The third overtone of a closed pipe of length ' L_c ' has the same frequency as the third overtone of the open pipe of length ' L₀ '. Both the pipes have same diameters. The ratio L_c : L₀ is equal to

Step-by-step solution

For a closed pipe, the frequency of the n -th overtone is given by f_c = (2n + 1) v 4L_c . For the third overtone ( n = 3 ), the frequency is f_c = 7 v 4L_c . For an open pipe, the frequency of the n -th overtone is given by f_o = (n + 1) v 2L₀ . For the third overtone ( n = 3 ), the frequency is f_o = 4 v 2L₀ = 2v L₀ . Given that the frequencies of the third overtones are equal: 7 v 4L_c = 2v L₀ L_c L₀ = 7 8 The ratio L_c : L₀ is 7:8 . Answer: 7:8