A sonometer wire under a tension of 64 N vibrating in its fundamental mode is in resonance with a vi

Q: Why is the beat frequency $f - f'$ and not $f' - f$?

Because the tuning fork is moving away from the stationary observer, its apparent frequency ($f'$) must be lower than its natural frequency ($f$). Since $f' < f$, the absolute difference $|f - f'|$ is written as $f - f'$ to ensure a positive value.

Q: Why does the speed of sound in air remain $300\text{ m/s}$ even though the tuning fork is moving?

The speed of sound is a property of the medium (air) through which it propagates. It depends on factors like temperature and density of the air, but is completely independent of the motion of the source or the observer.

Q: What would happen if the tuning fork moved towards the observer?

If the tuning fork moved towards the observer, its apparent frequency would increase ($f' > f$). The beat frequency would then be $f_b = f' - f = 1\text{ Hz}$, which gives $f' = 401\text{ Hz}$. The Doppler formula would also change to $f' = f \left( \frac{v}{v - v_s} \right)$.

Q: Why did we use the fundamental frequency formula $f = \frac{v}{2l}$ for the sonometer wire?

A sonometer wire is clamped at both ends, which means nodes are formed at the boundaries. In the fundamental mode of vibration, the wire forms a single loop with one antinode in the middle, corresponding to half a wavelength ($\lambda/2 = l$). Thus, $\lambda = 2l$, and using $f = v/\lambda$ gives $f = v/2l$.

Q: Does the motion of the tuning fork affect the frequency of the sonometer wire?

No, the sonometer wire is stationary. Its frequency of vibration is determined solely by its own physical properties: tension, length, and mass density. The motion of the tuning fork only affects the frequency of the sound waves emitted by the fork itself as perceived by the observer.

Question

A sonometer wire under a tension of $64	ext{ N}$ vibrating in its fundamental mode is in resonance with a vibrating tuning fork. The vibrating portion of the sonometer wire has a length of $10	ext{ cm}$ and mass of $1	ext{ g}$. The vibrating tuning fork is now moved away from the vibrating wire with a constant speed and an observer standing near the sonometer hears one beat per second. Calculate the speed with which the tuning fork is moved, if the speed of sound in air is $300	ext{ m/s}$.

Sigma Percentile · Accepted Answer

### Introduction: The Symphony of Waves

Imagine standing in a quiet room where two distinct sources of sound are playing.

One is a perfectly tuned sonometer wire, humming steadily.

The other is a vibrating tuning fork, moving slowly away from you.

As the fork recedes, a strange, rhythmic pulsing fills the air—a gentle "wah-wah-wah" sound.

This is the magic of **beats** and the **Doppler effect** working in unison.

In this journey, we will dissect this acoustic phenomenon step-by-step and find the exact speed of that receding tuning fork.

### Phase 1: Decoding the Sonometer Wire

Before we can understand the moving tuning fork, we must first master the stationary source: the sonometer wire.

The wire is stretched under a tension of $T = 64	ext{ N}$.

It has a length of $l = 10	ext{ cm}$ and a mass of $m = 1	ext{ g}$.

To find how waves travel along this wire, we must first calculate its **linear mass density**, $\mu$.

$$\mu = \frac{m}{l} = \frac{10^{-3}	ext{ kg}}{0.1	ext{ m}} = 10^{-2}	ext{ kg/m}$$

This tells us that every single meter of this wire weighs exactly $10	ext{ grams}$.

Now, the speed of a transverse wave on a stretched string depends on both the tension pulling it tight and its linear mass density.

$$v_{	ext{wire}} = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{64}{10^{-2}}} = 80	ext{ m/s}$$

With a wave speed of $80	ext{ m/s}$, we can now find the fundamental frequency of vibration.

Since the wire is vibrating in its fundamental mode, it forms a single loop with nodes at both ends.

This means the length of the wire is exactly half of the wavelength ($\lambda = 2l$).

$$f = \frac{v_{	ext{wire}}}{2l} = \frac{80}{2 	imes 0.1} = 400	ext{ Hz}$$

The sonometer wire is humming at a pristine frequency of **$400	ext{ Hz}$**.

