View source for Fourier Transform Solution for an Incident Wave Packet

{{complete pages}}

== Introduction ==

We present here a very general and simple method to solve for an incident wave packet in the time domian
using the freqency domain solutions for an incident wave.  The theory is present here in two dimensions
but can be extended to three dimensions (for plane incident waves) straightforwardly.

== Frequency Domain Solution ==

{{standard linear problem notation}}

{{standard linear wave scattering equations}}

The simplest case is for a fixed body 
where the operator is <math>L=0</math> but more complicated conditions are possible.

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution in the Time Domain using the Frequency Domain Solution ==

Due to linearity we can write the solution in the time domain as a superposition of the frequency
domain solutions. We restrict the potential to the free surface (remembering that the 
surface displacement is proportional to the potential).
<center>
<math>
\Phi(x,t) = \mathrm{Re} \left\{\int_0^{\infty} \bar{f}(\omega) \left\{\phi(x,\omega)e^{-\mathrm{i} \omega t}\right\}
\mathrm{d} \omega \right\}
</math>
</center>

For reasons which will become apparent we change the variable of integration to <math>k</math>
the wavenumber <math>ik =k_0</math> in our standard notation for the solutions of the
[[Dispersion Relation for a Free Surface]]. 

Therefore
<center>
<math>
\Phi(x,t) = \mathrm{Re} \left\{\int_0^{\infty} \hat{f}(k) \left\{\phi(\mathbf{x},\omega(k))e^{-\mathrm{i} \omega(k) t}\right\}
\mathrm{d} k \right\}
</math>
</center>

We now consider that we have an incident wave packet which has the shape <math>f(x)</math> at <math>t=0</math>
in the absense of any scattering. Therefore

<center>
<math>
f(x) = \mathrm{Re} \left\{\int_0^{\infty} \hat{f}(k) e^{\mathrm{i} k x} \right\}
\mathrm{d} k
</math>
</center>

We can therefore calculate <math>\hat{f}(k)</math>
as
<center>
<math>
\hat{f}(k) = \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) e^{-\mathrm{i} k x}
\mathrm{d} x
</math>
</center>

[[Category:Time-Dependent Linear Water Waves]]