#### 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 ().

**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

but since T = 1/f, we can write

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 = f or

where v = velocity of propagation, in centimetres per second

= 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) :

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 deg.C, then at temperature Tdeg. 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 (°C) |
Velocity of Sound in Air (m/sec) |

0 | 332.5 |

20 | 344 |

21 | 345 |

100 | 386 |

Table 1.3 Effect of variations in temp on velocity of sound in air. |

Medium |
Vel. Sound (m/sec) at 20°C |

Air | 343 |

H_{2}O |
1480 = 4 x Vsound in Air |

Quartz | 5486 |

Steel | 6096 |

Table 1.4 Effect of different mediums on Vsound at 20deg. C. |

** Wavelength Revisited**

We now have two expressions for the velocity of sound:

v = f and v = 20.1

therefore

f = 20.1 and = ( in metres)

Example. What is the wavelength of concert A (440Hz) at 20 deg. C?

Calculate what it is at 21 deg.C ( = 0.783m)

Here are some examples of wavelengths at 21 deg. C (294 deg. 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 V _{sound 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 (French Mathematician |
316 | Measured echoes over known distances. |

1660 | Borelli & Viviani (Italian) |
<<>>316 | Developed cannon boom technique. |

1708 | William Derham | 343 | Extended cannon boom technique of Borelli & Viviani. Time (cannon flash, at 20°C 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

where,

d = particle amplitude, in centimetres

u = particle velocity, in centimetres per second

f = frequency, in Hz.

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