Sensor selection. Inertial Measurement Unit.

Useful Wave Sensor Equation and Parameter Definitions

Purpose: To provide users with some of the commonly used equations for the Wave Sensor as well as provide other equations that may be helpful in implementing the sensor for the specific end-user application.
Last Updated: June 2020

\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)

Mean Wave Direction Spectrum- The mean distribution of wave energy in wave number and direction


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}

Wave Peak Direction- The wave direction at the frequency at which a wave energy spectrum reaches its maximum.

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)

Directional Width- indicates whether waves are coming from similar directions or a wide range of directions


Long Crestedness Parameter- measures the surface texture of the wave


For any additional questions, please email

Website maintenance has been scheduled for Sunday, April 2 from 7 am to 9 pm EDT.
The resource may be unavailable at this time. Please accept our apologies for any inconvenience.