An open organ pipe and a closed organ pipe, both of the same length L , are vibrating in their respective resonant modes. It is observed that the standing wave pattern in both the pipes contains exactly 4 displacement nodes. The ratio of the frequency of the open pipe to the frequency of the closed pipe is :

Step-by-step solution

For an open organ pipe, the p^ th harmonic contains p displacement nodes. Since the open pipe has 4 nodes, it is vibrating in its 4^ th harmonic. Its frequency is f_ open = 4 ( v 2L ) = 2v L . For a closed organ pipe, only odd harmonics are present. The q^ th harmonic (where q is an odd integer) contains q+1 2 displacement nodes. Given that the closed pipe has 4 nodes, we have q+1 2 = 4 q = 7 . Thus, it is vibrating in its 7^ th harmonic. Its frequency is f_ closed = 7 ( v 4L ) = 7v 4L . The ratio of their frequenc