# Wave simulations

There exists many different wave phenomena that can be observed in the nature. From a whispering sound to earthquake waves, the mechanical waves involved can be studied from a unified framework. Thus, scalar waves are studied in fields like Acoustics, and moreover, if shear elasticity is taken into account, more exotic phenomena can be observed and studied in Elastodynamics, Geophysics or Ultrasonics. In this case, the polarization of the waves and the particular boundary conditions leads to different wave types.

## Longitudinal waves

Simulations of longitudinal waves can be exerted by means of Finite-Differences Time-Domain methods. Thus, accurate predictions of scalar acoustic waves can be performed showing particle displacements in the direction of the propagation direction. This waves can be observed in both solid elastic and fluid acoustic media.

The propagation speed (speed of sound) of the acoustic waves in fluids can be expressed as: \[c_l=\sqrt{\frac{\kappa _0} {\rho _0}}\] where \(\kappa _0\) is the fluid compresibility and \(\rho _0\) the media density. In the case of solids, the longitudinal wave speed is \[c_l=\sqrt{\frac{\kappa _0 +\frac{4}{3}\mu _0} {\rho _0}}\] where \(\mu _0\) is the shear modulus.

## Shear waves

On the other hand, the particle dynamics for shear waves is transversal to the propagation direction of the wave.

Shear waves can be horizontal and vertically polarized, and the propagation speed can be expressed as \[ c_s=\sqrt{\frac{\mu _0} {\rho _0}}\]

## Rayleigh waves

Combinations of transversal and longitudinal waves under certain boundary condition can create a variety of wave dynamics.

Thus, on a fluid/solid interphase Rayleigh superficial waves appear with a propagation speed of about 0.95 \(c_s\). Note the particle trace describes a counterclockwise ellipse in the sagittal plane of the propagation direction.

## Gravity waves

Instead, gravity waves in the surface of the deep see water describes a clockwise ellipse in the sagittal plane of the propagation direction.

## Love waves

Under certain conditions, such a layered media, other wave phenomena can be observed.

In the case of a sound speed inversion, as communly occurs in the mantle of the surface of the earth, horizontally polarized surface waves can be trapped at a free elastic boundary. As the vertically polarized surface Rayleigh waves, the divergence of the wave is \(\frac{1}{\sqrt{r}}\), so love waves can propagate long distances.

## Lamb waves

Lamb waves exist in an elastic finite thickness plate. Transversal waves are trapped between the two interphases of the plate (upper and lower), and the relation thickness and wavelength induces different modes, called Lamb waves. Two main families of modes are observed, the extensional or symmetric modes \(S_n\) and the flexural or antisymmetrical \(A_n\) modes.

In the case of Lamb waves, the propagation speed strongly depends on the excited mode, the propagation in these layered media is dispersive. In the high frequency limit the speed converges to the Rayleigh wave speed.