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This animation considers the possibility of visualizing a sound wave in a hard sphere mono-atomic gas. It is well known that sound waves in gases are longitudinal temporal and spatial pressure variations and these are usually sinusoidal of the form:

`P(x,t)=P_0+deltaPcos(omegat-kx+phi_P)`

where `P_0` is the ambient pressure, the amplitude `deltaP` is a very small fraction of `P_0`, `omega` is the radian temporal frequency and `k` is the radian spatial frequency while `t` is time and `x` is spatial extent and `phi_P` is just a constant which adjusts the phase at x=0 and t=0. If there is no dispersion then an expression for `k` is `k=omega/c` where c is the speed of the sound wave.One of the goals of this animation is to determine the quantity `lambda=(2pi)/k=2pic/omega` which, for a given `omega`, effectively determines c.

The pressure fluctuations amplitudes are very small fractions of the average pressure so it is necessary to subtract off the mean pressure in order to see the very small pressure fluctuations. Even after this subtraction is done, we still have to consider the statistical variation, due to very limited numbers of atoms that can be handled by the computer, of the average pressure per longitudinal bin of the gas. For that reason, we have to resort to Fast Fourier Transform (FFT) in order to discern the true spatial frequency of the pressure variation. With the FFT we have the advantage that we can average the observed frequency content over very many piston cycles and thereby see the true pressure or density variation spectrum that is induced.

Top: Shows moving atoms (red) and moving piston (gray) for ideal sound wave.

Next Lower: Shows direct plot of the energy and density bin data as a function of position x as well as Fast Fourier transforms (FFTs) of energy (green) and density (red).

Next Lower: Shows moving atoms (red) and moving piston (gray) for the sound wave induced by the piston.

Bottom: A moving plot (red) of the density obtained by the time scan at the bin position given by the "x position of time scan" slider
as well as the accumulated FFT (green) of this time scan. The expected frequency of this time scan is marked by the green vertical line.

It should be noted that the peak of the expected frequency of the FFT of this scan washes out badly if the scan distance from the piston is any larger than about 100. This effect would have been expected for a spherical wave where the amplitude is expected to follow a 1/r^2 curve but not for this linear geometry.
The cause must be the inter-particle scattering, which is essentially diffusion of the density maxima.

The following parameters are learner adjustable:

1. Starting Energy of Each Atom (actually a disc since this is a 2D animation)

2. Total Number of Atoms

3. Piston angular Frequency (radians per second)

4. Piston Motion Amplitude

5. Number of Bins for the Energy or Density Distributions

6. Total Length of Cylinder

7. Number of Animation Frames for Bin Storage (without accumulation, bin data would be very noisy)

8. x Position of the Density Time Scan

9. Width of the Time Scan Bin