A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved w

Q: Why do we use the Doppler effect here? The tuning fork's frequency doesn't actually change!

You are absolutely right! The tuning fork itself still vibrates at its original frequency $f$. However, because it is moving, the sound waves get compressed (or stretched) in the air. The pipe 'hears' this modified frequency $f'$, and resonance depends on what the pipe receives, not what the fork emits.

Q: Does it matter if the pipe is open at both ends or closed at one end?

Interestingly, no! For both open and closed pipes, the fundamental frequency (and any harmonic) is inversely proportional to the length ($f = v/2L$ or $f = v/4L$). The constant of proportionality cancels out when we take the ratio $\ell_2/\ell_1$.

Q: Why did we assume the tuning fork is moving towards the pipe and not away?

We can assume either! If it moves towards the pipe, the length must decrease by $0.625\%$. If it moves away, the length must increase by $0.625\%$. The question asks for the *smallest value* of the percentage change, which refers to the magnitude, so it is $0.625$ in both cases.

Q: What if the tuning fork was moving perpendicular to the pipe?

If the tuning fork moved perfectly perpendicular to the line connecting it to the pipe, the component of its velocity towards the pipe would be zero. In that case, there would be no transverse Doppler effect in classical acoustics, so the frequency wouldn't change, and the percentage change in length would be zero.

Q: Can we use the binomial approximation for the percentage change?

In this specific problem, we don't even need an approximation! Because the length is inversely proportional to the frequency, and the Doppler formula for a moving source has the velocity term in the denominator, they perfectly cancel out to give an exact linear relation: $\ell_2 = \ell_1(1 \pm v_T/v)$.

Question

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2	ext{ ms}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320	ext{ ms}^{-1}$, the smallest value of the percentage change required in the length of the pipe is ______.

Sigma Percentile · Accepted Answer

### The Symphony of Resonance

Imagine a tuning fork vibrating right in front of an open pipe.

Initially, both the tuning fork and the pipe are perfectly at rest. They are in a state of beautiful acoustic harmony, which we call **resonance**.

This means the frequency of the tuning fork perfectly matches the natural frequency of the air column inside the pipe.

Mathematically, we can express this natural frequency as:

$$f = \frac{k}{\ell_1}$$

Here, $f$ is the frequency of the tuning fork, $\ell_1$ is the initial length of the pipe, and $k$ is a constant that depends on the speed of sound and whether the pipe is open or closed.

### The Doppler Shift: A Twist in the Tale

Now, let's introduce some motion into our system. We move the tuning fork towards the pipe with a speed of $v_T = 2 	ext{ ms}^{-1}$.

Because the source of the sound is now moving, the sound waves get compressed in the air before they reach the pipe. This is the classic **Doppler Effect**.

The pipe doesn't "hear" the original frequency $f$ anymore. Instead, it receives a new, higher frequency $f'$.

Using the standard Doppler formula for a moving source, we get:

$$f' = f \left( \frac{v}{v - v_T} ight)$$

### Re-tuning the Pipe

Because the incoming sound frequency has changed to $f'$, the old pipe length $\ell_1$ will no longer resonate. The system is out of tune!

To bring the magic of resonance back, we must change the length of the pipe to a new value, let's call it $\ell_2$.

This new length must have a natural frequency that exactly matches the Doppler-shifted frequency $f'$.

So, our new resonance condition becomes:

$$f' = \frac{k}{\ell_2}$$

### The Mathematical Elegance

Now comes the fun part. Let's equate our two expressions for the new frequency $f'$.

$$\frac{k}{\ell_2} = \frac{k}{\ell_1} \left( \frac{v}{v - v_T} ight)$$

Notice the beautiful cancellation here! The constant $k$ completely drops out of the equation.

This means it doesn't even matter if the pipe is open at both ends or closed at one end. The physics remains identical.

Rearranging this equation to find the ratio of the lengths, we get:

$$\frac{\ell_2}{\ell_1} = \frac{v - v_T}{v} = 1 - \frac{v_T}{v}$$

This is an incredibly elegant result. The ratio of the lengths depends entirely and exactly on the ratio of the velocities. We didn't even need to use any binomial approximations!

### The Final Calculation

Finally, we need to find the percentage change in the length of the pipe.

We can find the fractional change by subtracting $1$ from our length ratio:

$$\frac{\ell_2 - \ell_1}{\ell_1} = -\frac{v_T}{v}$$

To get the percentage, we simply multiply by $100$. The negative sign just indicates that the length must decrease.

Let's plug in the given values: $v_T = 2 	ext{ ms}^{-1}$ and $v = 320 	ext{ ms}^{-1}$.

$$	ext{Percentage Change} = \left| -\frac{2}{320} ight| 	imes 100 = 0.625\%$$

The smallest value of the percentage change required is exactly **$0.625$**.

### Food for Thought

What if the tuning fork was moved away from the pipe instead? The frequency would decrease, and the length would have to increase by the exact same percentage, $+0.625\%$.

But here is a more interesting question for you to ponder: What if the pipe was moving instead of the tuning fork?

In that case, the observer would be moving, and the Doppler formula would have the velocity term in the numerator instead of the denominator.

This would actually lead to a slightly different percentage change! Try calculating it yourself; it's a fantastic variation that could easily appear in your next exam.

Enter Numerical Value:

Visualized Solution

Initial Resonance

Doppler Effect

New Resonance Condition

Equating Frequencies

Length Ratio

Percentage Change

What if the pipe moved?

The Sigma Insight: Doppler Effect

The Symphony of Resonance

The Doppler Shift

Re-tuning the Pipe

The Mathematical Elegance

The Final Calculation

Food for Thought

Similar Questions

A train is moving on a straight track with speed $20 ms^{- 1}$ . It is blowing its whistle at the frequency of $1000 Hz$ . The percentage change in the frequency heard by a person standing near the track as the train passes him is close to (speed of sound = $320 ms^{- 1}$ )

An observer moves towards a stationary source of sound, with a velocity one-fifth of the velocity of sound. What is the percentage increase in the apparent frequency?