The student will:
- Understand the conditions for resonance.
- Solve problems with strings and pipes using the condition for resonance.
- natural frequency: The frequency at which a system vibrates normally when given energy without outside interference.
- resonance: Timing force to be the same as natural frequency.
Many systems have a tendency to vibrate. When the forced vibration frequency is the same as the natural frequency, the amplitude of vibration can increase tremendously. A well-known example of this situation is pushing a person on a swing, Figure below. We know from study of simple pendulums that without being pushed, the person in the swing rocks back and forth with a frequency that depends on gravity and the length of the chain.
This is one example of a natural frequency – the frequency at which a system vibrates normally when given energy without outside interference.
Pushing on the person in the swing will affect the amplitude of the swinging. This is called forced vibration – when a periodic force from one object (the person pushing) affects the vibration of another object (the person swinging). To get the most effect, the person pushing will start just at the very back of the swing. In other words, the frequency of how often they push is exactly the same as the frequency of the swing. Suppose they do not push at the right time, but instead push at some other frequency. That would mean that sometimes they are pushing forward when the swing is still going backward. In that case, the swing would slow down – i.e. the amplitude of the swing will be reduced.
Timing the pushes to be the same as the natural frequency is called resonance. For this reason, the natural frequency is also known as the resonant frequency. If the pushes are timed just right, then even if each individual push is small, the vibration will get larger with each push.
A classic example of an unfortunate consequence of a forced vibration at resonant frequency is what happened to the Tacoma Narrows Bridge, in 1940. See the link below.
In Figure below, the bridge is beginning to resonate, in part, due to the frequency of vibration of the wind gusts. In Figure below, we see that the bridge is no longer able to respond elastically to the tremendous amplitude of vibration from the forced vibration of wind energy (at its resonant frequency), and it is torn apart.
Modern bridges are built to avoid this effect, but through history there are a number of documented situations where a forced vibration at resonance had dire results. The Broughton Suspension Bridge (1831) and the Angers Bridge (1850) are two examples of bridges believed to have collapsed due to the effect of soldiers marching at a regular pace that caused resonance. The Albert Bridge in West London, England has been nicknamed The Trembling Lady because it has been set into resonance so often by marching soldiers. Though soldiers no longer march across the bridge, there still remains a sign of concern as shown in Figure below.
There is a typical classroom physics demonstration where one tuning fork is set into motion and an identical tuning fork, if placed closed enough, will also be set vibrating, though with smaller amplitude. The same effect occurs when tuning a guitar. One string is plucked and another, whose length is shortened by holding it down some distance from the neck of the guitar, will also be set into vibration. When this condition is met, both strings are vibrating with the same frequency. We call this phenomenon sympathetic vibration.
Resonance is a very common phenomenon, especially with sound. The length of any instrument is related to what note it plays. If you blow into the top of a bottle, for example, the note will vary depending on the height of air in the bottle. This plays an important role in human voice generation. The length of the human vocal tube is between 17 cm and 18 cm. The typical frequencies of human speech are in the range of 100 Hz to 5000 Hz.
By using the muscles in their throat, singers change the note they sing. A dramatic example of this is breaking glass with the human voice. By singing at exactly the resonant frequency of a delicate wine glass, the glass will resonate with the note and shatter.
The resonance of sound is also a mechanical analogue to how a radio set receives a signal. The Figure below shows one of the earliest radio designs, called a crystal radio because the element which detected the radio waves was a crystalline mineral such as galena.
An old crystal radio set.
Strings Fixed at Both Ends
A case of natural frequency that you can observe plainly is when you pluck a string or stretched rubber band. Normally, the string will vibrate at a single widest point in the middle. This is called the fundamental or first harmonic resonance of the string. This is the same as the natural frequency of a simple pendulum or mass on a spring. Because it vibrates all along its length, though, the string also lets us see further patterns of resonance.
By vibrating the end of the string rather than just plucking it, we can force vibration at frequencies other than the first harmonic. When the string is set into vibration, energy will travel down the string and reflect back toward the end where the waves are being generated. This steady pattern of vibration is called a standing wave. The points where the reflecting waves interfere destructively with the “generated’ waves are called nodes. The points where the reflecting waves interfere constructively with the generated waves are called anti-nodes.
Figure below shows a string fixed at both ends vibrating in its fundamental mode. There are two nodes shown and one antinode. The dashed segment represents the reflected wave.
Check Your Understanding
1. How many nodes and anti-nodes are shown in Figure above?
Answer: There are three nodes and two anti-nodes.
2. If the length of the unstretched string is 20 cm, what is the wavelength for the 10th harmonic?
Strings Fixed at One End and Opened at One End
Illustrative Example 1
1b. If the first harmonic has frequency of 261 Hz, what frequencies do the second and third harmonics have?
All 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
1. 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.
2. 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?
Open and Closed Pipes and Tubes
In our discussions of pipes, the length of the pipe will be assumed to be much greater than the diameter of the pipe.
There is a simple experiment your instructor may have you do in class that demonstrates resonance in an open tube. Roll two sheets of long paper into two separate tubes and use a small amount of tape to keep them rolled. Have the diameter of one tube just small enough to fit inside the other tube so the inside tube can freely slide back and forth. Hold a struck tuning fork (your instructor will make sure the frequency is adequate) close to the end of the outer tube while the inside tube is moved slowly. When the total length of the tubes is the proper length to establish resonance, you’ll hear a noticeable increase in the volume of the sound. At this moment, there are standing waves present in the tubes.
A closed pipe supporting the first harmonic (the fundamental frequency) will fit one-fourth of the wavelength, the second harmonic will fit three-fourths, and so on, as shown in Figure below. Compare these pictures to those in the figures above for a string fixed at only one end
A standard physics laboratory experiment is to find the velocity of sound by using a tuning fork that vibrates over a closed pipe as shown in Figure below. The water level in a pipe is slowly changed until the first harmonic is heard.
Illustrative Example 2
Resonance is established in a hollow tube similar to that shown in Figure above with a tuning fork of 512 Hz. The distance from the tube opening to the water level is 16.8 cm.
a. What is the velocity of sound according to this experiment?
Answer: We simply set the result of part A equal to the given equation: