Question

An organ pipe sounds two successive tones at 228.6 Hz and 274.3 Hz. Determine whether the pipe is open at both ends or open at one end and closed at the other. a. Open at one end and closed at the other end b. Open at both ends c. There is not enough information to answer this question. d. None of these

Answer 1

Answer:

The pipe is open on both ends .

Explanation:

In a closed organ pipe the nth  frequency is given as

$f_n=(2n+1)\frac{v}{4L}$

For the (n+1) th frequency is given as

$f_{n+1}=(2(n+1)+1)\frac{v}{4L}\\f_{n+1}=(2n+3)\frac{v}{4L}$

By taking ratios of both equations.

$\frac{f_{n+1}}{f_{n}}=\frac{(2n+3)\frac{v}{4L}}{(2n+1)\frac{v}{4L}}\\\\\frac{274.3}{228.6}=\frac{2n+3}{2n+1}\\\\274.3(2n+1)=228.6(2n+3)\\n=4.5$

As n has to be integer thus this is not a closed organ pipe.

In a open organ pipe the nth  frequency is given as

$f_n=\frac{nv}{2L}$

For the (n+1) th frequency is given as

$f_{n+1}=\frac{(n+1)v}{4L}$

$\frac{f_{n+1}}{f_{n}}=\frac{\frac{(n+1)v}{4L}}{\frac{nv}{4L}}\\\\\frac{274.3}{228.6}=\frac{n+1}{n}\\\\274.3(n)=228.6(n+1)\\n=5.0$

As n is an integer in this case, i.e. the pipe is open at both ends.