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Is the 2-D elevation wave spectrum (as a function of wavenumber and direction, with units of $m^4$) always positive? If so, why would that be the case?

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Yes, wave variance or energy spectrum, direcional or non-directional is positive-definite as @aretxabaleta said in the comment.

In linear water-wave theory, the surface elevation is described as a linear superposition of sinusoids:

$$ \eta(t) = \sum_{i-1}^{N}a_i \sin(f_i t + \phi_i) $$

where $a_i$, $f_i$ and $\phi_i$ are the amplitude, frequency and phase, respectively, of each wave component $i$.

The most commonly used wave spectrum is the wave variance spectrum. Wave variance is:

$$ \langle\eta^2\rangle = \dfrac{1}{2N}\sum_{i=1}^{N}a_i^2 = \sigma^2 $$

and wave variance spectrum $F(f)$ is defined such that:

$$ F(f)\Delta{f}=\dfrac{a_i^2}{2} $$

In the limit of $N \rightarrow \infty$ (continuous spectrum), the following holds:

$$ \int_{0}^{\infty}F(f)df = \sigma^2 $$

Being quadratic, both wave variance (spectrum integral) and individual discrete spectrum components are positive-definite.

Note that so far we implied non-directional frequency spectrum, i.e. spectrum defined in frequency space. It can be also defined in wavenumber $k$ space, and the following holds:

$$ \int_{0}^{\infty}F(k)dk = \int_{0}^{\infty}F(f)df = \sigma^2 $$

$$ F(k)\Delta{k} = F(f)\Delta{f} $$

$$ F(k) = F(f)c_g $$

where $c_g$ is group velocity of an individual component.

The non-directional spectrum is simply an integral of directional spectrum over all directions:

$$ \int_{0}^{\infty}F(k)dk = \int_{0}^{2\pi}\int_{0}^{\infty}F(k,\theta)dkd\theta $$

Be careful about units here. All spectrum integrals must come up at $m^2$. Thus, $F(k)$ has units of $m^3$ and $F(k,\theta)$ has units of $m^3$ $rad^{-1}$.

If you are considering polar (spectral bins scaling with $k\theta$ instead of $\theta$) directional wavenumber spectrum such that:

$$ \int_{0}^{2\pi}\int_{0}^{\infty}F(k,\theta)k\ dk\ d\theta = \sigma^2 $$

then $F(k,\theta)$ has units of $m^4$ $rad^{-1}$.

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  • $\begingroup$ Thank you milancurcic for your detailed explanation. What exactly does "positive-definite" mean? You used that term a couple times. I have seen it used to describe matrices, but I don't think its definition in matrix theory is one you have in mind. $\endgroup$ – user4624937 Mar 19 '15 at 2:01
  • $\begingroup$ It simply means the function is always real and greater than zero, see en.wikipedia.org/wiki/Positive-definite_function. For example, any quantity at even power (2, 4, 6, etc.) is positive definite; in physics, mass and energy are positive-definite quantities; in meteorology and oceanography, humidity and salinity are positive-definite. $\endgroup$ – milancurcic Mar 19 '15 at 2:51
  • $\begingroup$ Okay, that makes sense - thank you. By the way, would you have any suggestions for a reference that has in-depth coverage of the wave spectra? The book chapter aretxabalet mentioned is good but it doesn't cover 2D spectrum, for instance. $\endgroup$ – user4624937 Mar 19 '15 at 3:57
  • $\begingroup$ The wave book by Ian Young is excellent but very expensive store.elsevier.com/product.jsp?isbn=9780080433172. The book by Komen et al. is both broad and thorough, but more accessible cambridge.org/us/academic/subjects/…. There are many papers on the specifics of measuring or modeling directional wave properties, but I am not aware of any general or introductory text. $\endgroup$ – milancurcic Mar 19 '15 at 19:30
  • $\begingroup$ Alright thanks. If you ever write a book on the subject let us know :) $\endgroup$ – user4624937 Mar 20 '15 at 0:46

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