Sound Wave in Two Solid Media

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. Then change to slightly less aggressive variables and click Start again.
The idea here is to have the wave propagate through a region (slab) of higher or lower bonding force so we can see the change in wavelength. This is exactly analogous to the change in wavelength of an electromagnetic wave when it propagates through a medium of different index of refraction. Since the wavelength is the wave speed divided by the wave frequency this will also show the change in wave speed. A wave packet is initiated by applying a sinusoidal strain with gaussian envelope to the solid. The strain distribution is then

`s(x)=cosk_X(x-x_(Start))exp(-((x-x_(Start))/(w_X))^2)`

where `k_X` is the wave vector, `x_(Start)` is the center of the packet and `w_X` is the gaussian width at `1/e^2` point. The equation for the wavelength, `lambda`, in the slab is

`lambda_(Slab)=lambda_0sqrt(G_(slab)/G_0)`

where `G` is the shear modulus and `lambda_0=(2pi)/(k_X)`. This initial strain causes the wave packet to oscillate. Since the sinusoid has no time dependent directionality, the packet breaks up into two packets, one moving to the right and the other to the left. The packet going left soon reflects from the left hand boundary and continues to move rightward at the same speed as the other right going packet. When the packets get to the blue colored slab region their wavelength changes depending on the ratio of the slab stiffness to the nominal stiffness of the rest of the domain. Strain amplitude and reflection of the strain at the slab boundry also change dependent on this ratio. In this simulation the bonding is done with a Lennard-Jones (LJ) potential. The LJ potential is of the form

`V(r)=4V_(LJ)[(sigma/r)^2b-(sigma/r)^b]`

where `V_(LJ)` is the potential minimum, `sigma` is a length scaling parameter, `r` is the particle separation, and `b=6` in this simulation. The larger the value of `b`, the more abrupt is the force change with distance, `r`. 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 Lennard Jones potential tends to take into account the Pauli Exclusion Principle which causes much stronger repulsion when charges (Fermions) are forced together. 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. An obvious result of that set up is that alternate rows of particles have to be staggered left or right by 1/2 the spacing in the horizontal direction and the vertical spacing of rows is `sqrt(3)/2` times the vertical spacing of particles in any column. The initial configuration is that of a perfectly regular crystal that has 6-fold symmetry and that is by far the most stable 2D configuration.. A lattice site position represents the location of the nucleus of the atoms. The initial displacements cause repulsive or expansive forces between discs and therefore oscillations occur when Start is pressed. In general, frequency and propagation speed are proportional to square root of the stiffness divided by mass. The plot below and parallel to the graphic shows the time variation of the strain as a function of position. The centroid of the strain intensity, strain^2, is shown by a black vertical line, For constant `V_()` one can show that the restoring force between particles for b=6 is stress=`60V_(LJ)"strain"` where strain is the fractional displacement of the particles from the zero force distance. Thus the Young's modulus, `E`, is about `60V_(LJ)`. The speed of the shear wave depends on the shear Modulus, `G=E/(2(1+nu))` where `nu` is the Poisson ratio,usually less than 0.5, Since the particle mass here is unity, the expected shear wave speed is approximately the square root of the shear modulus or

`v_(Wave)=sqrt(60*VLJ/(2(1+nu)))`

or about `sqrt(20V_(LJ))`.

See:Speed of Sound in Solids. You will find that the longitudinal (compression) wave tends to wash out more than the transverse (shear) wave. I think this is due to domain end effects. These result in a compression at the left end of the domain and an expansion at the right end and that is possibly due to the recoil force needed to start the wave packets. A red vertical line shows this estimated position of the intensity centroid and it agrees reasonably well with the black vertical line. As always the learner can examine the Javascript code by hitting F12 in Windows or Ctrl Option j on Mac then choose Sources.