Question

A certain string is pulled taught between two supports, a distance L apart. When the string is
driven at a frequency of 850 Hz a standing wave is observed with n anti-nodes. When the string is
driven at 1190 Hz a standing wave is observed with n + 2 anti-nodes.
a) What is the fundamental frequency of the set-up?
b) What is the numerical value of n?
c) The distance between the supports is kept fixed, as is the linear mass density of the string, but
the tension in the string is increased from $\tau =\tau _{0}$ to $\tau =b\tau _{0}$ where $b$ is a positive constant greater
than 1. While the tension is increased the string is still driven at 1190 Hz, and at first it is observed
that the standing wave pattern disappear, but for certain values of $b$ a standing wave pattern is again
observed. What is the smallest value of $b$ that again produces standing waves? How many anti-nodes
are observed?

Answer 1

Let tension and linear mass density of the string is tau_o and mu respectively. Then speed of the wave in the string is V = squareroot T/mu = squareroot tau_o/mu ----(1) so if we have n antinodes on the string then total length of the string L = n lambda/2 ---(2) given at gamma_n = 850 Hz = > lambda_n = V/gamma_n n = n and at gamma_n + 2 = 1190 Hz = > lambda_n + 2 = V/gamma_n + 2, n rightarrow n + 2 gives equation (2) as L = n lambda_n/2 and L = (n + 2) lambda_n + 2/2 n V/gamma_n = (n + 2) V/gamma_n + 2 = > n gamma_n + 2 = (n + 2) gamma_n n(gamma_n + 2 - gamma_n) = 2 gamma_n = > n = 2 gamma_n/gamma_n + 2 - gamma_n = 2 times 850 Hz/(1190 - 850)Hz = 2 times 850/340 n = 2 times 85/39 n = 5 ---(4)