Fig.1.1
Music, Physics and Engineering - Harry F. Olson.
This is a very old philosophical dilemma which relies on using the word "sound" for two different purposes. One use is as a description of a particular type of physical disturbance:
"Sound is an organised movement of molecules caused by a vibrating body in some medium - water, air, rock or whatever."[1]The other is as a description of a sensation:
"Sound is the auditory sensation produced through the ear by the alteration ... in pressure, particle displacement, or particle velocity which is propagated in an elastic medium."[2]Both these definitions are correct, they differ only in the first being a cause and the second being an effect.
The Nature of a Sound Wave
Sound originates when a body moves back and forth rapidly enough to send a
coursing wave through the medium in which it is vibrating.
Demonstration: A simple form of sound wave is produced by an explosion of a small balloon of compressed air. By bursting the balloon, potential energy (energy of position) is converted to kinetic energy (energy of motion).
Fig.1.1
Music, Physics and Engineering - Harry F. Olson.
Fig.1.2
(Olson)
This is a spherical wave. It consists of one condensation and one rarefaction. The wavefront is a continuous spherical surface in which the variations for all parts of the surface have the same phase.
Sound Generators
Any motion that is repeated is called periodic motion. Examples of periodic
motion are the moon's orbit of the earth, a beating heart, the whirring tail of
a frightened rattlesnake, the wings of a hummingbird, and the movement of the
valve on a revolving bicycle tire. A vibrating body in contact with the
atmosphere will produce sound waves. One simple example is a vibrating
piston.
Figure 1.3 (from Olsen)
Fig. 1.4 (Olson)
Basic Features of Periodic Motion
There are thus three parameters which characterise the basic features of any
periodic motion.
1. The period, T is the time required for one complete cycle.
2. The frequency, f is the number of cycles occurring in a given time period.
3. The Amplitude, A. The "extent" of the motion.
Thus the frequency is the inverse of the period (and vice versa), i.e.:
| Motion Period (sec) | Period (sec) | Frequency (cps) |
|---|---|---|
| earth's rotation | 24x60x60= 86400 | 0.0000115 |
| heart beat | 1 | 1 |
| hummingbird's wings | 0.0160 | 62.5 |
| concert A | 0.0022727 | 440 |
Frequency used to be expressed in cycles per second (cps). The unit Hertz (abbreviated Hz) is now used. 1 cycle/second = 1 cycle sec.-1 = 1 Hz.
Frequency of a Sound Wave
Most sound generators produce recurrent waves which are generally similar to
each other. These waves are propagated at a definite velocity. This velocity
depends on the medium of propagation.
One cycle of a sound wave in air, consists of one compression of the air together with the subsequent rarefaction that occurs. The air molecules are forced together (compression or compaction) and then subsequently (in accordance with the 2nd law of thermodynamics) they immediately begin returning to their equilibrium state. The equilibrium state of the air molecules is the state in which they were before the compression under observation occurred. (Always taking into account that other disturbances of the atmosphere may have been occurring simultaneously with this compression.) In doing so they acquire momentum and thus become compressed again and so on.
Definition of Frequency
Frequency is the number of complete waves or oscillations or cycles of a
periodic quantity occurring in unit time (usually 1 second).
Note the difference between Frequency and Pitch. Frequency is a measure of the rate of disturbance whilst pitch is what our heads do with this phenomenon. In defining frequency, note the fundamental reliance on the concept of time.
Wavelength of a Sound Wave
The wavelength of a sound wave is the distance the sound travels to complete
one cycle. The symbol used to denote wavelength is the Greek letter lambda
([[lambda]]).
Velocity of Propagation of a Sound Wave
The preceding examples have shown that a sound wave travels with a definite
finite velocity. The actual velocity depends on the medium through which the
wave is travelling. In fact the following can be observed to be true:
V light wave >> V radio wave< V sound wave >V water wave >> V earth's rotation
Since a wave advances a distance of one wavelength in a time interval of one period, it follows that the velocity of a wave is given by
v = distance advanced time it takes to advance = [[lambda]] T
but since T = 1/f, we can write
v = _[[lambda]]
T
= f[[lambda]]
Now we know that the frequency of a sound wave is the number of cycles that pass an observation point per second. Thus the velocity of propagation of a sound wave is its wavelength times its frequency.
v = [[lambda]]f or
where v = velocity of propagation, in centimetres per second
[[lambda]] = wavelength, in centimetres ()
f = frequency, in Hz.
The frequency of a wave is independent of the waves' medium, however, the wavelength will depend on the wave velocity in the medium through which it is travelling.
