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Usually in acoustics, it is assumed that what we hear is the acoustic pressure. In far field - when the distance to the source is larger than the size of the source and than the wavelength - one has $p = \rho c v$ with $p$ the acoustic pressure, $\rho$ the density, $c$ the sound velocity and $v$ the speed of the air. So a good measure of the acoustic power intensity is $I = \overline{pv}$ and the acoustic power is $P = \int_{source} dS \ I(\mathbf{r})$.

Now, let us consider that a thin plate - like a sound board - is vibrating and radiating sound. Close to this plate, things are different.

  • "Active control of vibration", Fuller, Christopher C and Elliott, Sharon and Nelson, Philip A

Although the sound power radiated by the panel usefully quantifies the far-field pressure it generates, high levels of vibration in weakly radiated modes can give rise to signifi- cant pressure levels in the near field of the panel. It has been shown that the total kinetic energy of a panel provides a better measure of near-field pressure than radiated sound power, (11) and so if there is any possibility that listeners may be in close proximity to the panel, as well as being further away, then both of these criteria are important for active structural acoustic control.

  • "Active vibroacoustic control with multiple local feedback loops", Stephen J Elliott, Paolo Gardonio, and Michael J. Brennan

Reference (11) in the above excerpt refers to

  • M.E. Johnson and S.J. Elliott, ‘‘Active control of sound radiation using volume velocity cancellation,’’ J. Acoust. Soc. Am. 98, 2174 –2186, 1995.

The authors of this paper consider kinetic energy to be a good indicator, but they do not detail why.

Indeed, in near field - within a few wavelength to the panel - one has not $p(\mathbf{r}) \propto v(\mathbf{r})$ but

  • $v(\mathbf{r}) = \dot w(\mathbf{r})$ where $w$ is the tangential displacement of the plate

  • $p(\mathbf{r}) = \int_{plate} dS \ \frac{j \omega \dot{w} (\mathbf{r_s})e^{-j k |\mathbf{r - r_s}|}}{2 \pi |\mathbf{r - r_s}|}$

I think that in the near field, $\int_{plate} p^2$ must resemble to the kinetic energy $\int_{plate} v^2$, but I cannot figure out why.

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  • $\begingroup$ I am a little confused by a few things. You state that $P \propto v$ so would it not follow that $P^{2} \propto v^{2}$? I am also confused about what you call "near field." Is not the near field the region where you are within a few wavelengths of the source, i.e., near this panel (by the way, what are these panels to which you refer)? $\endgroup$ – honeste_vivere Jul 3 '18 at 13:26
  • $\begingroup$ "... one has not p ~ v". This is the issue. However I am also confused in that it seems that the OP convolves plate motion with fluid motion. At the surface a BC may constrain the movement but even within a few wavelengths away the full system of fluid equations (or the linearized wave equation) should hold. I think too many disparate sources are being referenced, perhaps out of context. $\endgroup$ – ggcg Jul 4 '18 at 11:49
  • $\begingroup$ Yes I don't really know until which distance the fluid speed can be considered the same as the plate's speed. What I really want to understand is the quote. I put the other sources to explain the problem and what I tried to do. $\endgroup$ – Cabirto Jul 4 '18 at 12:24
  • $\begingroup$ @Cabirto, which quote are you focused on? The one with a pink-ish background? $\endgroup$ – ggcg Jul 5 '18 at 22:49
  • $\begingroup$ Yes, I am focused on the quote in pink-ish background. $\endgroup$ – Cabirto Jul 6 '18 at 8:07

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