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# Sound and Acoustics

### Related Links (Outside this Site)

Longitudinal and Transverse Wave Motion:  Animations by  Dr. Dan Russell.

Wikipedia :   Sound

## Sound and Acoustics

Ike (Yahoo! 2008-09-11)   Speed of Sound in the Atmosphere
At q°C,  the speed of sound (in m/s) is roughly  V = 331.5 + 0.607 q
If q decreases by 1°C when the altitude z increases by 150 m, how long does it take for an airplane at 9000 m to be heard at sea level at 30°C ?

As q = 30 - z/150, we have  V = 349.71 - (0.607/150) z

dz/dt   =   -V   =   (0.607/150) z - 349.71   =   z / t0 / z - V0
dt   =   dz / ( z / t0 - V0 )   =   d ( t0 Log | z / t0 - V0 | )
Dt   =

(2015-02-28)   Sound is  not  an isothermal phenomenon!
Newton's prediction of the speed of sound is  15.48%  too low.

In proposition 49 of Book II of his Principia (1687)  Sir Isaac Newton (1643-1727)  gave the first analytical determination of the speed of sound and noted a discrepancy of about  15%  between his theoretical result and the observed value.  He ventured several explanations but missed the  main  one,  which would be given by  Laplace (1749-1827)  more than a century later,  when the budding science of  thermodynamics  came to the rescue.

Newton was right when he equated the square of the celerity of sound to the ratio of pressure and density variations.  However, he was wrong when he obtained the value of that ratio from an  isothermal  equation of state.

Sound is a rapid disturbance to which a fluid  (or a solid)  must react quickly.  It doesn't do so in the same way it would handle slow perturbations where the local gradient of temperature would have enough time to vanish.

Speed of Sound  by  Kevin S. Brown  (MathPages).

(2006-12-01)   Speed of Sound in a Fluid
Sound is the propagation of a  reversible  pressure disturbance.

Common experience indicates that sound of a reasonable amplitude propagates very well in the air, without appreciable losses for all frequencies in the  audio  range.  However, a membrane vibrating at an extremely high frequency is essentially hit randomly by gas molecules and its mechanical energy is thus entirely dissipated thermally, instead of being transmitted as a coherent wave.

Sound attenuation in the atmosphere
Laplace and the Speed of Sound  by  Bernard S. Finn  (1964).

(2006-12-01)   Speed of sound in a Van der Waals fluid

Acoustic cut-off frequency of the Sun & solar magnetic activity cycle  (2011)  by  A. Jimenez, R.A. Garcia, P.L. Palle.

(2006-12-01)   Acoustic Cutoff in the Sun's Atmosphere:  5.3 MHz

Acoustic cut-off frequency of the Sun & solar magnetic activity cycle  (2011)  by  A. Jimenez, R.A. Garcia, P.L. Palle.

(2006-12-02)   Measuring sound in Decibels  (dB)
Physical measurements and perceived loudness.

Both sound and light can be measured either in abolute terms related to the abolute power (in watts) carried by a wave, or in "human" terms involving a simplified model of the human senses.  For light, the power "perceived" by the retina is measured in  lumens,  not watts.  For a pure color  (single light frequency)  the two are proportional.  The cofficient of proortionalitiy is called the mechanical equivalent of light and is equal to 638 lm/W  (683 lumens per watt)  at 540 THz, by definition of the  lumen.

In the case of sound, one measure of power is the so-called  sound intensity,  which is the physical power flowing through a unit surface.  The SI unit of physical sound intensity is the watt per square meter (W/m).  The amplitude of sound is defined in terms of pressure differences and measured in  pascals (Pa).

From a physical standpoint, sound is best described in terms of the actual "root means square" (RMS) flux of mechanical power` carried by a soundwave, irrespective of the ability or inability of the human ear to gauge it in terms of perceived  loudness.  If that viewpoint is adopted, the 0 dB level corresponds simply to a  sound intensity  of  10-12 W/m This definition is preferred by physicists for theoretical computations...  The RMS power carried by a complicared wave is simply the sum of the RMS powers carried by each of its sinusoidal harmonic constituents.  Two sine waves of equal amplitude carry the same power even if they hace different frequencies.

Another approach exists which may assign different loudnesses to two sinusoidal sinewaves of the same amplitude:  A "sound pressure level" (SPL) of 0 dB level is assigned to the barely audible signal corresponding to a peak-to-peak pressure swing of .02 mPa (20 micropascals) in dry air for a a  1 kHz  sine wave.  (For water, the reference pressure swing seems [?] to be 1 micropascal.)  Peak-to-peak measurement cannot be used directly, except for a sine wave.  The power of other signals is the sum of the powers of all their harmonic components  (sinusoidal component).

This latter viewoint yields a purely physical basis ("dB-C" or "dB-SPL") which assigns exactly the same loudness to equally large sinewaves regardless of frequency (the conversion between the two scales is simply an additive shift involving the logarithm of the ratio between the power flux and the square of the amplitude, as discussed below).  However, different frequencies are usually given a different "A-weighting" which takes roughly into account the frequency-dependent sensitivity of a "typical" human hear.  The resulting "A-weighted" scale of sound magnitude may be identified as "dB-A" (which can be pronounced "dB audio" or "dB acoustic") and bears no direct relation with the physical (RMS) scale when a wide acoustic spectrum is considered.

So, what exactly is the peak-to-peak pressure swing of a 1 kHz sinusoidal soundwave at 0 dB RMS?

Numericana :   Measuring sound in decibels

(2006-12-01)   A ceiling to the loudness of atmospheric "sound"...
Consider a 1 kHz  sine wave  of 0 atm minimum and 2 atm maximum.

By definition, a standard atmosphere (atm) is 101325 Pa.

Of course, a pressure disturbance whose amplitude is 2 atm peak-to-peak would not qualify as "sound".  It would not be small enough to be  reversible  and it's difficult to envision how it could take the shape of a symmetrical wave, let alone a sinusoidal one.  Even if such a large disturbance was sinusoidal to begin with, its 100% swing in relative pressure would entail a nonlinear propagation that would soon distort it beyond recognition.

A pressure swing down to zero could be expected in the most violent nuclear or volcanic explosions.  Realistic nonsinusoidal shapes could possibly carry slightly more power than a sinewave with a zero minimum.  However, the latter still gives a good estimate of the highest mechanical energy which the atmosphere of the Earth can carry away in a "soundlike" way, namely:

194 dB-SPL   =   184 dB-RMS ???

The corresponding computation which follows is a good excuse to examine the exact basis which serves to measure the loudness of sound...  Using the "sound pressure level" (SPL) standard, the sine wave described above has 10132500000 times the amplitude of a 0 dB sound.  It has thus a 200 dB (SPL) magnitude  (more precisely: 200.114 dB).

(2006-12-01)   Acoustical Limits
Sound waves have a limited amplitude and frequency range.

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