http://www.ck12.org Chapter 11. Vibrations and Sound
FIGURE 11.22
Illustrative Example 1
1a. If the frequency of the first harmonic for a string fixed at both ends isf 1 , determine the frequency for successive
harmonics in terms off 1.
Answer:We know thatλn=^2 nLandv=λf. Combining, we havev=
( 2
nL)
f→fn= 2 nLv. Butvcan be expressed
v=λ 1 f 1 = 2 L f 1 and substituted intofn= 2 nLv, giving
fn= 2 nL( 2 L f 1 ) =n f 1 →fn=n f 11b. If the first harmonic has frequency of 261 Hz, what frequencies do the second and third harmonics have?
Answer:
Since
fn=n f 1 → 2 (261 Hz) =522 Hz
fn=n f 1 → 3 (261 Hz) =783 HzAll whole number multiples of the first harmonic (the fundamental) are called harmonics. String instruments, as
well as non-string instruments, can actually vibrate with many different frequencies simultaneously (called modes).
For example, a string may vibrate with frequencies 261 Hz, 522 Hz and 783 Hz simultaneously.
One of attributes of the “quality” or “timbre” of musical instruments depends upon the combination of the various
overtones produced by the instrument.
Check Your Understanding
- A tuning fork has a frequency of 512 Hz stamped on it. When it is struck, a student claims she can hear higher
frequencies from the tuning fork. Is this possible?
Answer: Yes, it is. The tuning fork may be producing harmonics, in which case the student may be hearing
frequencies in multiples of 512 Hz, such as 1,024 Hz and 1,536 Hz.
- A string with a fundamental frequency of 220 Hz vibrates in its third harmonic with a wavelength of 60 cm. What
is the wave velocity on the string?