wecopttool.derivative2_mat

derivative2_mat(f1, nfreq, zero_freq=True)

Assemble the second derivative matrix that converts the state vector of a response to the state vector of its second derivative.

For a state $x$ consisting of the mean (DC) component followed by the real and imaginary components of the Fourier coefficients (excluding the imaginary component of the 2-point wave) as $x=[X0,Re(X1),Im(X1),...,Re(Xn)]$, the state of its second derivative is given as $(DD)x$, where $DD$ is the second derivative matrix.

The time matrix has size (nfreq*2, nfreq*2).

If zero_freq = False (not default), the mean (DC) component X0 is excluded, and the matrix/vector length is reduced by 1.

Parameters:

• f1 (float) – Fundamental frequency f1 [$Hz$].

• nfreq (int) – Number of frequencies.

• zero_freq (bool | None) – Whether the first frequency should be zero.

Return type:

ndarray