**Figure 1.**
Oscillating surfaces filled with pseudo-gas from the abrasive granules.

**Figure 1.**
Oscillating surfaces filled with pseudo-gas from the abrasive granules.

**Figure 2.**
Collision of the moving granule with the surface of the processed part.

**Figure 2.**
Collision of the moving granule with the surface of the processed part.

**Figure 3.**
Change in the velocity component of the granule along the $X$ axis after its collision with the processed part ${F}_{fr}=N\left(t\right)f$.

**Figure 3.**
Change in the velocity component of the granule along the $X$ axis after its collision with the processed part ${F}_{fr}=N\left(t\right)f$.

**Figure 4.**
The results of the numerical solution of Equation (41).

**Figure 4.**
The results of the numerical solution of Equation (41).

**Figure 5.**
The decimal logarithm ${f}_{1}(d)$ (continuous curve) and the results of the numerical solution of Equation (41) (points).

**Figure 5.**
The decimal logarithm ${f}_{1}(d)$ (continuous curve) and the results of the numerical solution of Equation (41) (points).

**Figure 6.**
The function ${f}_{2}(d)$ (continuous curve) and the results of the numerical solution of Equation (41) (points).

**Figure 6.**
The function ${f}_{2}(d)$ (continuous curve) and the results of the numerical solution of Equation (41) (points).

**Figure 7.**
Horizontal and vertical component of the velocities of the pseudo-wave motion of the granules, the velocity modulus of the wave motion of the granules, the velocity of oscillatory movement of the granules under the action of an acoustic wave: (**a**) in the immediate vicinity of the reservoir working surface; and, (**b**) at a distance of 20 mm from the reservoir working surface.

**Figure 7.**
Horizontal and vertical component of the velocities of the pseudo-wave motion of the granules, the velocity modulus of the wave motion of the granules, the velocity of oscillatory movement of the granules under the action of an acoustic wave: (**a**) in the immediate vicinity of the reservoir working surface; and, (**b**) at a distance of 20 mm from the reservoir working surface.

**Figure 8.**
Dependence of the speed of granules movement $\langle {V}_{sda}\rangle $ on the distance to the reservoir working surface.

**Figure 8.**
Dependence of the speed of granules movement $\langle {V}_{sda}\rangle $ on the distance to the reservoir working surface.

**Figure 9.**
Behavior of function $F\left(k\right)$ as a function of the parameter $k=0.5\left(1+\beta \right)f$.

**Figure 9.**
Behavior of function $F\left(k\right)$ as a function of the parameter $k=0.5\left(1+\beta \right)f$.

**Figure 10.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude of ${A}_{r}$, mm and the oscillation frequency ${\omega}_{r}$, Hz of the reservoir working surface for different values of the vibration frequency of the processed part surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{d}$ = $0;16;34;52;70$ Hz); amplitude ${A}_{d}$ of the oscillation of the part surface 1 mm.

**Figure 10.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude of ${A}_{r}$, mm and the oscillation frequency ${\omega}_{r}$, Hz of the reservoir working surface for different values of the vibration frequency of the processed part surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{d}$ = $0;16;34;52;70$ Hz); amplitude ${A}_{d}$ of the oscillation of the part surface 1 mm.

**Figure 11.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude of ${A}_{r}$,mm and the oscillation frequency ${\omega}_{r}$, Hz of the reservoir working surface for different values of the amplitude of oscillations of the part surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{d}$ = $1;2;3;4;5$ mm); frequency of oscillations ${\omega}_{r}$ of the reservoir surface 16 Hz.

**Figure 11.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude of ${A}_{r}$,mm and the oscillation frequency ${\omega}_{r}$, Hz of the reservoir working surface for different values of the amplitude of oscillations of the part surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{d}$ = $1;2;3;4;5$ mm); frequency of oscillations ${\omega}_{r}$ of the reservoir surface 16 Hz.

**Figure 12.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude of ${A}_{d}$, mm and frequency ${\omega}_{d}$, Hz of oscillations of the processed part surface for various values of the amplitude of oscillations of the reservoir working surfaces: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{r}$ = $1;2;3;4;5$ mm); frequency ${\omega}_{r}$ of oscillations of the reservoir working surface 50 Hz.

**Figure 12.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude of ${A}_{d}$, mm and frequency ${\omega}_{d}$, Hz of oscillations of the processed part surface for various values of the amplitude of oscillations of the reservoir working surfaces: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{r}$ = $1;2;3;4;5$ mm); frequency ${\omega}_{r}$ of oscillations of the reservoir working surface 50 Hz.

**Figure 13.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude ${A}_{d}$, mm and frequency ${\omega}_{d}$, Hz of oscillations of the processed part surface for various values of the vibration frequency of oscillations of the reservoir working surfaces: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{r}$ = $0;16;34;52;70$ Hz); the amplitude ${A}_{r}$ of oscillation of the working reservoir surface is 1 mm.

