Integral formulas for supersonic reconstruction of the acoustic field

Figures & data

Figure 1. Setup for exterior radiation problem.

Figure 2. Half space radiation problem.

Figure 3. Surface element decomposition for the numerical experiments. (a) Spherical surface, (b) cylinder with spherical end-caps, (c) cylindrical box and (d) half-space cylinder with spherical end-caps.

Figure 4. View of the exact intensity over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$. (a) Intensity at 60 Hz, (b) 140 Hz, (c) 240 Hz, and (d) 500 Hz.

Figure 5. View of the spherical supersonic intensity over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$. (a) Intensity at 60 Hz, (b) 140 Hz, (c) 240 Hz, and (d) 500 Hz.

Figure 6. View of the calculated non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$. (a) Intensity at 60 Hz, (b) 140 Hz, (c) 240 Hz and (d) 500 Hz.

Table 1. Results from proposed supersonic intensity calculation over sphere.

freq (Hz)	$\mathrm{\Pi }(\mathrm{\Gamma },\mathit{\phi })$ (W)	$\mathrm{\Pi }(\mathrm{\Gamma },{\mathit{\phi }}^{(s)})$ (W)	% Relative error
60	8.52493	8.52005	0.05725
100	54.00441	54.02904	0.04559
140	159.21217	159.54925	0.21172
200	442.50394	444.42274	0.43362
240	768.11477	770.60661	0.32441
400	5425.05307	5520.77097	1.76437
500	13,837.04431	14,155.09496	2.29854
600	26,777.86938	27,657.30870	3.28420

Figure 7. 3D View of exact intensity and the corresponding 2D view at 60 Hz.

Figure 8. View of the calculated non-negative intensity using the power operator over the cylindrical surface. (a) Intensity at 60 Hz, (b) 140 Hz, (c) 240 Hz, and (d) 500 Hz.

Figure 9. 3D View of exact intensity and the corresponding 2D view at 500 Hz.

Table 2. Results from proposed non-negative intensity calculation over cylinder with spherical end-caps.

Freq (Hz)	$\mathrm{\Pi }(\mathrm{\Gamma },\mathit{\phi })$ (W)	$\mathrm{\Pi }\left(\mathrm{\Gamma },{\mathit{\phi }}^{(s)}\right)$ (W)	% Relative error
60	10.84644	10.96979	1.13721
100	68.11358	68.19500	0.11953
140	201.52913	198.44621	1.52976
200	607.55509	612.58618	0.82809
240	1176.43025	1141.88138	2.93675
400	9942.59951	10,504.16598	5.64808
500	19,218.04750	19,815.71634	3.10994
600	23,986.62309	25,144.01334	4.82515

Table 3. Results from proposed supersonic intensity calculation over half-space cylinder.

freq (Hz)	$\mathrm{\Pi }(\mathrm{\Gamma },\mathit{\phi })$ (W)	$\mathrm{\Pi }\left(\mathrm{\Gamma },{\mathit{\phi }}^{(s)}\right)$ (W)	% Relative error
60	0.58364	0.74105	26.97096
100	9.06277	10.69753	18.03816
140	52.75487	54.14611	2.63718
200	284.66673	287.14063	0.86905
240	586.91535	594.58866	1.30740
400	2806.62606	2862.62272	1.99516
500	10,092.91937	10,388.46125	2.92821
600	28,504.57161	29,537.48015	3.62366

Figure 10. View of the calculated non-negative intensity using the power operator over the half cylindrical plane. (a) Intensity at 60 Hz, (b) 140 Hz, (c) 240 Hz, and (d) 500 Hz.