Question

Assertion: In a standing wave on a string, the spacing between nodes is Δx. If the tension in string is increased wave same as before, then the separation between nearest node and antinode will be Δx. Reason: Spacing between nodes (consecutive) in the standing wave is equal to the wavelength of component waves.

Let us consider a familiar situation, a string fixed at either end or an air column in a pipe with either end closed in which reflection takes place at two or more boundaries. In a string, for example, a wave travelling in one direction will get reflected at one end, which in turn will travel and get reflected from the other end. This will go on until there is a steady wave pattern set up on the string. Such wave patterns are called standing waves or stationary waves. To see this mathematically, consider a wave travelling along the positive direction of x-axis and a reflected wave of the same amplitude and wavelength in the negative direction of x-axis. The Equations, with φ = 0, can be written as y1(x, t) = a sin (kx - ωt) y2(x, t) = a sin (kx + ωt) The resultant wave on the string is, according to the principle of superposition: y(x, t) = 2a sin kx cos ωt