Gas Acoustic Wave with Spherical Bounds An acoustic wave in a gas causes the gas density (molecules per unit volume) to vary sinusoidally as it travels outward from the transducer that generated it. For this animation we use reflective boundary conditions (normal momentum is reversed) at the containing sphere radius so we will expect to have a standing rather than a traveling wave. Further since we have spherical boundary conditions, we expect that any wave amplitude would follow the spherical Bessel function envelope. The learner needs to understand that this animation has spherical boundaries so that the reflections at the transducer and at the outer boundary are NOT always at the bounds shown on the screen since some of the atom positions have a larger z (in/out of the screen) component. In this animation the motion of the spherical transducer, as it expands and contracts, is shown and this causes the density variation near the transducer. But the density wave is difficult to see so the best way that we can confirm that the density is varying is to compute the density versus time in a spherical shell that is located at a greater radius than the transducer and then compute the Fourier transform of this density temporal scan. In the present case it is sufficient and convenient to use the Cooley-Tukey fast fourier transform (FFT). The temporal scan density in the chosen spherical shell is plotted in black. The sum of all FFTs of the density scans, subsequent to getting a full width scan, is plotted in red. It is seen that, for radial scans close to the transducer, the FFT's main peak is at the same frequency as the transducer frequency. Shells further away from the transducer, either require a very large nunber of FFTs to point to the transducer frequency or never show it at all. This writer believes that this is a natural sound attenuation effect due to the diffusion of the density maxima toward the density minima. It should be mentioned that the zero frequency component of the FFT has been suppressed by removing the average of the density scan prior to taking the FFT. Otherwise the zero frequency component would dominate the FFT result by at least a factor of 10. What is the spatial wavelength for this animation? For this we first have to compute the wave's phase propagation speed, c. For the wave we have here where no heat is being added as it propagates this speed is given by: c=sqrt(gamma*P/rho). Here gamma is the ratio of heat capacities cP/cV, where cP is the heat capacity at constant pressure and cV is the heat capacity at constant volume. P is the gas pressure, and rho is its density. For a mono atomic gas the pressure can be expressed as 2/3(E/V) where E is the total translational kinetic energy, V is its volume, and gamma is 5/3. rho=M/V where M is the total mass of the gas. Therefore we can re-write c as: c=sqrt(10/9(E/M). The equation for the wavelength is c/f where f is the transducer frequency. In the program, the transducer frequency, fT, is expressed in cycles per time scan. The total time of the time scan is the number of bins in the time scan times the frame time of a single video update, usually 1/60 of a second. In the animation, the speed of particles is shown by the number of pixels the particle progresses per computer frame time. As an example, take the startup transducer frequency, fT=4, and startup atom energy 5. Then c=2.35 pixels per video frame time. The transducer frequency expressed in cycles per video frame time is f=fT/nBins where nBins at startup is 256. Therefore f=4/256=1/64. The resulting wavelength is then c/f=150.84 pixels while, the startup containing sphere radius is 200 pixels. Therefore we should have about 1 wavelength between the transducer and outer sphere and that is a reasonable condition for resonance at startup. For visualization, a measure of the expected wavelength is shown in green. It important to understand that the average kinetic energy of the atoms will increase cyclically as the animation progresses. This is due to the fact that the transducer radial speed is a signficant fraction of the average atom speed so the that number of atoms hitting the transducer as it moves outward is greater than the number as it moves inward. It is also important to understand that the visual picture shown of the atoms includes all of them, including those that are actually displaced inward or outward from the screen (i.e. those that have a non-zero value of z). If a learner wants to see the javascript code that is used here, and is using the Chrome browser in Windows, they only have to press F12 to see the Chrome debugger. Then select "Sources" from the menu and press F5 and they can select a soruce file to view. The file names are very descriptive of what is their function so it should not be hard to find the code segment they seek. The learner has access to many sliders to adjust the parameters of the animation. These include 1. Starting Atom Average Energy 2. Total Number of Atoms 3. Transducer Frequency 4. Tansducer Amplitude 5. Number of Density Bins in Time Scan 6. Outer Sphere Radius 7. Transducer Nominal Radius 8. Atom Radius 9. Radial Position of Center of Time Scan Bins 10. Radial Width of Time Scan Bin
xTime xTimeWidth E nAtoms fT amp nBins R rT rad Single Step