Effect of Temperature on Vsound in air
The frequency of a wave is determined by the frequency of the source, therefore the frequency is generally known or at least unchanged by the medium through which it is travelling.
The velocity of a sound wave travelling through air does not vary with the air pressure, but it does depend on the temperature of the air. The pressure of the air is caused by the velocity of the air molecules. The square of this velocity is proportional to the temperature in kelvins (i.e. in degrees measured with respect to absolute zero (0 deg.K = -273 deg.C) :
v[2 ][[proportional]] Tk
The velocity of a sound wave is thus proportional to the velocity of the molecules of the air through which the wave travels. Thus the velocity of a sound wave at a temperature T is
If v=344 m/s at 20 [o]C, then at temperature T[[ring]] kelvin
where TA is the absolute temperature and the constant 20.1 is determined from the basic properties of air. Tk= TC + 273deg., where TC is the temperature in Centigrade. Thus for Fahrenheit temperatures (TF), Tk = 273 + 5/9(TF - 32).
In a cold medium molecules move more slowly and this reduces the velocity at which sound is transmitted.
Temperature ( deg. C) Velocity of Sound in Air (m/sec) 0 332.5 20 344 21100 345386
Table 1.3 Effect of variations in temp on velocity of sound in air.
Medium Vsound (ms-1) at 20deg. C Air 343 H2O 1 480 = 4 Vsound in air QuartzSteel 5 4866 096
Table 1.4 Effect of different mediums on Vsound at 20[o] C.
Wavelength Revisited
We now have two expressions for the velocity of sound:
v = f[[lambda]] and v = 20.1
therefore
f[[lambda]] = 20.1
and [[lambda]] =
([[lambda]] in metres)
Example. What is the wavelength of concert A (440Hz) at 20[o]C?
Calculate what it is at 21[o]C ( = 0.783m)
Here are some examples of wavelengths at 21[o]C (294[o]K):
Note Frequency (Hz) Wavelength (metres)
C1 32.5 10.5
C4 261.6 1.32
C8 4186 0.0082
Extension Topic: History of Measuring Vsound in air
Vsound in air was first measured in about 1640 by the French mathematician Marin Mersenne by computing the time for echoes to return over a known distance to the sound source.
Mersenne's estimate: Vsound in air <=> 316 m/s
Date Investigator Estimated V (m/s) Method
1640 Marin Mersenne 316 Measured echoes
(French over known
mathematician) distance.
1660 Borelli & Viviani <<>>316
Developed cannon
(Italian). boom technique.
1708 William Derham 343 Extended cannon
boom technique of
Borelli &
Viviani. Time
(cannonflash, at
20oC explosion
heard). Included
averaged wind
effects.
Some Extension Topics
Pressure in a Sound Wave
It is variations of pressure which is what affects our ears - i.e. what our ears physically respond to. A sound wave consists of pressures above and below the normal undisturbed pressure in the gas. At a point in space:
Pinstantaneous = Ptotal instantaneous - Pstatic
i.e. the instantaneous sound pressure at a point is the total instantaneous pressure at that point minus the static pressure. Static pressure is the normal atmospheric pressure in the absence of sound.
The effective sound pressure ("sound pressure") at a point is the root mean square value of the instantaneous sound pressure over a complete cycle at that point. The unit is the dyne per square centimetre. The maximum variation of pressure above or below its normal value is sometimes called the pressure amplitude.
The sound pressure in a spherical sound wave falls off inversely as the distance from the sound source.
Particle Displacement and Particle Velocity in a Sound Wave
The passage of a sound wave passing through a gas medium produces a displacement of the particles or molecules of gas from their normal positions, i.e. their positions in the absence of the wave in question.
The particle displacement of the medium through which the sound waves of speech and music pass is a very small fraction of a millimetre. For e.g. in normal conversational speech at a distance of 3 metres from the speaker, the particle amplitude or displacement of the air is of the order of a 2 millionth of a centimetre.
The particle or molecule in the medium (for e.g. air) oscillates at the frequency of the sound wave. The velocity of such a particle or molecule which is being displaced is termed the particle velocity.
The relation between sound pressure and particle velocity is given by:
Psound = pcu
where Psound = sound pressure, in dynes per square centimetre.
p = density of air, in grams per square centimetre
c = velocity of sound, in centimetres per second
u = particle velocity, in centimetres per second
The amplitude or displacement of the particle from its position in the absence of a sound wave is given by
d = u
2[[pi]]f
where d = particle amplitude, in centimetres
u = particle velocity, in centimetres per second
f = frequency, in Hz.
[2] Music, Physics and Engineering, Harry F Olsen. Dover Publications, Inc. N.Y., 2nd ed. 1967. This book forms the basis of most of this chapter.