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Figure 6.7 Time sequence of the positions of the particles of a vibrating string.

(In practice, there is no such a thing as a completely rigid support and a small amount of energy will 'escape' from the string at each reflection.)

6.4.5 Standing waves

A string of fixed length has a number of normal modes of vibration corresponding to exact numbers of half wavelengths which 'fit exactly' into that length of string. The amplitude of vibration must be zero at both ends of the string.

The frequencies of the normal modes are called the natural frequencies (fn) of the string, and

A mathematical treatment can be found in Appendix 6.3. It does not give any more physical insight, but it does make a very clear distinction between travelling and standing waves and allows us to calculate the position of any particle at any time.

6.5 Forced oscillations and resonance

We can use an oscillator to produce waves on a string or in other materials, such as air. If the string or other system has natural frequencies of vibration, its subsequent behaviour depends on whether the frequency of the oscillator matches one of these frequencies.

6.5.1 Forced oscillations

Suppose that an oscillator drives a string to vibrate. The wave will be directed down the string and reflected at the far end, which is a node because it is fixed. As the incoming and reflected waves meet, nodes separated by distances of A/2 will be formed. The reflected wave returns to the oscillator end, which is also a node (assuming the amplitude of the waves generated by the oscillator to be small). In Figure 6.8 the positions of nodes created when reflections take place at each of the ends illustrate the condition for the formation of standing waves on a string.

If the oscillator frequency is not one of the natural frequencies of the string, the fixed end at B is not exactly one or more half wavelengths away from A and the condition for standing waves is not met. The reflected wave, which travels back up the string, is out of phase with the waves pumped in by the oscillator. So, although the oscillator keeps

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Nodes must match for standing waves to form.

Figure 6.8 Nodes and the condition for standing waves.

Nodes must match for standing waves to form.

Figure 6.8 Nodes and the condition for standing waves.

pumping energy into the string, this energy is absorbed mainly at the ends of the string. The waves themselves are travelling waves of small amplitude. This type of behaviour can also be seen in electrical circuits which have characteristic modes of vibration. The energy is absorbed by the resistance of the circuit.

6.5.2 Natural frequencies of vibration and resonance

When someone plucks a stretched string, the disturbance propagates in both directions and is reflected from the ends. The incident and reflected waves may combine to give a standing wave at one of the natural frequencies of the string. Now suppose that an oscillator is coupled to one end of the string. If the oscillator frequency matches one of the natural frequencies, each new wave from the oscillator reinforces the wave reflected from the far end of the string. The total amplitude of the wave is higher than at other frequencies; we refer to this as resonance. The amplitude of vibration is limited by elasticity and by other constraints on the system.

All flexible mechanical systems which are constrained in some way, like suspension bridges, guitar strings or the air in an organ pipe, have natural frequencies at which they will vibrate if prompted by an impulse.

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