Speed of Sound in a 2D Numerically Modeled Solid Lattice

I have deliberately allowed slider variable ranges that can result in a break up of the solid. It that happens, the learner just needs to refresh the page, which in Windows means hitting the F5 key. In this simulation the bonding is done with a Lennard-Jones (LJ) potential. LJ potentials have extremely large repelling forces when the particles get too close and longer range attractive forces when the particles are farther apart. The LJ potential is of the form V(r)=4VLJ[(sigma/r)^2b-(sigma/r)^b] where VLJ is the potential minimum, sigma is a length scaling parameter, r is the particle separation, and b=6 here. The larger the value of b, the more abrupt is the force change with distance, r. LJ potentials are much more realistic as bonding means than are simple springs because close spacing between sites is strongly repelled by Pauli Exclusion Principle forces. The solid is initially set up so that all the particles are separated by a distance that causes the nearest neighbor forces to be zero. The force between a site and its second nearest neighbor is not initially zero and this effect can easily be seen when starting with no excited columns. An obvious result of that set up is that alternate rows of particles have to be staggered by 1/2 the spacing in the horizontal direction and the vertical spacing of rows is squareroot(3)/2 times the horizontal spacing of particles in any column. The initial configuration is that of a perfectly regular crystal. Some of the left side (learner adjustable) columns have non-zero initial kinetic energy and remaining columns have zero initial kinetic energy. The object of this animation is to compute longitudinal and transverse (shear) wave speeds, c. To induce a wave, the left column of discs is oscillated either vertically (transverse wave) or horizontally (longitudinal wave). The waves can easily be seen in the patterns of motion of the discs but are also amplified and plotted in blue. The speed of the waves can be obtained either from their wavelengths (lambda) since c=lambda*(transducer frequency) or from estimating the speed of the modes that are shown in blue. The node speed estimates require that we obtain the progress of a single node over one iteration step but that is complicated by the need to make sure it is the SAME node whose progress is being estimated. Therefore we have resorted to using the wavelength for speed estimates. The two methods that are used are the spacing between nodes and the Fourier Tranform of the x or y motion of the columns. With a longitudinal wave, the x motion waves of the columns often do not cross zero so node spacings are often not a reliable way of estimating wavelength. Therefore I take a Fast Fourier Transform (FFT) of the wave data since the first harmonic of the FFT frequency does not depend on wave zero crossings. The results of this transform are plotted with a horizontal scale of wave cycles per full width of the graphic. For the longitudinal wave, probably the best estimate of speed is to use that associated with the quote of the longest wavelength. Wave speeds are very important for quantum models of thermal properties of solids since the cutoff frequency for these models is the wave speed divided by the lattice constant or particle spacing. The animation is programmed to stop before too much reflection from the right hand end contaminates the wavelength data.