is in terms of cos thus I am confused.

Reply 8

RichE

Original post by Shaonx

Yes I know the Fourier series, but after I put the initial condition in I get the sine Fourier series, but my initial condition is in terms of cos thus I am confused.

It's just another function f. It could be the exponential or a polynomial. Any continuous function on that interval has a Fourier sine series. Just work out the requisite integrals that RDK posted earlier.

Reply 9

DFranklin

Original post by Shaonx

Yes I know the Fourier series, but after I put the initial condition in I get the sine Fourier series, but my initial condition is in terms of cos thus I am confused.

As RichE says, just because your initial condition is written in terms of cos doesn't mean it doesn't have a sine Fourier series.

I'd also point out that the actual initial condition is zero outside of |x-c|<h/2. Your analysis needs to account for this.