Question 79: Waves

Question

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $$\left(\frac{a-1}{a}\right)$$ then the value of $$a$$ is _________.

Accepted Answer

Step 1: Write the formulas for the frequencies of open and closed organ pipes.
The frequency of an open organ pipe is $$f_o = \frac{v}{2l}$$, where
v
is the speed of sound and
l
is the length of the pipe.
The frequency of a closed organ pipe is $$f_c = \frac{v}{4l}$$.
Step 2: Find the frequency of the seventh overtone for each pipe.
The seventh overtone in an open pipe is the 8th harmonic (since the first overtone is the 2nd harmonic). So, its frequency is: $$f_{o_7} = 8 	imes \frac{v}{2l}$$.
In a closed pipe, only odd harmonics are present. The seventh overtone is the 15th harmonic, so: $$f_{c_7} = 15 	imes \frac{v}{4l}$$.
Step 3: Find the ratio of the seventh overtone frequencies.
The ratio of the frequencies is:

$$\frac{f_{c_7}}{f_{o_7}} = \frac{15 \frac{v}{4l}}{8 \frac{v}{2l}}$$.
Simplify the ratio:

$$\frac{15}{4l} \div \frac{8}{2l} = \frac{15}{4l} 	imes \frac{2l}{8} = \frac{30}{32} = \frac{15}{16}$$.
Step 4: Connect the answer to the value of
a
The given ratio is $$\frac{a-1}{a}$$, so:

$$\frac{a-1}{a} = \frac{15}{16}$$.
Solve for $$a$$: $$a = 16$$.

Waves

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right) then the value of a$ is _________.

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