## Abstract

The integrated amplitude and phase distribution of the sound field propagating from sinusoidally vibrating objects in air have been recorded by time-averaged TV holography. The behavior of the sound field is observed in real time by use of dynamic phase modulation. The magnitude and direction of the field are found by acoustic phase stepping and image processing. The field is three dimensional, and recordings of several cross sections are necessary for a complete description, but in many cases valuable information can be obtained by two-dimensional projections. The technique has been used to study sound propagation and details in the sound emission from extended sources.

© 1994 Optical Society of America

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### Equations (10)

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(1)
$$\frac{n-1}{\rho}=\text{const}\phantom{\rule{0.1em}{0ex}}.\phantom{\rule{0.2em}{0ex}},$$
(2)
$$\Delta \phantom{\rule{0em}{0ex}}m=\frac{d}{\lambda}\phantom{\rule{0.1em}{0ex}}({n}_{0}-1)\phantom{\rule{0.1em}{0ex}}(\frac{{p}_{\upsilon}}{{p}_{0}}-1)\phantom{\rule{0.2em}{0ex}},$$
(3)
$${I}_{\mathit{\text{TA}}}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)\approx {{J}_{0}}^{2}\phantom{\rule{0.2em}{0ex}}[\phantom{\rule{0.1em}{0ex}}\frac{2\pi}{\lambda}g\phantom{\rule{0.1em}{0ex}}(\varphi ){a}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.2em}{0ex}}]\phantom{\rule{0.2em}{0ex}},$$
(4)
$${I}_{\mathit{\text{TA}}/\mathit{\text{PM}}}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)\approx {{J}_{0}}^{2}\phantom{\rule{0.1em}{0ex}}\left(\frac{2\pi}{\lambda}\phantom{\rule{0.1em}{0ex}}{\{\phantom{\rule{0.1em}{0ex}}{[g\phantom{\rule{0.1em}{0ex}}(\varphi )\phantom{\rule{0.1em}{0ex}}{a}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.1em}{0ex}}]}^{2}+{{a}_{r}}^{2}-2g\phantom{\rule{0.1em}{0ex}}(\varphi )\phantom{\rule{0.1em}{0ex}}{a}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.1em}{0ex}}{a}_{r}\times cos\phantom{\rule{0.1em}{0ex}}[{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)-{\phi}_{r}]\phantom{\rule{0.1em}{0ex}}\}}^{1/2}\right)\phantom{\rule{0.2em}{0ex}}.$$
(5)
$${I}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)={I}_{b}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)-k\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y){a}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)cos{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.2em}{0ex}},$$
(6)
$${I}_{90}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)={I}_{b}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)+k\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y){a}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)sin{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.2em}{0ex}},$$
(7)
$${I}_{180}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)={I}_{b}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)+k\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.1em}{0ex}}{a}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)cos{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.2em}{0ex}},$$
(8)
$${I}_{270}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)={I}_{b}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)-k\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.1em}{0ex}}{a}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)sin{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)\phantom{\rule{0.2em}{0ex}},$$
(9)
$${\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},\phantom{\rule{0.1em}{0ex}}y)=arctan\phantom{\rule{0.1em}{0ex}}\left[\phantom{\rule{0.1em}{0ex}}\frac{{I}_{90}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)-{I}_{270}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)}{{I}_{180}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)-{I}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)}\right]\phantom{\rule{0.2em}{0ex}}.$$
(10)
$${a}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)=\frac{{\{\phantom{\rule{0.1em}{0ex}}{[{I}_{180}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)-{I}_{0}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)]}^{2}+{[{I}_{90}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)-{I}_{270}\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)]}^{2}\}}^{1/2}}{2k\phantom{\rule{0.1em}{0ex}}(x,\phantom{\rule{0.1em}{0ex}}y)}\phantom{\rule{0.2em}{0ex}}.$$