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

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

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

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

<p>The most commonly used wave spectrum is the wave variance spectrum. Wave variance is:</p>

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

<p>and wave variance spectrum $F(f)$ is defined such that:</p>

<p>$$
F(f)\Delta{f}=\dfrac{a_i^2}{2} 
$$</p>

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

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

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

<p>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:</p>

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

<p>$$
F(k)\Delta{k} = F(f)\Delta{f}
$$</p>

<p>$$
F(k) = F(f)c_g
$$</p>

<p>where $c_g$ is group velocity of an individual component.</p>

<p>The non-directional spectrum is simply an integral of directional spectrum over all directions:</p>

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

<p>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 $m^3$ $rad^{-1}$.</p>

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

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

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

CC BY-SA 3.0
2015-03-19T02:01:53.070Z
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.
user4624937
CC BY-SA 3.0
2015-03-19T02:51:47.003Z
It simply means the function is always real and greater than zero, see http://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.
milancurcic
CC BY-SA 3.0
2015-03-19T03:57:30.027Z
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.
user4624937
CC BY-SA 3.0
2015-03-19T19:30:27.537Z
The wave book by Ian Young is excellent but very expensive http://store.elsevier.com/product.jsp?isbn=9780080433172. The book by Komen et al. is both broad and thorough, but more accessible http://www.cambridge.org/us/academic/subjects/earth-and-environmental-science/oceanography-and-marine-science/dynamics-and-modelling-ocean-waves. 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.
milancurcic
CC BY-SA 3.0
2015-03-20T00:46:48.617Z
Alright thanks. If you ever write a book on the subject let us know :)
user4624937
CC BY-SA 3.0
2015-03-13T16:45:56.507Z
2015-03-13T16:45:56.507Z
milancurcic
<p>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?</p>

CC BY-SA 3.0
2015-03-13T14:57:26.747Z
All spectra are always positive, by definition. It considers the energy (elevation squared) and thus it has to be positive. An introduction to wave spectra can be found in: http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_03.htm
arkaia
<waves><wave-modeling>
Is wave spectrum always positive and why?
user4624937
# Is wave spectrum always positive and why?

(Asked by: username_0 on 2015-03-13)

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?

**Tags:** waves, wave-modeling

> **Comments:**
> **username_1**: All spectra are always positive, by definition. It considers the energy (elevation squared) and thus it has to be positive. An introduction to wave spectra can be found in: http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_03.htm  

---

## 1 answers

### Answer by username_2 on 2015-03-13. Score: 9 | Accepted answer

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 $m^4$ $rad^{-1}$.

> **Comments:**
> **username_0**: Thank you username_2 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.  
> **username_2**: It simply means the function is always real and greater than zero, see http://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.  
> **username_0**: 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.  
> **username_2**: The wave book by Ian Young is excellent but very expensive http://store.elsevier.com/product.jsp?isbn=9780080433172. The book by Komen et al. is both broad and thorough, but more accessible http://www.cambridge.org/us/academic/subjects/earth-and-environmental-science/oceanography-and-marine-science/dynamics-and-modelling-ocean-waves. 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.  
> **username_0**: Alright thanks. If you ever write a book on the subject let us know :)  

---

