String instruments such as (a) violins and (b) guitars use resonance in their sounding boxes to amplify and enrich the sound created by their vibrating strings. The bridge and supports couple the string vibrations to the sounding boxes and air within. (credit a: modification of work by Feliciano Guimares; credit b: modification of work by Steve Snodgrass)Resonance has been used in musical instruments since prehistoric times. This marimba uses gourds as resonance chambers to amplify its sound. (credit: “APC Events”/Flickr)
We have emphasized sound applications in our discussions of resonance and standing waves, but these ideas apply to any system that has wave characteristics. Vibrating strings, for example, are actually resonating and have fundamentals and overtones similar to those for air columns. More subtle are the resonances in atoms due to the wave character of their electrons. Their orbitals can be viewed as standing waves, which have a fundamental (ground state) and overtones (excited states). It is fascinating that wave characteristics apply to such a wide range of physical systems.
Summary
Some musical instruments can be modeled as pipes that have symmetrical boundary conditions: open at both ends or closed at both ends. Other musical instruments can be modeled as pipes that have anti-symmetrical boundary conditions: closed at one end and open at the other.
Some instruments, such as the pipe organ, have several tubes with different lengths. Instruments such as the flute vary the length of the tube by closing the holes along the tube. The trombone varies the length of the tube using a sliding bar.
String instruments produce sound using a vibrating string with nodes at each end. The air around the string oscillates at the frequency of the string. The relationship for the frequencies for the string is the same as for the symmetrical boundary conditions of the pipe, with the length of the pipe replaced by the length of the string and the velocity replaced by
Conceptual questions
How does an unamplified guitar produce sounds so much more intense than those of a plucked string held taut by a simple stick?
Consider three pipes of the same length (
L ). Pipe
A is open at both ends, pipe
B is closed at both ends, and pipe
C has one open end and one closed end. If the velocity of sound is the same in each of the three tubes, in which of the tubes could the lowest fundamental frequency be produced? In which of the tubes could the highest fundamental frequency be produced?
When resonating at the fundamental frequency, the wavelength for pipe
C is 4
L , and for pipes
A and
B is 2
L . The frequency is equal to
Pipe
C has the lowest frequency and pipes
A and
B have equal frequencies, higher than the one in pipe
C .
Pipe
A has a length
L and is open at both ends. Pipe
B has a length
L /2 and has one open end and one closed end. Assume the speed of sound to be the same in both tubes. Which of the harmonics in each tube would be equal?
A string is tied between two lab posts a distance
L apart. The tension in the string and the linear mass density is such that the speed of a wave on the string is
A tube with symmetric boundary conditions has a length
L and the speed of sound in the tube is
What could be said about the frequencies of the harmonics in the string and the tube? What if the velocity in the string were
?
Since the boundary conditions are both symmetric, the frequencies are
Since the speed is the same in each, the frequencies are the same. If the wave speed were doubled in the string, the frequencies in the string would be twice the frequencies in the tube.
If a wind instrument, such as a tuba, has a fundamental frequency of 32.0 Hz, what are its first three overtones? It is closed at one end. (The overtones of a real tuba are more complex than this example, because it is a tapered tube.)
What are the first three overtones of a bassoon that has a fundamental frequency of 90.0 Hz? It is open at both ends. (The overtones of a real bassoon are more complex than this example, because its double reed makes it act more like a tube closed at one end.)
How long must a flute be in order to have a fundamental frequency of 262 Hz (this frequency corresponds to middle C on the evenly tempered chromatic scale) on a day when air temperature is
? It is open at both ends.
(a) Find the length of an organ pipe closed at one end that produces a fundamental frequency of 256 Hz when air temperature is
. (b) What is its fundamental frequency at
?
A sound wave of a frequency of 2.00 kHz is produced by a string oscillating in the
mode. The linear mass density of the string is
and the length of the string is 1.50 m. What is the tension in the string?
Consider the sound created by resonating the tube shown below. The air temperature is
. What are the wavelength, wave speed, and frequency of the sound produced?
A student holds an 80.00-cm lab pole one quarter of the length from the end of the pole. The lab pole is made of aluminum. The student strikes the lab pole with a hammer. The pole resonates at the lowest possible frequency. What is that frequency?
A string on the violin has a length of 24.00 cm and a mass of 0.860 g. The fundamental frequency of the string is 1.00 kHz. (a) What is the speed of the wave on the string? (b) What is the tension in the string?
By what fraction will the frequencies produced by a wind instrument change when air temperature goes from
to
? That is, find the ratio of the frequencies at those temperatures.
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