How can we predict whether, a wave ( say a one dimensional string wave) travelling in any direction (say +x) after reflection from the fixed end, forms a standing wave in the medium or not (here string)? I do not understand why the length of stirng should be integral multiple of wavelength( lymbda) for the formation of standing wave. Please help me.
Hi PrakashPhy! (have a lambda: λ ) Draw a wave, cut it at any point, and turn one of the bits round to fit on top of the other bit (with or without reflection) … you'll quickly find that you can't match the two bits unless you make the cut at a top middle or bottom position.
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 travelling 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.
What happens if two ends are at max amplitude (sum of either amplitudes) rather than zero. I mean why should the ends be at zero amplitude to "maintain" the standing wave pattern?
The same as if there were nodes at each end. We still have a double constraint and the possible wavelengths of standing waves are 2L/n in which L is the string length and n = 1, 2, 3... . This what you get for sound waves in a pipe open at each end - a so-called 'open pipe'! If there'e a node at one end and an antinode at the other (as for sound in a pipe closed at one end and open at the other - a 'closed pipe'), it's easy to see that the possible wavelengths are 4L, 4L/3, 4L/5 ... .
Because the ends are "fixed." Meaning whatever else happens to the string, the ends stay at their original positions, hence the wave amplitude at the ends has to work out to be zero.