6. Surface Waves on Water: Here we explore another situation, closely related to

Question

6. Surface Waves on Water: Here we explore another situation, closely related to blackbody radiation in cavities or phonons in solids, where the low-energy excitation of a system can be treated as harmonically oscillating modes, or quantum mechanically, as non-conserved, non-interacting bosons Consider waves on the surface of a fluid, like (non-breaking) waves in the ocean, or ripples in your cup of coffee. For sufficiently large wavelengths, the restoring force will be dominated by gravity, while for sufficiently small wavelengths and/or thin films of liquid, the restoring force is primarily due to surface tension. (n between, both effects will be important, and the dispersion relation is somewhat complicated

Explanation / Answer

Solution:-

Given

For shorter wavelength surface tension waves. the dispersion relation between the linear frequency

v = /2*pi()

And the wavelength = 2*pi()/k for small amplitude waves on the surface of a shallow fluid of mass desnity and surface tension is approximately

v^2 = 2*pi()* /^3

So first we have to convert to the circular frequency

= 2*pi()*v

and waveength number = k = 2*pi()/

First of all here we have to consider surface waves,

so k is only 2-dimensional.

Actually the oscillations are in the height of the fluid as these waves are horizontally propogates.

Because of above reason these waves are actually transverse waves. Also we know that there is actually only one possible polarization for any value of k.

Since we know that = 2* (spacing between the atoms)

And here the total area A = L^2

From above equations, expression for the surface energy E is:-

E =   (deltaA)   (2*Pi)   (1 to 2) I(,) d cos d dA

The low and high temperature limits 0 to maximum values.

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