# Math Help - Fourier Transform for gravity waves

1. ## Fourier Transform for gravity waves

Hi,
I've been trying to replicate the results I've seen in a paper, in order to understand it and take it a step further. However, I am not having much luck.

We are given equation (1), which is:
Z(x,z=0,t=0) = A{Ai(-x+1)*x/2*exp[-(x-2)/2]}

where Ai is the Airy function and x is the horizontal position. The amplitude of the forcing is A. I have this plotted without a problem.

However, it then states, the Fourier Transform of (1) provides the wavenumber (k) spectrum of the forcing (equation (2))

Z_hat(k,0,0) = 1/(2*pi)*integral from -inf to inf of (Z(x,0,0)*exp(i*k*x)dx)

I think I'm getting lost on this step - I don't quite understand what it means or how it should be implemented. If anyone could help me, it would be greatly appreciated!

If it is helpful to know the next step it is: water phase speed = c = sqrt(gh) where g is 9.8m/s^2 and h~4000m such that c=-200 m/s. Which implies for every k in the spectrum defined by (2), there is a corresponding wave frequency omega=-200*k

Now, the full-wave model for each omega-k pair in the spectrum. The vertical velocity spectrum is calculated as W_hat(k,0,0)=i*omega*Z_hat(k,0,0)=-i*200*k*Z_hat(k,0,0).

A discrete fourier transform is then used to evaluate the surface displacement (Z_hat) and its vertical velocity spectrum (W_hat).

If any knows how to set this problem up - it would be very helpful. Thank you so much!

2. Originally Posted by jschmid2
Hi,
I've been trying to replicate the results I've seen in a paper, in order to understand it and take it a step further. However, I am not having much luck.

We are given equation (1), which is:
Z(x,z=0,t=0) = A{Ai(-x+1)*x/2*exp[-(x-2)/2]}

where Ai is the Airy function and x is the horizontal position. The amplitude of the forcing is A. I have this plotted without a problem.

However, it then states, the Fourier Transform of (1) provides the wavenumber (k) spectrum of the forcing (equation (2))

Z_hat(k,0,0) = 1/(2*pi)*integral from -inf to inf of (Z(x,0,0)*exp(i*k*x)dx)

I think I'm getting lost on this step - I don't quite understand what it means or how it should be implemented. If anyone could help me, it would be greatly appreciated!

If it is helpful to know the next step it is: water phase speed = c = sqrt(gh) where g is 9.8m/s^2 and h~4000m such that c=-200 m/s. Which implies for every k in the spectrum defined by (2), there is a corresponding wave frequency omega=-200*k

Now, the full-wave model for each omega-k pair in the spectrum. The vertical velocity spectrum is calculated as W_hat(k,0,0)=i*omega*Z_hat(k,0,0)=-i*200*k*Z_hat(k,0,0).

A discrete fourier transform is then used to evaluate the surface displacement (Z_hat) and its vertical velocity spectrum (W_hat).

If any knows how to set this problem up - it would be very helpful. Thank you so much!
This is decomposing Z(t,0,0) into its monochromatic (single frequency) components so that subsequently theory for sine waves can be applied for the speeds of the individual components.

CB