Waves - Power spectral density to amplitude

[chipmeisterc]

Hi all,

I am trying to simulate ocean waves on a 3d grid of points. I have recently discovered the Pierson Moskowitz spectrum which will give a wave spectral density (ms/hz) for a given wind condition.

I am trying to represent this by using a large number (say 64) Gerstner waves. Each wave I am assigning a unique frequency with 64 even intervals between 0.05 hz and 20hz which seems to cover most of the useful spectrum and discreetly sampling the equation:
http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/Images/eq16-30.gif

I am picking a random phase and direction.

For amplitude on my first attempt I tried to directly use the spectral density directly multiplied by a constant however this seems to give very poor results - what's more as my amplitudes are summed the amplitudes get bigger and bigger with more and more waves so my question is how do I get from wave spectral density to amplitude correctly?

From reading various papers my understanding is that spectral density represents amplitude variance? I had no idea what variance meant until about 15 minutes ago, however after watching Khan Academy I know know this is Standard deviation^2 (another new one to me! ) or the squared average of each point minus the mean. I can therefore understand variance and how it relates to a set of data but struggling to wrap my head around it in the context of a single frequency or how I use this to pick sensible amplitudes for each wave?

Any help greatly appreciated!

Edit just to add that using the above equation the values returned do not seem to have the same y-axis scaling as the following diagram, which would also suggest the output units of the above equation are slightly different?? :

Thanks

 

[jedishrfu]

This looks like homework. I moved it to the Biology, chemistry and Earth homework forum.

In the future, please select the proper forum and you will be provided with a homework template to fill in.

This will help us answer your questions more quickly and effectively.

 

[SteamKing]

The units in the spectrum plot shown are m²/Hz, not ms/Hz.

The various wave spectral formulations, of which the P-M is one of the oldest, were never intended to give a full description of sea conditions, like the heights of individual waves. These spectral formulations were developed after analyzing various observational data, like those obtained from buoys, in order to gauge roughly the amount of energy is a seaway given a steadily blowing wind which had been present long enough for the sea waves to come into equilibrium.

In short, while the analysis of actual wave data can determine a spectrum of all the actual wavelengths present, it was never intended to turn the crank backwards, i.e. take a given wave spectrum and try to re-create all of the different wave components.

For water waves, there is also no direct correlation between wavelength and wave amplitude. A wave of a given wavelength can have many different amplitudes. The height of a wave is twice its amplitude.

There are various proportions which seem to fit most observed waves, like wave height = wavelength / 20, or wave height = 1.1(wavelength)^1/2, but these are rough approximations. When water waves exceed a certain slope, their heights cannot increase further, and the wave will break. This phenomenon occurs when a deep water wave hits shallow water.

To get a better feel for wave spectra, this link is helpful:

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_01.htm

Beware though ... it's going to take more than a few Khan Academy videos to understand it all.