Question

An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by $100 \mathrm{~Hz}$ than the fundamental frequency of the open pipe. What is the fundamental frequency of the open pipe? (A) 240 (B) 200 (C) 300 (D) 480

Short answer

The fundamental frequency of the open pipe is 200 Hz. The correct choice is (B) 200.

Step-by-step solution

Recall the formulas for the fundamental frequency of open and closed pipes.

The formula for the fundamental frequency (1st harmonic) of an open pipe is: \(f_o = \dfrac{v}{2L}\) Where \(f_o\) is the fundamental frequency of the open pipe, \(v\) is the speed of sound, and \(L\) is the length of the pipe. The formula for the fundamental frequency (1st harmonic) of a closed pipe is: \(f_c = \dfrac{v}{4L}\) Notice that the frequency of the 3rd harmonic of a closed pipe is three times the fundamental frequency: \(3f_c = \dfrac{3v}{4L}\) Now, we are given that the frequency of the 3rd harmonic of the closed pipe is 100 Hz higher than the fundamental frequency of the open pipe: \(3f_c = f_o + 100\)

Express the fundamental frequency of the open pipe in terms of the closed pipe frequencies.

We can replace \(3f_c\) and \(f_o\) in the frequency relationship equation using the formulas for the fundamental frequency of open and closed pipes: \(\dfrac{3v}{4L} = \dfrac{v}{2L} + 100\)

Solve the equation for the fundamental frequency of the open pipe.

To solve for \(f_o\), divide both sides of the equation by \(v\): \(\dfrac{3}{4L} = \dfrac{1}{2L} + \dfrac{100}{v}\) Next, subtract \(\dfrac{1}{2L}\) from both sides of the equation: \(\dfrac{1}{4L} = \dfrac{100}{v}\) Now, multiply both sides by \(4L\) : \(1 = \dfrac{400L}{v}\) Divide both sides by 400: \(\dfrac{1}{400} = \dfrac{L}{v}\) Finally, substitute the formula for the fundamental frequency of the open pipe: \(f_o = \dfrac{v}{2L}\) Since \(\dfrac{1}{400} = \dfrac{L}{v}\), the equation becomes: \(f_o = \dfrac{1}{\dfrac{1}{200}}\) Which simplifies to: \(f_o = 200\) So, the fundamental frequency of the open pipe is 200 Hz. The correct choice is (B) 200.