Other equations

Math behind the simulations

Motion equations

The vector of phonon motion is given by:

$$x = sin(\theta)·abs(cos(\phi)) \\y = cos(\theta)·abs(cos(\phi))\\\\z = sin(\phi)$$

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:

$$\Delta x = sin(\theta)·abs(cos(\phi))·v·dt \\ \Delta y = cos(\theta)·abs(cos(\phi))·v·dt \\ \Delta z = sin(\phi)·v·dt$$

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

$$\alpha = arccos(cos(\phi)·cos(\theta))$$

Vertical wall

$$\alpha = arccos(cos(\phi)·sin(abs(\theta)))$$

Inclined wall of the triangle

For triangle facing up

$$\alpha = arccos(cos(\phi)·cos(\pi/2-|\theta|+\beta))$$

For triangle facing down

$$\alpha = arccos(cos(\phi)·cos(\pi/2-|\theta|-\beta))$$

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

Wall of the circular hole

$$\alpha = arccos(cos(\phi)·cos(\theta+sign(y-y_0)·\theta_{tangent}))$$

where

$$\theta_{tangent} = arctan((x-x_0)/(y-y_0))$$

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:

$$p = exp(-16 \pi ^2 \sigma^2 cos^2(\alpha) / \lambda ^2)$$

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.