Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I need your help. Suppose I have a function $V(x)$. Now suppose I vary it with time as $W(x,t) = V(x+\cos(\omega \cdot t)$. I need to find out What is the fourier transform of $W(x,t)$ with respect to $t$. In other words I need to derive $\int{W(x,t)e^{i\omega t}dt} = \int{V(x+\cos(\omega\cdot t))e^{i\omega t}dt}$

Really appreciate your help.

share|improve this question
    
Welcome to MSE! Do you have any thoughts and have tried some things so responders don't rehash those items? Regards –  Amzoti Jun 2 '13 at 23:08
    
Since $\cos(\omega t)$ is periodic in $t$, so is $W(x,t)$. Namely $W(x,t + 2\pi / \omega) = W(x,t)$. This means that function will not be absolutely integrable on the line, as a function of $t$, and the Fourier integral will fail to converge in general. On the other hand, since $W(x,t)$ is periodic with period $2\pi / \omega$, perhaps you want to instead consider its corresponding Fourier series? –  Willie Wong Jun 3 '13 at 16:22

1 Answer 1

Just a possible idea:

If V(x) is a periodic function in x with period $\tau$, take the exponential Fourier series of $V(x)$:

$V(x) = \sum_{n =-\infty}^{\infty}c_n e^{inx}$

where $c_n = \frac{1}{\tau}\int_0^\tau V(x)e^{-inx}$

and plug in $W(x, t) = V(x + \cos(\omega t))$

$W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}e^{i n\cos(\omega t)}$

Use the general form of the Jacobi-Anger expansion to get:

$W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}\sum_{k =-\infty}^{\infty}i^kJ_k(n)e^{ik\omega t}$

or in terms of cosines:

$W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}(J_0(n) + 2\sum_{k =-\infty}^{\infty}i^kJ_k(n)\cos(k \omega t)) = $

$\sum_{n =-\infty}^{\infty}c_n e^{inx}J_0(n) + 2\sum_{n =-\infty}^{\infty}c_n e^{inx}\sum_{k =-\infty}^{\infty}i^kJ_k(n)\cos(k \omega t)$

Where $J_k$ is the Bessel function of order k.

Fourier transform both sides:

$\mathscr{F} W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}J_0(n)\int_{-\infty}^{\infty} 1 e^{-2\pi i s t}dt + 2\sum_{n =-\infty}^{\infty}c_n e^{inx}\sum_{k =-\infty}^{\infty}i^kJ_k(n)\int_{-\infty}^{\infty}\cos(k \omega t)e^{-2\pi i s t}dt =$

$\sum_{n =-\infty}^{\infty}c_n e^{inx}J_0(n)\delta(s) + \sum_{n =-\infty}^{\infty}\sum_{k =-\infty}^{\infty}c_n e^{inx}i^kJ_k(n)((\delta(s - k f_0) + \delta(s + k f_0))$

Where $\omega = 2 \pi f_0.$

share|improve this answer
    
Thank you for the quick response Birtex! however I fail to understand it.. $V(x)$ is not periodic with time it is a function of $x$ (not $t$) I define a new function $W(x,t) = V(x+\cos(\omega\cdot t))$ and I want to find the fourier transform of $W(x,t)$. I do not understand what do you mean by $V(W(x,t))$... Thanks again! –  Grisha Jun 3 '13 at 7:58
    
@Grisha Sorry, I did this when I was tired and made some mistakes. I have edited my answer. –  Bitrex Jun 3 '13 at 15:39

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.