# Wave kinematics within HydroDyn

Dear forum,

I know that HydroDyn can output the incident wave elevation in several points besides the origin and I would like to reproduce this feature in a custom code i have for hydrodynamic calculations.
Can you provide me some information about that, like references to the source code or documents with theory backgrounds?

Alessandro

Hi Alessandro,

HydroDyn computes the first-order wave surface elevation by first calculating the Discrete Fourier Transform (DFT) of the wave elevation at (0,0), which includes the wave amplitudes (wave spectra), wave phases, and wave directions of the various frequency components. The time series at (0,0) can be found from the Inverse DFT (IDFT). The first-order wave elevation at points (X,Y) on the free surface other than (0,0) can be found by multiplying the DFT by

EXP( -jk(omega)( XCOS(beta) + YSIN(beta) )

before computing the IDFT, where:
j = imaginary number
k(omega) = frequency-dependent wave number
beta = wave direction

The wave number satisfies the dispersion relationship in finite water depth:

k(omega)*TANH( k(omega)*h ) = omega^2/g

where:
h = water depth
g = gravity

The approach, including extensions to second order, are presented in our OMAE 2014 paper: nrel.gov/docs/fy14osti/61161.pdf.

Best regards,