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 | )
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
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 isothermalequation 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 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.
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
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/m2 ).
The amplitude of sound is defined in terms of pressure differences and measured in
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/m2 .
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?
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).
Acoustical Limits Sound waves have a limited amplitude and frequency range.