A string with mass density mu and tension T is fixed at the points z = 0 and z =

Question

A string with mass density mu and tension T is fixed at the points z = 0 and z = L, and supports standing waves of the form f(z, t) = A sin kz cos omega t. Since these standing waves are produced by counterpropagating travelling waves (due to all of the multiple reflections), the relationship omega = kv still applies, where v is the velocity of the travelling waves. In terms of L, what are the allowed values of the wavenumber k which satisfy f(0, t) = 0 and f(L, t) = 0 for all time? Given L = lm, T=300N and mu = 8g/m, what are the three lowest standing-wave frequencies, in Hz, at which the string could vibrate?

Explanation / Answer

a) Allowed values of the wavenumber k

=>   k = L , 2L , 3L , .....nL

where, n = integer

b) Here, f = sqrt(T/d)/2L

=> f1 = sqrt(300/0.008)/2

              = 96.82 Hz

=> f2 = 2 * 96.82 = 193.65 Hz

=> f3 = 3 * 96.82 = 290.46 Hz

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