MHT CET · Physics · Waves and Sound

A pipe open at both ends and a pipe closed at one end have same length. The ratio of frequencies of their $P^{t h}$ overtone is

A $\frac{P + 1}{2 p}$
B $\frac{p + 1}{2 p + 1}$
C $\frac{2 (p + 1)}{2 p + 1}$
D $\frac{p}{2 p + 1}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{2 (p + 1)}{2 p + 1}$

Step-by-step Solution

Detailed explanation

Frequency of p th overtone of open organ pipe

= (p + 1) \frac{v}{2 L}

…. (i)
where, L = length of organ pipe
and that of closed organ pipe is

= (2 p + 1) = \frac{v}{4 L}

…. (ii)
So, the ratio of p th overtone of open to closed

\frac{(p + 1) \frac{v}{2 L}}{(2 p + 1) \frac{v}{4 L}}

(using Eqs. (i) and (ii))

= \frac{2 (p + 1)}{(2 p + 1)}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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A pipe open at both ends and a pipe closed at one end have same length. The ratio of frequencies of their Pth overtone is

Step-by-step Solution

A pipe open at both ends and a pipe closed at one end have same length. The ratio of frequencies of their $P^{t h}$ overtone is