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Standing waves and resonance

Standing waves are an example of superposition. They occur when identical waves travelling in opposite directions.
In the diagram below, P and Q represent points along a rope. At the instant shown, two wave trains travelling in opposite directions are just about to overlap at point P. Points L1 to L4, R1 to R4 represent peaks or troughs along the wave train.
formation of a standing wave
Formation of a standing wave
Superposition at point P causes P to oscillate with amplitude 2A, since peaks L1 and R1 arrive there simultaneously, followed half a cycle later by troughs L2 and R2, etc. Careful inspection shows that superposition at Q will result in Q remaining stationary in space all the time.
Standing waves in bounded systems
The edges of any solid object act as boundaries to waves. Superposition of waves travelling towards the boundary with those reflected from it can lead to standing waves, if the object is vibrated at an appropriate frequency (unless the vibrations are damped). In a similar way, standing waves can be set up in fluids if they are contained (air in a trumpet, water in the bath). The same ideas are used to explain the energy levels of atoms.
These more complex standing waves have the following features in common with waves on a string:
1 There is a series of definite modes of oscillation, corresponding to different frequencies, at each of which the response is large (resonance).
2 The patterns developed depend on the frequency, there being more nodes at higher frequencies.
3 The standing waves must 'fit' into the system.