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.
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Free-end Reflectors
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fundamental frequency |
Fixed-end Reflectors
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1st overtone |
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2nd overtone |
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For a string that is fixed at one end and open at the other, the fundamental natural frequency has a wavelength equal to four times the length of the string. One method of illustrating this phenomena is to hold one end of a rope and tie a a very light string to the other. When you flick the rope, part of the pulse travels forward through the light string but part is also reflected [in-phase], back to the starting point. Water lapping on the side of a swimming pool when a person jumps into the pool also acts like a free-end - sloshing up the wall and then reflecting in-phase back to the original position of the disturbance. This linked video shows a free-end reflection on a wave machine. Resonance is illustrated by Edward A Zobel with his animation of a thumb piano. 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 resonance states can be seen in an animation, programmed by Edward A. Zobel, using a violin [zonalandeducation.com]. CK Ng shows in his animation 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.
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