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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 11 months ago)
Reply 1
Original post by Shaonx
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.

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
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
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.
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.

IMG_3135.jpeg
Reply 5
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.

IMG_3135.jpeg


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

DCC8E893-A910-46FE-BCE4-F99E485F0E79.jpeg
Reply 6
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
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.

BC057C61-DAC6-4827-B049-F6F4FBE8D09D.jpeg
Reply 8
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.

BC057C61-DAC6-4827-B049-F6F4FBE8D09D.jpeg

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
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.

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