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The waves and the wind. Calculation of wave characteristics

Calculation of wave characteristics
Timur2015-11-02 10:56:50

The user gave us a request - , where asked to create a calculator "calculation of the wave height and intervals between waves (frequency)?".

Intuition suggests that there is some relationship between the force of the wind and waves. Since I don't know much about the wave theory I've got to study it.
The result of my study is a calculator below along with my thoughts on the subject. The calculator does not calculate, or more precisely, does not predict the height of the waves - is a separate issue which is reviewed here - The waves and the wind. Wave height statistical forecasting.

Wave performance calculationCreative Commons Attribution/Share-Alike License 3.0 (Unported)
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Theory

It is obvious enough that the waves on the sea can not be described by a single sine wave, as they are formed as a result of imposing a plurality of waves with different periods and phases. For example, look at the picture below, which shows the wave resulting from the imposition of three different sine waves.

"Wave disp" by Kraaiennest - Own work. Licensed under GFDL via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Wave_disp.gif#mediaviewer/File:Wave_disp.gif


Source: "Wave disp" by Kraaiennest - Own work. Licensed under GFDL via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Wave_disp.gif#mediaviewer/File:Wave_disp.gif

Therefore, for analysis of the sea state energy spectrum is usually built, that is energy units lotted on the Y-axis, and frequency on the X-axis, thus obtaining energy density - the amount of energy carried by waves with a corresponding frequency range. And, as it turned out, under the influence of the wind, the shape of the energy spectrum is changed, the stronger the wind, the more strongly peak in the spectrum expressed - waves of certain frequencies carrying the most energy. In the picture below have drawn its aproximate look as best I could.

The energy distribution of the frequency spectrum, depending on the wind strength

The frequencies, where a peak is observed are called dominant. Accordingly, you can make your life easier and calculate the characteristics of the waves only for the dominant frequency. The practice has shown that it will be enough to give a good approximation to the reality.

What concerns the characteristics of waves, the linear wave theory comes to help, namely, the calculation of gravitational waves in the linearized approximation. To make it clearer what I mean, let's give some definitions further from
wikipedia:

Capillary waves —the name of various waves generated at the interface between a liquid and a gas or liquid and liquid. The lower part of waves is called trough, higher— crest.

Gravity waves on the water — kind of waves on the surface of the liquid in which the force returning the deformed surface of the liquid to a state of balance are simply the force of gravity related to the height difference of crests and troughs in the gravitational field.

Wave dispersion — in the wave theory the difference in phase velocities of linear waves according to their frequency. That is, different wavelengths (respectively of different frequency) have different speed in an environment that has clearly demonstrated experience with the refraction of light in a prism. This is important for further discussion

Wavenumber — is the 2π radians to the wavelength ratio: k \equiv \frac{2\pi}{\lambda}.Wavenumber can be represented as the difference in wave phase (in radians) at the same time in the spatial points at a distance per unit length (one meter) or a number of spatial periods (crest) of waves arriving at 2π meters.

Using the definition of the wave number we can write the following formula:

Wavelength
\lambda=\frac{2\pi}{k}

Phase velocity (crest velocity )
c=\frac{\omega}{k}

Wave period(expressed in terms of angular frequency)
T=\frac{2\pi}{\omega}

Picture to draw attention - кred dot shows the phase velocity, green - the group velocity (the velocity of the wave packet)

"Wave group" by Kraaiennest - Own work. Licensed under GFDL via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Wave_group.gif#mediaviewer/File:Wave_group.gif


Source: "Wave group" by Kraaiennest - Own work. Licensed under GFDL via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Wave_group.gif#mediaviewer/File:Wave_group.gif

The dispersion law

The key element in the calculation of the waves characteristics is the concept of the dispersion law or dispersion relation (ratio)

The dispersion law or dispersion equation(ratio) in the wave theory - is the relationship of frequency and wave vector (wave number).

In general terms, this relationship is written as
\omega=f(k).

This ratio of water derived in the linear wave theory for the so-called free surface, ie the surface of the liquid, not limited to the walls of the vessel or bed, and looks as follows:
\omega^2=gk \tanh(kh),
where
g - acceleration of free fall,
k - wave number,
tanh - hyperbolic tangent,
h - distance from the liquid surface to the bottom.

