A uniform string of length L is fixed at both ends and supports stationary waves. Let the displacement travelling waves that superpose have amplitude A, angular frequency ω, and wave number k.
(i) Derive an expression for the positions xm of the nodes measured from the left end in terms of the integer m (m=0,1,2,…) and the wavelength λ.
(ii) Using your result, obtain the separation between two successive antinodes.
(iii) The string has length 1.80m and supports a mode that displays exactly seven antinodes. Calculate the wavelength and the frequency of this mode if the speed of transverse waves on the string is 90ms−1.