Ocean wave spectrum

Remarks

Example: the process $\eta(t) = a \cos(\omega_0 t + \phi)$, with $a > 0$ and a phase $\phi$ uniform on $[0, 2\pi)$, is stationary with $R(\tau) = \tfrac12 a^2 \cos(\omega_0 \tau)$. Its spectrum is $S(\omega) = \tfrac12 a^2 \, \delta(\omega - \omega_0)$, a point mass, and the variance is $\tfrac12 a^2$, as for a single sinusoid. Here $R$ is not integrable, so the spectrum is a measure rather than a function; a real sea has a spread of frequencies.

Non-example: a sea that is growing under a rising wind has a variance $E[\eta(t)^2]$ that increases with $t$, so $\eta$ is not stationary and has no time-independent spectrum. In practice the spectrum is estimated over records of 20 to 30 minutes, during which the sea is close to stationary.

The moments $m_j = \int_0^\infty \omega^j S(\omega) \, d\omega$ give standard sea-state parameters, such as the significant wave height $H_s \approx 4 \sqrt{m_0}$. In linear wave theory each frequency belongs to a water wave whose wavenumber is fixed by the dispersion relation, $\omega^2 = g k$ in deep water, so $S$ also gives the wavenumber spectrum. The definition uses the Fourier transform of the autocovariance (the Wiener–Khinchin theorem).