**Figure 13.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitude ${A}_{d}$, mm and frequency ${\omega}_{d}$, Hz of oscillations of the processed part surface for various values of the vibration frequency of oscillations of the reservoir working surfaces: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{r}$ = $0;16;34;52;70$ Hz); the amplitude ${A}_{r}$ of oscillation of the working reservoir surface is 1 mm.

**Figure 14.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on frequencies of oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surface and the processed part for different values of the amplitude of the oscillations of the part: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{r}$ = $1;2;3;4;5$ mm); the amplitude ${A}_{r}$ of oscillation of the reservoir working surface is 1 mm.

**Figure 14.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on frequencies of oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surface and the processed part for different values of the amplitude of the oscillations of the part: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{r}$ = $1;2;3;4;5$ mm); the amplitude ${A}_{r}$ of oscillation of the reservoir working surface is 1 mm.

**Figure 15.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on frequencies oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surface and the processed part for different values of the amplitude of the oscillations of the reservoir working surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{d}$ = $1;2;3;4;5$ mm); the amplitude ${A}_{d}$ of oscillations of the part surface 1 mm.

**Figure 15.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on frequencies oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surface and the processed part for different values of the amplitude of the oscillations of the reservoir working surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${A}_{d}$ = $1;2;3;4;5$ mm); the amplitude ${A}_{d}$ of oscillations of the part surface 1 mm.

**Figure 16.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surface and the processed part for different values of frequencies of oscillations of the part surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{d}$ = $0;16;34;52;70$ Hz); the oscillation frequency ${\omega}_{r}$ of the reservoir working surface is 50 Hz.

**Figure 16.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surface and the processed part for different values of frequencies of oscillations of the part surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{d}$ = $0;16;34;52;70$ Hz); the oscillation frequency ${\omega}_{r}$ of the reservoir working surface is 50 Hz.

**Figure 17.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surface and the processed part for different values of frequencies of oscillations of the working reservoir surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{r}$ = $0;16;34;52;70$ Hz); the frequency ${\omega}_{d}$ of oscillations of the part surface 50 Hz.

**Figure 17.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surface and the processed part for different values of frequencies of oscillations of the working reservoir surface: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at ${\omega}_{r}$ = $0;16;34;52;70$ Hz); the frequency ${\omega}_{d}$ of oscillations of the part surface 50 Hz.

**Figure 18.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the frequencies of oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surfaces and the processed part for different values of the distance between them: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $Y$ = $40;80;120;160;200$ mm); the amplitudes of the oscillations of the reservoir surface ${A}_{r}$ and the processed part ${A}_{d}$ planes are 1 mm.

**Figure 18.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the frequencies of oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surfaces and the processed part for different values of the distance between them: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $Y$ = $40;80;120;160;200$ mm); the amplitudes of the oscillations of the reservoir surface ${A}_{r}$ and the processed part ${A}_{d}$ planes are 1 mm.

**Figure 19.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surfaces and the processed part for different values of the distance between them: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $Y$ = $40;80;120;160;200$ mm); the frequencies of oscillations of the reservoir ${\omega}_{r}$ working surfaces and the part ${\omega}_{d}$ surface 50 Hz.

**Figure 19.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surfaces and the processed part for different values of the distance between them: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $Y$ = $40;80;120;160;200$ mm); the frequencies of oscillations of the reservoir ${\omega}_{r}$ working surfaces and the part ${\omega}_{d}$ surface 50 Hz.

**Figure 20.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surfaces and the processed part for different values of diameters of the abrasive granules: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $d$ = $10;15;20;25;30$ mm); the frequencies of oscillations of the reservoir ${\omega}_{r}$ working surfaces and the part ${\omega}_{d}$ surface 50 Hz.

**Figure 20.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the amplitudes of oscillations ${A}_{r}$, mm and ${A}_{d}$, mm of the reservoir working surfaces and the processed part for different values of diameters of the abrasive granules: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $d$ = $10;15;20;25;30$ mm); the frequencies of oscillations of the reservoir ${\omega}_{r}$ working surfaces and the part ${\omega}_{d}$ surface 50 Hz.

**Figure 21.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the frequencies of oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surfaces and the processed part surface for different values of the diameters of the abrasive granules: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $d$ = $10;15;20;25;30$ mm); amplitudes of oscillations of the reservoir ${A}_{r}$ working surfaces and the part ${A}_{d}$ 1 mm.

**Figure 21.**
Dependence of the metal removal $Q$, mg/h·cm^{2} on the frequencies of oscillations ${\omega}_{r}$, Hz and ${\omega}_{d}$, Hz of the reservoir working surfaces and the processed part surface for different values of the diameters of the abrasive granules: (${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, ${Q}_{4}$, ${Q}_{5}$—metal removal at $d$ = $10;15;20;25;30$ mm); amplitudes of oscillations of the reservoir ${A}_{r}$ working surfaces and the part ${A}_{d}$ 1 mm.