2.2 WAVELENGTH, FREQUENCY, AND WAVE NUMBER

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There is a single frequency ( f ) associated with a simple harmonic wave or

sinusoidal wave. This frequency depends on the frequency of vibration of

Basics of Acoustics 13

FIGURE 2-1 Sound waves in materials.

Copyright © 2003 Marcel Dekker, Inc.

the source of sound and is independent of the material through which the

sound is transmitted for non-dissipative sound transmission. The period (_)

for a wave is defined as the time elapsed during one complete cycle for the

wave, or the time elapsed between the passage of the successive peaks for a

simple harmonic wave, as shown in Fig. 2-2. The frequency is the reciprocal

of the period, f ј 1=_. The unit for the frequency is hertz (Hz), named in

honor of the German physicist Heinrich Rudolph Hertz, who conducted

pioneering studies in electromagnetism and in elasticity (Timoshenko, 1983).

The unit hertz is the same as the unit cycle/sec.

To get a physical understanding of the magnitude of the frequency of

sound waves usually considered in noise control, we may note that the range

of audibility for the undamaged human ear is from about 16 Hz to about

16 kHz. Frequencies below about 16 Hz are considered infrasound, and frequencies

above 16 kHz are ultrasound. The standard musical pitch (frequency)

is A-440, or the note ‘‘A’’ above middle ‘‘C’’ has an assigned

frequency of 440 Hz. The soprano voice usually ranges from about middle

‘‘C’’ ( f ј 261:6 Hz) to approximately ‘‘C’’ above the staff ( f ј 1046:5 Hz).

Thus, the female voice has a frequency on the order of 500 Hz. The baritone

voice usually ranges from about 90 Hz to 370 Hz, so the male voice has a

frequency on the order of 200 Hz.

The wavelength (_) of the sound wave is an important parameter in

determining the behavior of sound waves. If we take a ‘‘picture’’ of the wave

at a particular instant in time, as shown in Fig. 2-2, the wavelength is the

distance between successive peaks of the wave. The wavelength and speed of

sound for a simple harmonic wave are related by:

_ ј c=f (2-5)

Another parameter that is encountered in analysis of sound waves is

the wave number (k), which is defined by:

k ј

2_

_ ј

2_f

c

(2-6)

Example 2-1. A sound wave having a frequency of 250 Hz is transmitted

through air at 258C (298.2Kor 778F). The gas constant for air is 287 J/kg-K,

and the specific heat ratio is _ ј 1:40. Determine the speed of sound, wavelength,

and wave number for this condition.

The speed of sound is found from Eq. (2-1):

c ј рgc_RTЮ1=2 ј Ѕр1Юр1:40Юр287Юр298:2Ю_1=2

c ј 346:1m=s ј 1136 ft=sec ј 774 mph

14 Chapter 2

Copyright © 2003 Marcel Dekker, Inc.

The wavelength for a frequency of 250 Hz is:

_ ј

c

f ј

346:1

250 ј 1:385m ј 4:543 ft

The wave number is:

k ј

2_

_ ј

2_

1:385 ј 4:538m_1

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