# Water Waves' velocity, wavelengh and amplitude

1. Oct 11, 2008

### tom_backton

I'm trying to make a simple water simulation for a platform game (some of you will probably think "why should I care what you need it for?" but the fact it's a platform game tells you what point of view the game has). When a body touches the surface of the water, it creates two waves, one for each direction (right & left), with a single wavelengh. I could easily make a simple formula to calculate the wave's properties, for instance Amplitude = body's vertical velocity multiplied by a constant. But I want to know how it works in the real world...The body has a known shape, velocity, mass, etc. what I would like to know is how the body's parameters affect the wave's velocity(v), amplitude(A) and wavelengh($$\lambda$$).

2. Oct 11, 2008

### arildno

Well, if we only look at a linearized, gravity-driven wave-train propagating over a flat bottom of height "h" , the phase velocity "c" of the wave is:
$$c=\sqrt{gh}\sqrt{\frac{tanh(kh)}{kh}}, k=\frac{2\pi}{\lambda}$$
wher "k" is the wave-number, and "tanh" is the hyperbolic tangent function.
Since the phase velocity generally depends on "k", we say that these waves are "dispersive", since an initial signal composed of wave components of different wave-lengths would become twisted and distorted due to the different component velocities we'd have.

In the "shallow water" limit, i.e kh<<1 (ratio height/wavelength tiny), this reduces to the non-dispersive relationship $$c=\sqrt{gh}$$
That is, in shallow water, an initial signal RETAINS its form as it propagates.

As for amplitude-effects:
In general, this is what we call a "non-linear" effect, which means that we can only see them if we retain the non-linear terms of the constitutive differential equations (typically, potential flow with a free surface is sufficient here, i.e, neglecting foremost the effects of friction (either from internal viscosity or interaction with, say, wind/water)/vorticity)

Typically, we will also see amplitude dispersion as well, along with wavenumber dispersion, and the more energetic high-amplitude waves travel faster than low-amplitude waves.

But, an interesting case is given by the so-called solitons, an isolated trough or peak, in which the dispersive effects cancels out, so that the initial signal, although composed of different wave-lengths, can propagate undistorted.

Further, effects of surface tension comes into play at tiny wave-lengths, the shortest travelling faster
(this is a "linear" effect)

They and other waves are, of course, subject to the forces of friction,which slowly "burn off" the energy contained in them.
But, the observable effect of friction is amplitude-reduction, rather than reduction of phase velocity.

Finally:
As for the intrusive body's parameter's effects on the initial wave signals' shapes, I wouldn't hazard to guess at all. I have a very strong suspicion that nobody would..

Last edited: Oct 11, 2008
3. Oct 11, 2008

### tom_backton

Thanks for the help! :)
I'll just let A and v be functions of the body's velocity...Trying the longer formula of c could be interesting, but in this game I'm using square waves instead of sine waves, so constant c will be good enough.

4. Oct 12, 2008

### arildno

Indeed.
If people start dissing you for not taking into account, say that you obviously had in mind a scenario fulfilling the shallow water assumption. Thatll shut'em up.

However:
Let "A" be a function of the body's velocity if you like, but as for the phase velocity c (your v), you should use the shallow water assumption, it is unphysical to assume that the wave some distance away from your object would traverse at a different speed than the square root of g*h.