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Length
f 1

+
f 1 Fundamental
frequency
f 2 =

+

f 2 First f 3
overtone
FIGURE 5.23 A combination of the fundamental and overtone
frequencies produces a composite waveform with a characteristic
sound quality.
f 3 Second
overtone
and the overtones when the string length and velocity of waves

on the string are known. The relationship is
FIGURE 5.22 A stretched string of a given length has a
nv _
number of possible resonant frequencies. The lowest frequency f n =

is the fundamental, f 1 ; the next higher frequencies, or overtones, 2L
shown are f 2 and f 3 .
equation 5.8
where n = 1, 2, 3, 4 . . ., and n = 1 is the fundamental frequency
Standing waves occur at the natural, or resonant, frequencies of and n = 2, n = 3, and so forth are the overtones.
the string, which are a consequence of the nature of the string,
the string length, and the tension in the string. Since the stand- EXAMPLE 5.8
ing waves are resonant vibrations, they continue as all other
What is the fundamental frequency of a 0.5 m string if wave speed on
waves quickly fade away.
the string is 400 m/s?
Since the two ends of the string are not free to move, the ends
of the string will have nodes. Th e longest wave that can make a
standing wave on such a string has a wavelength (λ) that is twice SOLUTION
the length (L) of the string. Since frequency (f) is inversely pro-

The length (L) and the velocity (v) are given. The relationship between
portional to wavelength (f = v/λ from equation 5.3), this lon- these quantities and the fundamental frequency (n = 1) is given in
gest wavelength has the lowest frequency possible, called the equation 5.8, and

fundamental frequency. The fundamental frequency has one
L = 0.5 m nv _ where n = 1 for the

antinode, which means that the length of the string has one-half a f n = fundamental frequency
2L
wavelength. The fundamental frequency (f 1 ) determines the pitch v = 400 m/s

1 × 400 m∙s

of the basic musical note being sounded and is called the fi rst har- f 1 = ? f 1 = __

monic. Other resonant frequencies occur at the same time, how- 2 × 0.5 m
_
400 m _
1 _

ever, since other standing waves can also fit onto the string. A = ×

m
s
higher frequency of vibration (f 2 ) could fi t two half-wavelengths 1

between the two fixed nodes. An even higher frequency (f 3 ) = 400 m _
could fit three half-wavelengths between the two fi xed nodes s⋅m

1 _
( Figure 5.22). Any whole number of halves of the wavelength = 400

s
will permit a standing wave to form. The frequencies (f 2 , f 3 , etc.)

of these wavelengths are called the overtones, or harmonics. It is = 400 Hz
the presence and strength of various overtones that give a musi-
cal note from a certain instrument its characteristic quality. Th e
fundamental and the overtones add to produce the characteristic EXAMPLE 5.9
sound quality, (Figure 5.23) which is different for the same-pitch What is the frequency of the fi rst overtone in a 0.5 m string when the

note produced by a violin and by a guitar. wave speed is 400 m/s? (Answer: 800 Hz)
Since nodes must be located at the ends, only half wave-
lengths (1/2 λ) can fit on a string of a given length (L), so the

fundamental frequency of a string is 1/2 λ = L, or λ = 2L. Sub- The vibrating string produces a waveform with overtones,

stituting this value in the wave equation (solved for frequency f ) so instruments that have vibrating strings are called harmonic

will give the relationship for finding the fundamental frequency instruments. Instruments that use an air column as a sound
130 CHAPTER 5 Wave Motions and Sound 5-16