Why Do Water Waves Behave Mathematically as They Do?

[nhmllr]

I don't really understand the math of these things, so maybe one of you could help me. :)
Answer any of these questions that you can.
1. Why is the energy deposited by a wave when it hits a wall at a position proportional to the height squared?
2. Are the waves sinusoidal, and if so what's the good mathematical reason for it?

For reference, I know some basic differential and integral calculus.

Thanks a ton

 

[espen180]

Do you know any wave theory? Do you know what the terms phase velocity, group velocity and dispersion mean?

 

[Acut]

1) Because the energy in a wave is proportional to the square of its "height" (better call it amplitude).

(There's a full analogy between waves and simple harmonic motion. Take a look at: http://en.wikipedia.org/wiki/Simple_harmonic_motion#Energy_of_simple_harmonic_motion )

2) They need not be sinusoidal, but any (sufficiently well behaved) periodic function can be described as a sum of sines and cosines. The mathematical reason behind this is called Fourier analysis. Take a look at http://en.wikipedia.org/wiki/Fourier_series .

 

[olivermsun]

1. Why is the energy deposited by a wave when it hits a wall at a position proportional to the height squared?

Beyond the links that have already been given, you may try to think of the wave intuitively in the following way:

For a wave of a given frequency to have a larger amplitude, the water has to move faster during each cycle? Kinetic energy goes as velocity squared.

2. Are the waves sinusoidal, and if so what's the good mathematical reason for it?

2. Water waves are often expressed mathematically as sums of sine waves (Fourier series). In practice they are generally not observed to be very sinusoidal in appearance (even though very long waves may be close).

 

[nhmllr]

Beyond the links that have already been given, you may try to think of the wave intuitively in the following way:

For a wave of a given frequency to have a larger amplitude, the water has to move faster during each cycle? Kinetic energy goes as velocity squared.

Hm... perhaps. But for simple trajectory motion, the maximum height acheived by the projectile is v²/2g (unless I did it wrong), so it would seem there that
h ~ v², and KE ~ v², so shouldn't h ~ KE?
(Although I realize that simple projectile motion might not describe water waves.)

 

[ZealScience]

I think that usually such waves can be modeled by harmonic motions. Practically between short periods, the pattern is quite close to simple harmonic motion (when the daping is quite small). In harmonic motions energies are proportional to amplitudes.

And sinusoidal patterns are usually the solutions to harmonic patterns. Possibly you could get that analyzing the force experienced by small objects at the surface of the water.

 

[olivermsun]

Hm... perhaps. But for simple trajectory motion, the maximum height acheived by the projectile is v²/2g (unless I did it wrong), so it would seem there that
h ~ v², and KE ~ v², so shouldn't h ~ KE?
(Although I realize that simple projectile motion might not describe water waves.)

No, it's a good question. As you said, PE ~ h, and PE and KE are conserved during the motion.

The difference between the projectile and the wave is that the projectile is an object with a certain mass m, so PE = mgh. The wave has a potential energy PE = ∫ ρgz dz, where the integral is over all the water "involved" in the wave. The limits of integration are from 0 to the height h of the wave, so PE = ρgh^2.

 

[nhmllr]

The wave has a potential energy PE = ∫ ρgz dz, where the integral is over all the water "involved" in the wave. The limits of integration are from 0 to the height h of the wave, so PE = ρgh^2.

Quick question- what does ρ represent?
Also, how did you obtain that integral, ∫ ρgz dz?

Thanks

EDIT: Ah, ρ is the period. Although I'm still confused about the integral.

 

[olivermsun]

Quick question- what does ρ represent?
Also, how did you obtain that integral, ∫ ρgz dz?

ρ is mass density, e.g., in kg / m^3.

Hence there is an analogy between mgh (potential energy for a particle) and∫ ρgz dz, which is just potential energy (per unit Area) for the surface wave.