Results 1 to 2 of 2

Math Help - Fourier Transform for gravity waves

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by jschmid2 View Post
    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fourier transform of x(t)=1
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 17th 2011, 07:38 PM
  2. Laplace transform and Fourier transform what is the different?
    Posted in the Advanced Applied Math Forum
    Replies: 8
    Last Post: December 29th 2010, 11:51 PM
  3. Fourier transform
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: September 5th 2009, 09:18 AM
  4. Replies: 0
    Last Post: April 23rd 2009, 06:44 AM
  5. from fourier transform to fourier series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 1st 2008, 07:35 AM

Search Tags


/mathhelpforum @mathhelpforum