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Cyclical temperature change in a sound wave

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klugesmith

Wed Aug 30 2017, 02:20AM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1714

New thread here, instead of burdening Physikfan's microphone topic. Over there I wondered how much temperature "vibration" goes with the 124 dB sound pressure in a pistonphone for calibrating microphones.

Worked it out and came to share the answer. And to practice making it look easy. By good luck, the numbers are so nice that artificial calculating aids (even pencil and paper) can be skipped until the third line from bottom.

Let's start at 94 dB SPL, a popular level in microphone calibrators and microphone calibration standards.

The reference pressure (0 dB ) is 20 micropascals, so 100 dB is 2.0 Pa, and 94 dB is 1.0 Pa.

Took me a while to confirm that it generally means 1 pascal RMS. That eliminates a clutter term when converting pressures to things like sound power density ( in W/m^2 ). So our sound pressure peak value is 1.4 pascals. In air at standard pressure (101.325 kPa), the peak and valley pressures are 101326.4 and 101323.6 Pa. Each extreme is 14 parts per million away from the middle.

The next step involves what textbooks call "adiabatic expansion of ideal gases". Different gases, equally ideal, behave differently. Distinguished by a factor called gamma, explained very well by Julius Smith at CCRMA:

The value gamma=1.4 is typical for any diatomic gas. Monatomic inert gases, on the other hand, such as Helium, Neon, and Argon, have gamma approx 1.6. Carbon dioxide, which is triatomic, has a heat capacity ratio gamma=1.28. We see that more complex molecules have lower values because they can store heat in more degrees of freedom.

Back to our 94 dB sound. When air pressure is reduced by 14 ppm at constant temperature, its volume increases by 14 ppm, as taught by Boyle hundreds of years ago. But when a 14 ppm pressure reduction happens adiabatically, the volume increase is only 10 ppm, while the temperature goes down by 4 ppm. The values of gamma and 1-gamma for air play well in this example.

The adiabatic model is appropriate for sound waves. a) No time for heat conduction between warmed and cooled zones. b) the expansion performs mechanical work, accelerating the air mass. c) It's reversible. On the opposite slope of acoustic wave, the same air mass has its kinetic energy converted back to P-V-T changes, with the same gamma and 1-gamma ratios.

If ambient temperature is 300 K, that 8 ppm temperature excursion (peak to peak) is 2.4 millikelvins.

If we crank the sound up by 30 dB, the cyclical variations are larger by sqrt(1000). Pressure variation is 31.6 Pa (RMS). With respect to ambient, the P,V,T changes are 885, 632, and 253 ppm, peak to peak. That's 0.076 kelvins, or Â°C, at +124 dB.

Somewhere between 152 dB and 153 dB, the alternating temperature variations reach 1 degree above and below ambient, and the pressure extremes reach + and - 12 mbar. Comparable to the natural changes when an air mass moves up and down as it passes over geological hills and valleys.

I hope somebody else likes this stuff.

Finn Hammer

Wed Aug 30 2017, 08:34PM

Registered Member #205 Joined: Sat Feb 18 2006, 11:59AM
Location: SkÃ¸rping, Denmark
Posts: 741

Rick,

Does this mean you can calculate the sound pressure of a lightning bolt based on the temperature in the discharge channel?

Cheers, Finn Hammer

Uspring

Tue Nov 07 2017, 09:47AM

Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711

Nice calculation, klugesmith. I just remembered your post, when I was thinking on related issues in trying to understand TC arc dynamics. Here's the connection:

TC arcs in consecutive bursts often follow a similar paths. I believe this is due to the heat generated by the previous burst. That makes the air expand, making it thinner and thus lowering the voltage required for breakdown in the next burst. So the next arc will often follow the path of the previous one.

The dynamics of a single arc is also influenced by this. Initially the air won't reach plasma temperatures, when it grows. But it will become warm enough to lower breakdown voltages. A temperature rise of e.g. 300K to 600K will reduce the air density by a factor of 2 and thus also the breakdown voltage by a factor of 2.

TC arcs grow very fast, though, so the question arises, whether the heated air can expand fast enough, so that its density will actually be lowered. The inertia of the air surrounding the arc channel will keep the arc channel compressed to some extent. This must be the case, otherwise the arc would not cause a sound pressure wave causing the loud bang.

This compression effect does have 2 consequences: It will slow down the growth speed of the arc and it might cause the arc to branch, since the not yet expanded air will still have high density and therefore not lowered breakdown voltage. This might be the cause for the still not understood observation, that it takes slowly ramped up TCs in order to grow straight sparks.

I'd like to get a quantitative handle on the compression effect to see, whether it is valid. Sound waves follow the wave equation:

(1/r) * d/dr (r*dp/dr) = k*p

for some material constant k in cylindrical coordinates. I'd like to find an analytical solution for that. There's probably one, but I can't find it. Can anybody help out there?

Plasma

Tue Nov 07 2017, 11:01AM

Registered Member #61406 Joined: Thu Jan 05 2017, 11:31PM
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Like kludgesmith said it's related to k, but the only thing that effect arc is self incuce magnetic field, that makes the path more resent and, the k constant just makes the sound waves propagation different, but it gives rise to a faster velocity, because it is squared to k, for example a hydrogen powered rocket can make 1200isp the same mass off hydrogen burning oxygen can reach 400 tops

Uspring

Wed Nov 08 2017, 09:12AM

Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711

I believe magnetic fields in an TC arc are too weak to have much effect in comparison with the electric fields. The only issue, that comes to my mind is the skin effect. But since the arcs are quite resistive, skin depth is large in comparison to the arc channel and therefore not important.

Wolfram Alpha helped with the diff equation. Sadly the solution involves Bessel functions. I've instead looked at the simpler planar waves solution for a ballpark estimate. In a sound wave, the pressure is linearly related to the speed of the air movement of the wave. When that speed approaches the speed of sound, the pressure reaches atmospheric pressure.

A TC arc radius is maybe 2mm, and that being reached in about 50us. The speed of expansion is thus somewhat lower than the speed of sound. This implies a mostly unhindered expansion of the hot channel air. But the values are not too far apart.

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