An open organ pipe of length 40 cm and a closed organ pipe of length 50 cm are sounded together. If the 1^ st overtone of the open pipe resonates with the n^ th overtone of the closed pipe, what is the value of n ? (Assume the speed of sound is the same in both pipes and ignore end corrections).

Step-by-step solution

For an open organ pipe of length L_o , the frequency of the 1^ st overtone (which is the 2^ nd harmonic) is given by: f_o = 2 ( v 2L_o ) = v L_o For a closed organ pipe of length L_c , the frequency of the n^ th overtone (which is the (2n+1)^ th harmonic) is given by: f_c = (2n+1) ( v 4L_c ) Given that these two frequencies are equal: v L_o = (2n+1)v 4L_c Substitute L_o = 40 cm and L_c = 50 cm : v 40 = (2n+1)v 4 50 1 40 = 2n+1 200 2n+1 = 200 40 = 5 2n = 4 n = 2 Answer: 2