PrakashPhy said:
I do not understand why the length of string should be integral multiple of wavelength for the formation of standing wave.
Please help me.
Your intuition is correct. Standing waves, in general, do not require string lengths to be integral multiples of wavelength.
When any sinusoidal wave is reflected back on itself from a single reflecting surface or point the result is two superimposed waves of equal wavelength traveling in opposite directions. The two superimposed waves will always sum to give a standing wave.
The amplitude of the sum of the two waves at the exact reflecting point is zero.
The amplitude of the sum of the two waves rises up to a maximum at one quarter wavelength from the reflecting point, drops down to zero again at half a wavelength from the reflecting point ... and so on.
The standing wave pattern extends for whatever overall distance the two waves continue to overlap, with zero amplitude nodes at regular half wavelength intervals.
This will occur for
any wavelength.
In the special case of a piece of string fixed at each end the thing that makes a difference is that the string is fixed at
two points, not just one point. For a standing wave to be maintained with two fixed points
both those points must be zero amplitude nodes.
As described above, the distance from node to node in a standing wave is half the wavelength of the original sinusoidal wave. Therefore if the string is fixed at each end the length of the string must be an integral multiple of that distance.