It is possible to further simplify the formula, based on the graph of the hyperbolic tangent. Note that for kh, tends to zero, the hyperbolic tangent can be approximated by its argument, ie, kh value, and if kh, tends to infinity, the hyperbolic tangent kh tends to one. The latter case, obviously, relates to a very great depth. Is it possible to evaluate how they need to be big? If you take the hyperbolic tangent of pi, then its value is approximately equal to 0.9964, which is already quite close to one (the number Pi is taken for the convenience of the formula). Then
kh\geq\pi \Rightarrow \frac{2 \pi h}{\lambda}\geq\pi \Rightarrow h\geq\frac{\lambda}{2}.
That is, to calculate the characteristics of the wave can water be considered as deep if the depth is greater than at least half the wavelength, and in most places of the world ocean, this condition is met.

In general, based on the graph of a hyperbolic tangent, the following classification of waves for the relative depth is used (ratio of depth to wave length).

1. Waves in deep water
The depth is more than half of wavelength, the hyperbolic tangent approximates by one
h\geq\frac{\lambda}{2}, \tanh(kd)\approx1

2. Waves on the transitional depths
The depth from one-twentieth to one-half wavelength, the hyperbolic tangent can not be approximated
\frac{\lambda}{20} \leq h \leq \frac{\lambda}{2}, \tanh(kd)=\tanh(kd)

3. Waves in shallow water
The depth less than one-twentieth of the wavelength, the hyperbolic tangent approximated by its argument
h\leq\frac{\lambda}{20}, \tanh(kd)=kd

Consider the ratio for these cases

The case of shallow water

The equation takes the form
\omega^2=gk(kh)=ghk^2,
whence
c=\sqrt{gh}
\lambda=T\sqrt{gh}

The group velocity for the case of shallow water
c_g=c=\sqrt{gh}

That is, according to the theory, in shallow water, the wave does not have to have the dispersion because the phase velocity is independent of frequency. However, we must remember that in shallow water nonlinear effects are beginning to work associated with the increase in wave amplitude. Nonlinear effects impact, when the amplitude of the wave is comparable to its length. One of the characteristic effects in this mode is the appearance of fractures on the wavestops. In addition, there is a posibility of wave breaking - a well-known surf. These effects are not yet amenable to precise analytic calculation.

Transitional depths case

The equation is not simplified, and then
c=\frac{gT}{2\pi}\tanh(\frac{2\pi d}{\lambda})
\lambda=\frac{gT^2}{2\pi}\tanh(\frac{2\pi d}{\lambda})

The group velocity for transitional depths
c_g=\frac{1}{2}(1+\frac{4 \pi \frac{d}{\lambda}}{\sinh(4 \pi \frac{d}{\lambda})})c

Note that the equation of the wavelength is transcendental and find its solution should be found numerically. For example, usingFixed-point iteration method

Deep water case

The equation takes the form
\omega^2=gk,
whence
c=\frac{gT}{2\pi}
\lambda=\frac{gT^2}{2\pi}

The group velocity in the deep water case
c_g=\frac{1}{2}c=\frac{gT}{4\pi}

So, by measuring the period of a wave, with sufficient accuracy, we can calculate the phase velocity, group velocity, and wavelength.And the measurement of the wave period can be made, for example, by timing the passage time of wavecrests with a stopwatch, ie the period - this is the most reasonable thing that can be measured without special instruments. If you are somewhere near the coast - it is necessary to imagine the depth, if the depth obviously big, you can use formulas for the deep water, in which the depth, as a parameter is not included. Since we have a computer processing power at hand, the calculator does not use the simplified formula, finding wavelength by iteration method (method will converge, as the derivative of the function is less than one).

Now, back to the wind. Actually, constantly blowing in one direction wind is what generates the wave, that reports energy to wave.
And quite obvious, in order to report energy to waves, the wind should blow faster, or at least at a rate equal to the phase velocity of the wave.

Here comes fully developed sea formulation). Fully developed sea - wave, which achieved maximal values at a given wind. That is, the wave is in equilibrium with respect to energy - as much energy is reported, as much goes into the motion. Not every wave reaches such a state, as it's required for the wind to constantly blow over the entire surface, which the wave passes for some time. And the stronger the wind, the more time and distance required for the formation of such a wave. But certainly when it's formed, it will catch up with the phase velocity of the wind speed.

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