# Acoustic pressure & particle velocity phase - plane vs spherical waves

1. Sep 13, 2014

### mikelotinga

Hi there.

I've been looking around for a decent physical explanation of the differences in the phase relationships between acoustic pressure and particle velocity in different types of waves.

Mathematical analyses abound, e.g. http://physics.stackexchange.com/questions/24091/why-is-there-a-90%CB%9A-phase-angle-between-particle-velocity-and-sound-pressure-in-s?newreg=bffb79157e614fa8ba8031efb96dde42 but I have failed to find an intuitive explanation.

In a plane wave, most sources seem to agree pressure and velocity is in phase (one source I have read asserts that in a plane wave the phase relationship is either fully in or fully out, see page 13 of http://www.microflown.com/files/media/library/books/microflown_ebook/ebook_2_sound_and_vibration.pdf under the heading 'The far field (plane waves)', and please comment if this is correct, and explain why). If acoustic pressure is the variation in total pressure from ambient, it must include static and dynamic components. It seems to me that, when static pressure, which is a measure of the wave potential energy, is maximum, particle velocity must be minimum (i.e. static pressure and particle velocity must be out of phase). If this is correct, does this mean that in a plane wave the dynamic component of pressure, which is a measure of the wave kinetic energy, is dominant, and this is why sound (total) pressure and velocity can be in phase?

What is happening when the phase relationship changes, i.e. in a spherical wave, in which the phase relationship is not in?

2. Sep 13, 2014

### elegysix

I wouldn't put too much trust in the ebook you linked too. The section where he covers this is really short, in a hand-waiving kind of way. However, I do agree with him that the far-field (plane waves) the ratio of the two phases should be constant. I can't think of a reason to assume that they have to be either in or out of phase though, so I wouldn't go that far.

But because plane waves' phases ratio is constant, that leaves us to conclude the changes in phase for the spherical wave have to be due to the geometry. As for technical details on proving this, it's beyond me.

3. Sep 22, 2014

### mikelotinga

Thankyou for your reply. I think that yes, the geometry is the reason. And solutions to the wave equation in 3-dimensions show that the velocity solution includes an r^(-2) factor whereas the pressure solution contains only r^(-1) factors (see Bies and Hansen, 2009 "Engineering Noise Control" for example). But they (and all the authors I've managed to read so far) do not offer the intuitive physical description in terms of energy and mechanics that I'm searching for. Hopefully someone can take this step...

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