Standing Waves on a String - Confined Waves and Line Spectra

The relationship between frequency, wavelength, and energy absorption for standing waves on a stretched string is discussed.

Discussion

When waves are confined by reflecting boundaries, they may interact to establish standing waves, waves for which the vibration pattern is stationary, in that nodal and antinodal positions do not move. If, as in this movie, we attach one end of a string to a stationary point (the clamp on the right) and the other end to a wave driver (on the left) that is oscillating up and down with small amplitude displacements at a fixed frequency, this closely approximates a string with two fixed ends. Each end acts as a boundary for waves traveling on the string.

Waves generated by the wave driver at the left hand end of the string interact with waves reflected by the fixed right-hand end. In such a situation we find that at certain frequencies, called resonant frequencies or resonances stationary wave forms or standing waves occur. In contrast with traveling waves, which can be of any frequency, standing waves exist only at resonant frequencies.

When the driving frequency is a resonant frequency, stable stationary wave forms are established and wave amplitudes are large; at other, non-resonant frequencies, stable stationary wave forms are not established, and wave amplitudes are small. This observation means that energy from the wave driver is most efficiently absorbed by the string at resonant frequencies. In the same way, the wobble of an out-of-balance tire or the buzz of a misadjusted muffler will be most noticeable at certain speeds, at which the frequency of rotation of wheel or vibration of muffler is resonant with an energy source, such as the suspension or drive train of the car.

The relationship between frequency and energy absorption is reflected in an absorption spectrum, a plot of wave amplitude vs. wave frequency. An absorption spectrum provides a measure of the amount of energy absorbed by the string from the wave driver as a function of frequency. The figure below represents what an absorption spectrum for the stretched string used in this set of movies might look like.

Absorption Spectrum for Standing Waves on a Stretched String

This spectrum is an example of a line or discrete spectrum, where absorption occurs at discrete resonant frequencies, here at 20, 40, 60, and 80 Hz. A line absorption or emission spectrum is a characteristic property of a confined wave; if a line absorption or emission spectrum is observed, it means that a wave confined by boundaries is absorbing or emitting energy.

In contrast, the absorption spectrum for traveling waves on a very long string (so long that there are no reflected waves) might look something like the figure below. This spectrum is an example of a continuous spectrum, with energy absorbed over a continuous range of frequencies, and is characteristic of an unconfined wave absorbing or emitting energy. This difference between confined and unconfined waves is an important clue to understanding the properties of atoms and molecules.

Absorption Spectrum for Traveling Waves on a Very Long String

For standing waves on a string there are simple relationships between the length of the string, the wave frequency, and wavelength. On a string of length L that is fixed at each end, standing waves must have N + 1 nodes and N antinodes, where N = 1, 2, 3,.... The corresponding wavelengths are given by the expression λ = 2L/N.

The speed of a wave on a stretched string can be set by adjusting the tension in the string. Given the relationship between wave speed, frequency, and wavelength, v = λ/T = ƒλ, one can choose an arbitrary driving frequency and then adjust the tension so that a standing wave is formed. For these demonstrations, the tension was adjusted so that a standing wave of maximum amplitude with one antinode was established at approximately 20 Hz. Under these conditions, standing waves of maximum amplitude also occurred at about 40 Hz (two antinodes), 60 Hz (three antinodes), 80 Hz (four antinodes), and 100 Hz (five antinodes). In these movies there is significant standing wave amplitude over a 5-10 Hz range of driving frequencies, with amplitudes peaking near the frequency at the middle. Between these 5-10 HZ resonances there is very little wave amplitude.

The length L of the string in these movies was 0.435 m; with this information and the resonant frequencies, 20, 40, 60, 80, and 100 Hz, you can calculate the wave speed on the string (approximately 18.5 m/s).