Resource Lesson Resonance in Strings
Every object, substance, has a natural frequency at which it is "willing" to vibrate. When an external agent applies a forced vibration that matches this natural frequency, the object begins to vibrate with ever increasing amplitude, or resonate.
• For a swing, that natural frequency depends on its length, T = 2π√(L/g). If the swing is pushed at a frequency which either matches the swing's natural frequency or is a sub-multiple of that natural frequency, then the swing's amplitude builds, and we say that it is in resonance.
• Sometimes in mountain streams, you can see standing waves behind large boulders in rapids where the water is reflected off the surface of the boulder and traps objects so that they cannot continue their journey downstream.
• When a spring stretched between a fixed end and an harmonic oscillator sets up standing waves as shown in the picture at the bottom of this page.
Problems involving resonating springs usual focus on string that are fixed on one end and free on the other, or strings that are fixed on both ends. The open ends act as free-end reflectors (producing antinodes, A) and the closed ends act as fixed-end reflectors (producing nodes, N). Free-end reflectors reflect the waveforms in-phase; that is, a crest is reflected as a crest or a trough as a trough, while fixed-end reflectors reflect the waveforms 180º out-of-phase; that is, a crest is reflected as a trough. A worksheet on free and fixed-end reflectors is provided here.

Free-end Reflectors
 N A
fundamental frequency
Fixed-end Reflectors
 N A N
 N A N A
1st overtone
 N A N A N
 N A N A N A
2nd overtone
 N A N A N A N

For a string fixed at both ends, its fundamental natural frequency has a wavelength equal to twice the length of the string since L = ½λ or λ = 2L. Its 1st overtone, or 2nd harmonic, has a wavelength equal to the length of the string, L = λ. Its 2nd overtone, or 3rd harmonic, has a wavelength equal to 2/3λ since L = 3/2λ. Remember that since both ends are fixed-end reflectors, the ends are nodes. These resoance states can be seen in this example using a violin.

CK Ng shows in his physlet the relationship between the specific frequencies that will resonate along a string. Slowly change the frequency and watch the amplitude build as you approach each resonance state. You can alter the length of the string by sliding the stand. By showing the ruler, you can measure the length of a loop and calculate the wave speed. See if you can predict when the next resonance state will occur.

Shown below is an animated gif (you may need to refresh your page to restart the animation) of the 3rd overtone of a standing wave along a string. Note that the wavelength is ½ the length of the string. The same waveform along a spring can be seen in the accompanying picture which was taken during class.

Refer to the following information for the next four questions.

Two identical vibrating strings are 2.0 meters long and uniform in density. One is fixed on both ends, the other is free on one end.
 What is the wavelength of the fundamental frequency along the fixed-end string?

 What is the wavelength of the fundamental frequency along the free-end string?

 What is the wavelength of the first overtone along the free-end string?

 What is the wavelength of the second overtone along the fixed-end string?

Refer to the following information for the next five questions.

Let the speed of the waves along the 2.0 meter strings in the first question group equal 24 m/sec.
 What is the value of the fundamental frequency in the fixed-end string?

 What is the value of the fundamental frequency in the free-end string?

 What would be the frequency of the 4th overtone in the fixed-end string?

 What would be the frequency of the 4th overtone in the free-end string?

 What would be the frequency of the 4th harmonic in the free-end string?