FFT wave simulation

The derivation of this formula 、FFT The principle implementation and the quotation of a picture on Zhihu come from this 【 Learning notes 】Unity be based on GPU FFT The realization of the ocean - Theory Chapter - You know (zhihu.com)

According to the study notes above ,FFT The implementation of has gone through three stages , Namely ：

        Generate spectrum --->IDFT（ inverse discrete Fourier transform ）, Get the offset map ---> normal 、 Foam texture , Rendering

The main process is as follows ： so , The first thing is to generate a high spectrum .

 

The first stage of generating spectrum ：

        The formula for generating the height spectrum is as follows ：

       

  so , To get a high spectrum , We need not only Gaussian random numbers , It also needs to be phillips spectrum （ Directional spectrum ）, In the formula k The arrow is the wavelength vector ,k Is the mode of the wavelength vector . Last , We can generate the corresponding x\z Displacement spectrum .

Two Gaussian random numbers are independent of each other and obey a mean of 0, The standard deviation is 1 The random number .

The following figure shows Gaussian random numbers and phillips The process of generating high frequency spectrum .

  The following is based on the height spectrum FFT The whole process of wave simulation .

  Use of the second stage IDFT：

utilize IDFT The process of generating an offset map from a spectrum diagram . among dx\dz\dy It's the offset graph ,displacement Is the offset texture synthesized by the offset map .

 FFT The derivation of

 

 

 

  According to the derivation, it is known that , We need butterfly operations , And sampling points （ Or wave number ） yes 2 Omega to an integer power . At the same time, it should be noted that , Butterfly operation will be input in sequence 、 Output in reverse order , Reverse input 、 Sequential output . So use Stockham Algorithm to achieve sequential input and output .

 

  The third stage is the generation of foam and sea surface rendering ：