### Phase 2: The Power of Resonance

The problem tells us that the tuning fork is initially in resonance with the sonometer wire.

**Resonance** is one of nature's most beautiful phenomena.

It occurs when the frequency of an external driving force matches the natural frequency of the system it acts upon, leading to large-amplitude oscillations.

Because they are in resonance, the natural frequency of the tuning fork must be identical to the fundamental frequency of the wire.

$$f_{	ext{fork}} = 400	ext{ Hz}$$

Now we have our two sound sources: the sonometer wire vibrating at $400	ext{ Hz}$, and the tuning fork also vibrating at $400	ext{ Hz}$.

### Phase 3: Enter the Doppler Effect

Now, the tuning fork begins to move away from the stationary observer.

As the source of sound recedes, each successive wave crest is emitted from a position further away from the observer than the previous crest.

This stretches the distance between wave crests, effectively increasing the wavelength.

Because the wavelength increases while the speed of sound in air remains constant, the frequency perceived by the observer, $f'$, must decrease.

This is the **Doppler effect**.

We can express this mathematically using the classic Doppler formula:

$$f' = f \left( \frac{v}{v + v_s} ight)$$

Here, $v = 300	ext{ m/s}$ is the speed of sound in air, and $v_s$ is the speed of the moving tuning fork.

Since the denominator ($v + v_s$) is larger than the numerator ($v$), the apparent frequency $f'$ will indeed be less than the original frequency $f$.

### Phase 4: The Phenomenon of Beats

The observer is now receiving two sound waves simultaneously: one from the stationary sonometer wire at $f = 400	ext{ Hz}$, and one from the moving tuning fork at $f' < 400	ext{ Hz}$.

When two sound waves of slightly different frequencies superimpose, they interfere constructively and destructively over time.

This periodic variation in intensity is heard as **beats**.

The number of beats heard per second, known as the beat frequency $f_b$, is simply the absolute difference between the two frequencies.

$$f_b = |f - f'| = 1	ext{ Hz}$$

Since $f'$ is less than $f$, we can write:

$$f' = f - f_b = 400 - 1 = 399	ext{ Hz}$$

The observer hears the moving tuning fork at a slightly lower pitch of **$399	ext{ Hz}$**.

### Phase 5: Solving the Master Equation

We now have all the pieces of our puzzle.

Let us substitute our values into the Doppler equation and solve for the source velocity, $v_s$.

$$399 = 400 \left( \frac{300}{300 + v_s} ight)$$

Let us solve this elegantly.

Divide both sides by $400$:

$$\frac{399}{400} = \frac{300}{300 + v_s}$$

Taking the reciprocal of both sides gives:

$$\frac{300 + v_s}{300} = \frac{400}{399}$$

We can simplify the left side:

$$1 + \frac{v_s}{300} = 1 + \frac{1}{399}$$

Subtracting $1$ from both sides yields:

$$\frac{v_s}{300} = \frac{1}{399}$$

Finally, solving for $v_s$:

$$v_s = \frac{300}{399} \approx 0.75	ext{ m/s}$$

The tuning fork is moving away at a gentle speed of **$0.75	ext{ m/s}$**.

### Conclusion: The Beauty of Wave Mechanics

Look at how beautifully different concepts of wave mechanics—standing waves, resonance, the Doppler effect, and beats—interlock to solve this problem!

By understanding the physical mechanism behind each step, we turn a complex-looking problem into a simple, logical sequence.

Keep exploring, keep asking questions, and let the physics of waves continue to amaze you!

Visualized Solution

Visualizing the Setup

Identifying Sonometer Parameters

Calculating Linear Mass Density $μ$

Wave Velocity on the Wire

Fundamental Frequency of Sonometer

Frequency of the Tuning Fork

Understanding the Doppler Shift

The Doppler Formula

Relating Beats to Frequencies

Finding Apparent Frequency $f^{'}$

Substituting into the Doppler Equation

Solving for Source Velocity $v_{s}$

The Way Forward

The Sigma Insight: Doppler Effect

Introduction

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

Conclusion

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