The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an ope

Question

The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is $5 / x$. The value of $x$ is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

The frequency of the $n^{	ext{th}}$ harmonic (where $n$ is an odd integer) of a closed organ pipe of length $L_c$ is $f_c = n\frac{v}{4L_c}$. Thus, its fifth harmonic is $f_c = 5\frac{v}{4L_c}$. The frequency of the first harmonic (fundamental) of an open organ pipe of length $L_o$ is $f_o = \frac{v}{2L_o}$. Being in unison means their frequencies are equal: $5\frac{v}{4L_c} = \frac{v}{2L_o}$. Rearranging to find the ratio of their lengths gives $\frac{L_c}{L_o} = \frac{5 	imes 2}{4} = \frac{5}{2}$. Comparing this with the given ratio $5/x$, we get $x = 2$.