When a wave is reflected at a fixed end of a string, a phase change of phase occurs. A node of the stationary wave exists at a fixed end. A free end can also reflect waves but without a phase change.

When a wave passes through a point on a string, that point oscillates. If the wave later returns to the same point (for example, after reflection), interference will occur between the existing oscillation and the reflected wave. The interference will be constructive if the "effective distance" moved by the wave nl a whole number of wavelengths.

A phase change of 180° makes a wave "look as if" it has travelled an extra half wavelength. So an effective distance of l could be due to the wave actually moving l/2 and experiencing a phase change of 180° (prads).

All points oscillate in phase but with different amplitudes of oscillation.

Consider the string to be disturbed at A (the centre).
Waves travel towards B (and C) and are reflected with with 180° phase change.
If the "effective distance" travelled by the waves is l, then resonance occurs.

This means that, for the first resonance, the distance A ® B ® A (or A®C®A) must be equal to l/2.

This means that = l/2.

Conclusion: The distance between adjacent nodes of a stationary wave is equal to half the wavelength (l/2) of the travelling waves which are producing the stationary wave.

If the distance A®B®A (= ) is l/2 then the time taken by the waves to move this distance is half of a time period (T/2).

If the wave velocity is v then

and, since in general, f = 1/T we have:

Now, the velocity of waves on a string under tension is given by:

Where µ is the mass per unit length of the wire.

So, the fundamental frequency is given by:

and the frequencies of the higher harmonics are given by f = nf_o where n = 1,2,3… etc

The diagram below represents the second harmonic.

Assuming that the tension is still the same this vibration (resonance) has a frequency 2f_o.

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