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the absolute difference in frequency of two interfering waves

a b with slightly different frequencies, or
f b = f 2 – f 1
a + b equation 5.6

5.5 ENERGY OF WAVES
A b a
All waves transport energy, including sound waves. Th e vibrat-
ing mass and spring in Figure 5.2 vibrate with an amplitude that
depends on how much work you did on the mass in moving it
from its equilibrium position. More work on the mass results
a b
in a greater displacement and a greater amplitude of vibration.
A vibrating object that is producing sound waves will produce
more intense condensations and rarefactions if it has a greater
a + b

amplitude. The intensity of a sound wave is a measure of the
energy the sound wave is carrying (Figure 5.18). Intensity is
defi ned as the power (in watts) transmitted by a wave to a unit
b a
area (in square meters) that is perpendicular to the waves. Inten-
B sity is therefore measured in watts per square meter (W/m ) or
2
FIGURE 5.16 (A) Constructive interference occurs when two _
power
intensity =
equal, in-phase waves meet. (B) Destructive interference occurs area
when two equal, out-of-phase waves meet. In both cases, the wave
P _
displacements are superimposed when they meet, but they then I =

pass through one another and return to their original amplitudes. A
equation 5.7

frequency is the difference between the frequencies of the two
sources. A 352 Hz source and a 350 Hz source sounded HOW LOUD IS THAT SOUND?
together would result in a beat frequency of 2 Hz. Th us, the Th e loudness of a sound is a subjective interpretation that varies
frequencies are closer and closer together, and fewer beats from person to person. Loudness is also related to (1) the energy of
will be heard per second. You may be familiar with the phe- a vibrating object, (2) the condition of the air that the sound wave
nomenon of beats if you have ever flown in an airplane with travels through, and (3) the distance between you and the vibrat-

two engines. If one engine is running slightly faster than the ing source. Furthermore, doubling the amplitude of the vibrating

other, you hear a slow beat. The beat frequency (f b ) is equal to source will quadruple the intensity of the resulting sound wave,
but the sound will not be perceived as four times as loud. Th e rela-
tionship between perceived loudness and the intensity of a sound

f 1

f 2
Constructive Constructive
interference Destructive interference Condensations
interference

Area:
f b 1.0 m 2

joule
Power = = watt
s
Resulting Resulting Resulting Intensity of sound = watts (W)
2
condensation rarefaction condensation area (m )
FIGURE 5.17 Two waves of equal amplitude but slightly FIGURE 5.18 The intensity of a sound wave is the rate of
different frequencies interfere destructively and constructively. The energy transferred to an area perpendicular to the waves. Intensity
2
result is an alternation of loudness called a beat. is measured in watts per square meter, W/m .
5-13 CHAPTER 5 Wave Motions and Sound 127