# Other equations

Math behind the simulations

### Motion equations

The vector of phonon motion is given by:

where θ is the angle between the projection to *x-y* plane and *y*-axis, and φ is the angle to the horizontal plane. Thus, the phonon moves step-by-step with the speed *v* in the assigned direction according to the following equations:

where *dt* is the duration of one time step.

### Angles to the walls

At each collision with walls, we calculate the angle between the phonon motion vector and the normal vectors to the walls of different kinds:

**Horizontal wall**

**Vertical wall**

**Inclined wall of the triangle**

For triangle facing up

For triangle facing down

where β is the half-angle of the tip of the triangle.

**Wall of the circular hole**

where

is the normal to the surface of the circle, with *x0* and *y0* as circle center coordinates.

### Specular scattering probability

Then, specular scattering probability, determined by Soffer's equation:

where *p* is the specularity probability (number between zero and one), σ is the surface roughness, α is the angle to the surface, and λ is the phonon wavelength.

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