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Solving wave equation with Fourier transform

  • Thread starter fluidistic
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fluidistic
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Homework Statement


Use Fourier transforms to calculate the motion of an infinitly large stretched string with initial conditions [itex]u(x,0)=f(x)[/itex] and null initial velocity. The displacements satisfy the homogeneous wave equation.


Homework Equations



[itex]\frac{\partial ^2 u }{\partial t^2 }-c\frac{\partial ^2 u }{\partial x^2 }=0[/itex].
[itex]\mathbb{F} (u)=F(\omega )=\int _{-\infty }^{\infty }u(t)e^{i\omega t }dt[/itex].
[itex]\mathbb{F} \left ( \frac{d^nf}{dx^n} \right )=(-i\omega )^n \mathbb{F} (u)[/itex].

The Attempt at a Solution


So my idea is to take the Fourier transform of the wave equation. I guess I have the choice to take it with respect to either x or t?
Taking it with respect to x, I obtain [itex]\frac{d^2}{dt^2}\mathbb{F} (u)+\underbrace {c\omega ^2}_{\geq 0 } \mathbb{F}(u)=0[/itex].
So that [itex]\mathbb{F} (u)=A\cos (c\omega ^2 t )+B \sin (c\omega ^2 t )[/itex]. Now I don't really know how to proceed.
I don't know if I should take some inverse Fourier transform or use the initial conditions, namely [itex]u(x,0)=f(x)[/itex] and [itex]\frac{\partial u}{\partial t } (x,0)=0[/itex]. I'm not confident so far in what I've done... could someone help me?
 

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