Improved integral formulae for supersonic reconstruction of the acoustic field

Figures & data

Figure 1. Setup for exterior radiation problem.

Figure 2. Half space radiation problem.

Figure 3. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the Dirichlet eigenvalue frequency 171.5 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Figure 4. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the Dirichlet eigenvalue frequency 446.69 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Table 1. Results from non-negative intensity calculation for Dirichlet eigenvalues.

Frequency (Hz)	$\mathrm{\Pi }$ (W)	${\mathrm{\Pi }}_s$ (W)	${\mathrm{\Pi }}_d$ (W)	${\mathrm{\Pi }}_c$ (W)
171.50	4966	2309.1 (53.5%)	4954.6 (0.23%)	4970.8 (0.09%)
245.30	10,647	10,644 (0.04%)	9832.1 (7.65%)	10,687 (0.37%)
314.63	20,885	3858.3 (81.53%)	20,981 (0.46%)	20,933 (0.23%)
381.47	30,679	30,651 (0.25%)	30,647 (0.10%)	30,662 (0.05%)
446.69	40,649	20,566 (49.04%)	40,983 (0.82%)	40,939 (0.71%)

Table 2. Results from non-negative intensity calculation for Neumann eigenvalues.

k	$\mathrm{\Pi }$ (W)	${\mathrm{\Pi }}_s$ (W)	${\mathrm{\Pi }}_d$ (W)	${\mathrm{\Pi }}_c$ (W)
113.63	82.736	82.972 (0.28%)	$24.081(70.9\%)$	$82.973(0.28\%)$
182.45	341.32	342.35 (0.30%)	$198.33(41.9\%)$	$342.5(0.35\%)$
245.30	825.16	812.14 (1.57%)	$529.54(35.8\%)$	$828.75(0.43\%)$
246.43	837.85	843.04 (0.62%)	$524.74(37.37\%)$	$841.78(0.47\%)$
308.25	1897.2	1905.9 (0.45%)	$1179.8(37.81\%)$	$1908.3(0.58\%)$

Figure 5. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the Neumann eigenvalue frequency 245.3 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Figure 6. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the Neumann eigenvalue frequency 308.25 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Figure 7. Plot of the multi-frequency acoustical response that results from the proposed point forces. The lower part of the pictures shows the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ view of the acoustical pressure measurement hologram at the surface ${\mathrm{\Gamma }}_0$ for different frequencies.

Figure 8. Elastic shell data. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the resonant frequency 1009 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Figure 9. Elastic shell data. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the resonant frequency 1346 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Figure 10. Elastic shell data. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the non-resonant frequency 1080 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Figure 11. Elastic shell data. Image of the non-negative intensity using the power operator over the spherical plane $\mathit{\theta }$, $\mathit{\varphi }$ at the non-resonant frequency 2160 Hz. (a) Exact intensity, non-negative intensity, (b) $I_s^{ss}$, (c) $I_d^{ss}$, and (d) $I_c^{ss}$.

Table 3. Results from proposed non-negative intensity calculation for the elastic sphere experiment.

k	$\mathrm{\Pi }$ (MW)	${\mathrm{\Pi }}_s$ (MW)	${\mathrm{\Pi }}_d$ (MW)	${\mathrm{\Pi }}_c$ (MW)
740.5	0.14	$0.14(0.62\%)$	$0.14(0.51\%)$	$0.14(0.58\%)$
775	7.16	$7.25(0.51\%)$	$7.26(1.40\%)$	$7.26(1.41\%)$
880	8.97	$9.11(1.59\%)$	$9.15(2.09\%)$	$9.15(2.08\%)$
1009	4.44	$4.53(2.08\%)$	$4.59(3.52\%)$	$4.59(3.49\%)$
1080	0.11	$0.11(0.26\%)$	$0.11(0.16\%)$	$0.11(0.27\%)$
1165	1.24	$1.31(1.41\%)$	$1.35(5.43\%)$	$1.35(5.38\%)$
1346	2.95	$3.04(3.01\%)$	$3.17(7.28\%)$	$3.17(7.22\%)$
1551	4.67	$4.83(3.32\%)$	$5.06(8.34\%)$	$5.06(8.27\%)$
1778	4.67	$4.87(4.21\%)$	$5.08(8.68\%)$	$5.08(8.64\%)$
2025	39.95	$41.80(4.62\%)$	$43.33(8.38\%)$	$43.30(8.38\%)$
2160	0.21	$0.21(1.70\%)$	$0.21(1.63\%)$	$0.21(1.93\%)$
2289	12.59	$13.28(5.47\%)$	$13.61(8.06\%)$	$13.61(8.04\%)$
2567	4.51	$4.77(5.92\%)$	$4.85(7.76\%)$	$4.85(7.75\%)$
2859	4.52	$4.84(6.98\%)$	$4.89(8.16\%)$	$4.89(8.13\%)$
3161	4.81	$5.17(7.50\%)$	$5.21(8.29\%)$	$5.21(8.30\%)$
3471	1.07	$1.16(8.70\%)$	$1.17(9.12\%)$	$1.17(9.11\%)$

Figure A1. Elastic Shell model. Interior and Exterior domain are homogeneous acoustic mediums and shell contains elastic properties.