# List of variables

The environmental variables used throughout the website are listed below. The definitions include mathematical formulations that are likely to scare users that don’t possess a high scientific education. If you don’t understand the equations, don’t panic! Just skip them. You should be able, anyway, to understand the main and most important concepts.

#### Fetch: $X$

The fetch is the distance over which waves propagate under the wind forcing. It is measured in km.

#### Iribarren number: $\xi_0$

The Iribarren number is the ratio between the beach slope $\beta$ and the square root of the wave steepness. It is a dimensionless parameter.$$\xi_0 = \frac{\beta}{\sqrt{H_s/L_0}},$$where $L_0$ is the deep water wave length.

#### Mean directional spread: $\sigma_{\theta}$

The mean directional spread is the distribution of wave energy with direction. The smaller the directional spread, the larger the amount of wave energy concentrated around the mean wave direction. It is measured in degrees.
$$\sigma_{\theta}= \Big\{ 2 \Big[ 1- \Big( \frac{a^2+b^2}{E^2} \Big) ^{1/2} \Big] \Big\} ^{1/2},$$where $E=\int \int S(f,\theta) \, df \, d\theta.$

#### Mean wave direction: $\theta_m$

The mean wave direction is the direction from which wave energy is coming. It is measured in degrees from North.
$$\theta_m = \mathrm{atan} \Big( \frac{b}{a} \Big),$$$$a=\int \int\mathrm{cos}(\theta)S(f,\theta) \, df \, d\theta,$$$$b=\int \int\mathrm{sin}(\theta)S(f,\theta) \, df \, d\theta.$$

#### Mean wave period: $T_m$

The mean wave period is the weighted average of the periods of the wave components that form the spectrum. The weighting factor is the energy of the wave components. It is measured in seconds.
$$T_m=\frac{\int \int S(f,\theta) \, df \, d\theta}{\int \int f S(f,\theta) \, df \, d\theta}.$$

#### Peak wave period: $T_p$

The peak wave period is the wave period of the most energetic wave component. It is measured in seconds.

#### Significant wave height: $H_s$

The significant wave height corresponds to the average of the largest third of the recorded wave heights. It is measured in m. $$H_s=\frac{1}{N/3}\sum_{n=1}^{N/3}H_n,$$where $N$ is the size of the set in which waves are sorted from the largest wave,$H_1$, to the smallest wave $H_N$

#### Wave spectrum: $S(f,\theta)$

The wave spectrum defines the distribution of wave energy with respect to frequency and direction. It has been introduced to describe a real, irregular sea state made up of a large number of individual wave components. It is measured in $\mathrm{m}^2/\mathrm{Hz}$ $$S(f,\theta) \, df \, d \theta=\frac{1}{2}A^2(f,\theta),$$ where $A(f,\theta)$ is the amplitude of the wave component with frequency $f$ and direction $\theta$.