Fourier Transform of the Wave Eq.

[Haths]

Given;

<br /> \frac{d^{2}u}{dx^{2}} = \frac{1}{c^{2}} \frac{d^{2}u}{dt^{2}}<br />

and;

<br /> u(x,0) = \phi (x)<br />

<br /> \frac{d^{2}u(x,0)}{dt^{2}} = \theta(x)<br />

Show that the Fourier Transform of the u(x,t) w.r.t. to x is;

<br /> \tilde{u}(k,t) = \tilde{\phi} (k) cos(ckt) + \frac{\tilde{\theta(k)}}{ck} sin(ckt)<br />





My attempt at the question

Well I know I can FT both sides in the following way;

<br /> \tilde{F}[ \frac{d^{2}u}{dx^{2}} ] = \frac{1}{c^{2}} \tilde{F}[ \frac{d^{2}u}{dt^{2}} ]<br />

L.H.S.

<br /> \tilde{F}[ \frac{d}{dx} \frac{du}{dx} ] = ik \tilde{F}[ \frac{du}{dt} ]<br />

<br /> \therefore<br />

<br /> \tilde{F}[ \frac{d}{dx} \frac{du}{dx} ] = -k^{2} \tilde{F}[ u ]<br />

Where i was the imaginary number i. Thus leaving me with;

<br /> -k^{2} \tilde{F}[ u ] = \frac{1}{c^{2}} \tilde{F}[ \frac{d^{2}u}{dt^{2}} ]<br />


From here I'm not sure what to do (to be frank I'm not even sure I know precisely what the questuon is asking so bear with me). Oh the d's should be partial derivatives, hence I reason that because t is just a parameter value I can take the second derivative out of the R.H.S. of the last eq. thus leaving me with an F;

<br /> -k^{2} \tilde{F}[ u ] = \frac{1}{c^{2}} \frac{d^{2}}{dt^{2}} \tilde{F}[ u ]<br />

<br /> -k^{2}c^{2} = \frac{d^{2}}{dt^{2}}<br />

Which looks so tantilisingly close to the wave equation it appears be on the right track. Of course I want the wave equation because I know that the general solution is given in terms of a sum of a sin and cos wave;

<br /> \tilde{u}(k,t) = A cos(kct) + B sin(kct)<br />

Now because k is the my Fourier coeffecient. I guess that makes the second time derivative 1 equation back on the R.H.S. one with respect to k. Hence why I can write the above equation. Yes or no?

If Yes, how do I present a mathamatical argument that get's me from my A and B constants to the functions of theta and phi?

If no, why is this wrong, and how do I go about getting to the solution?


Thanks.

PS: Why aren't the formula coming up quite right?

 

[vela]

-k^{2} \tilde{F}[ u ] = \frac{1}{c^{2}} \frac{d^{2}}{dt^{2}} \tilde{F}[ u ]

This is a differential equation you can solve for \tilde{u}(k,t), giving you...

\tilde{u}(k,t) = A \cos(kct) + B \sin(kct)

where A and B don't depend on t but can depend on k. So now use the initial conditions to determine what A and B are.

 

[vela]

\frac{d^{2}u(x,0)}{dt^{2}} = \theta(x)

Is this really supposed to be the second derivative? I think it should be only the first.