\begin{gathered} A_{N f}=\alpha_{N f}+i \beta_{N f}\\\\ A_{W f}=\alpha_{W f}+i \beta_{W f}\\\\ A_{V f}=\alpha_{V f}+i \beta_{V f} \end{gathered}
\begin{gathered} C_{i j_f}=\overline{A_{i f}} \cdot \overline{A_{j f}}=\alpha_{i f} \alpha_{j f}+\beta_{i f} \beta_{j f}\\\\ Q_{i j_f}=\overline{A_{i f}} \times \overline{A_{j f}}=\alpha_{i f} \beta_{j f}-\beta_{i f} \alpha_{j f} \end{gathered}
Fourier coefficient may be calculated for each frequency bin
\begin{gathered} a_1\left(f_i\right)=\frac{Q_{n v_f}}{\sqrt{\left(C_{n n_f}+C_{w w_f}\right) C_{v v_f}}} \\\\ a_2\left(f_i\right)=\frac{C_{n n_f}-C_{w w_f}}{C_{n n_f}+C_{w w_f}} \end{gathered}
\begin{gathered} b_1\left(f_i\right)=\frac{-Q_{w v_f}}{\sqrt{\left(C_{n n_f}+C_{w w_f}\right) C_{v v_f}}} \\\\ b_2\left(f_i\right)=\frac{-C_{n w_f}}{C_{n n_f}+C_{w w_f}} \end{gathered}
Wave Mean Direction- The mean direction from which the waves are coming or going as a function of frequency
\theta_0=\operatorname{atan}\left(\sum_i E\left(f_i\right) \cdot \frac{b_i\left(f_i\right)}{a_i\left(f_i\right)}\right)
\theta_1(f)=\operatorname{atan}\left(\frac{b_i\left(f_i\right)}{a_i\left(f_i\right)}\right)
Principal Wave Direction Spectrum- Equation of principal wave direction is similar to mean wave direction, but it is calculated from other directional Fourier series coefficients.
\theta_2(f)=0.5 \cdot \operatorname{atan}\left(\frac{b_2\left(f_i\right)}{a_2\left(f_i\right)}\right)
Significant Frequency- the frequency at which the harmonic components of the real waveform begin to drop off faster than 1/f.
\left(f_{p e a k}\right)
Wave Fourier Coefficients
\begin{gathered} a_1=\frac{Q_{n v}}{\sqrt{\left(C_{n n}+C_{w w}\right) C_{v v}}} \\\\ a_2=\frac{C_{n n}-C_{w w}}{C_{n n}+C_{w w}} \\\\ b_1=\frac{-Q_{w v}}{\sqrt{\left(C_{n n}+C_{w w}\right) C_{v v}}} \\\\ b_2=\frac{-C_{n w}}{C_{n n}+C_{w w}} \end{gathered}
D=\operatorname{atan}\left(\frac{b_1}{a_1}\right)=\operatorname{atan}\left(-\frac{Q_{w v}}{Q_{n v}}\right)
Wave Peak Period- The wave period associated with the most energetic waves in the total wave spectrum at a specific point
T=\frac{1}{f_{p e a k}}
Mean Spreading Angle- Angle at which even spreading is assumed across the entire frequency range
\theta_k=\operatorname{atan}\left(\frac{0.5 \cdot b_1^2\left(1+a_2\right)-a_1 b_1 b_2+0.5 \cdot a_1^2\left(1-a_2\right)}{a_1^2+b_1^2}\right)
\sigma=\sqrt{a_1^2+b_1^2}
\tau=\sqrt\frac{1-\sqrt{a_1^2+b_1^2}}{1+\sqrt{a_1^2+b_1^2}}=\sqrt{\frac{1-\sigma}{1+\sigma}}\