A musician produce the sound of second harmonics from open end flute of 50 cm. The other person move

Q: Why is the formula for the second harmonic of an open pipe $v/L$ and not $v/2L$?

For an open pipe, the fundamental frequency (first harmonic) is $v/2L$. The second harmonic is simply twice the fundamental frequency, so $2 \times (v/2L) = v/L$.

Q: Why do we use a positive sign in the numerator of the Doppler formula?

When the observer moves towards the source, they encounter more wave crests per unit time, which increases the apparent frequency. The positive sign in the numerator mathematically ensures the resulting frequency $f'$ is greater than $f_s$.

Q: What if the flute was closed at one end?

If the flute were closed at one end, it would only produce odd harmonics. The fundamental would be $v/4L$, and the next available harmonic (the third harmonic) would be $3v/4L$.

Q: Why did we have to convert the observer's speed to m/s?

In physics equations, all quantities must be in consistent units. Since the speed of sound is given in m/s, the observer's speed must also be in m/s to correctly calculate the dimensionless ratio in the Doppler formula.

Q: Does the speed of the sound wave itself change because the observer is running?

No, the speed of sound relative to the medium (air) remains constant at 330 m/s. The observer's motion only changes the relative speed at which the wave fronts hit their ear, altering the perceived frequency, not the actual wave speed in the air.

Question

A musician produce the sound of second harmonics from open end flute of 50 cm. The other person moves toward the musician with speed 10 km/h from the second end of room. If the speed of sound 330 m/s, the frequency heard by running person will be

Options:

(A) 666 Hz

(B) 500 Hz

(C) 753 Hz

(D) 333 Hz

Sigma Percentile · Accepted Answer

### The Symphony of Physics: Setting the Stage

Imagine you are standing in a large, quiet room. At one end, a musician is playing a flute. But this isn't just any melody; the musician is skillfully producing the **second harmonic** of the flute. You, the eager listener, decide to run towards the musician to get a closer look, moving at a steady speed of $10 	ext{ km/h}$.

As you run, the pitch of the note you hear seems slightly higher than what the musician is actually playing. This fascinating phenomenon is a beautiful interplay of two fundamental concepts in wave physics: the resonance of air columns and the **Doppler Effect**.

In this problem, we are tasked with finding the exact frequency that you, the running observer, will hear. Let's break down this symphony into its atomic mathematical steps.

### The Flute's Melody: Finding the Source Frequency

Before we can figure out what you hear, we must first determine what the flute is actually emitting. The flute acts as an **open organ pipe**—a tube open at both ends.

When air is blown into an open pipe, it sets up standing waves. The fundamental frequency (or first harmonic) is formed when the length of the pipe $L$ is equal to half a wavelength ($\lambda/2$). Therefore, the fundamental frequency is $f_1 = \frac{v}{2L}$, where $v$ is the speed of sound.

However, our musician is playing the **second harmonic**. The second harmonic has a frequency twice that of the fundamental:
$$f_2 = 2 	imes f_1 = 2 	imes \left( \frac{v}{2L} ight) = \frac{v}{L}$$

We are given the length of the flute $L = 50 	ext{ cm} = 0.5 	ext{ m}$ and the speed of sound $v = 330 	ext{ m/s}$. Let's substitute these values into our master equation for the source frequency $f_s$:
$$f_s = \frac{330}{0.5} = 660 	ext{ Hz}$$

The flute is steadily broadcasting sound waves at a frequency of $660 	ext{ Hz}$. If you were standing still, this is exactly what you would hear. But you are running!

### The Doppler Effect: A Moving Observer

As you run towards the flute, you are rushing into the oncoming sound waves. Because of your motion, you encounter wave crests more frequently than if you were stationary. This relative increase in the rate of wave interception causes you to perceive a higher pitch. This is the essence of the **Doppler Effect**.

The general formula for the Doppler effect when the source is stationary and the observer is moving is:
$$f' = f_s \left( \frac{v \pm v_o}{v} ight)$$

Here, $f'$ is the apparent frequency, $v$ is the speed of sound, and $v_o$ is the speed of the observer.

**Sign Convention Check:** Since you are moving *towards* the source, the apparent frequency must increase. To make the fraction greater than $1$, we must use the **positive sign** in the numerator.
$$f' = f_s \left( \frac{v + v_o}{v} ight)$$

### The Final Calculation: Mind the Units!

Before we plug in the numbers, there is a classic trap waiting for us: inconsistent units. The speed of sound is given in $	ext{m/s}$, but your running speed is given in $	ext{km/h}$. We must convert $v_o$ to $	ext{m/s}$ by multiplying by the conversion factor $\frac{5}{18}$:
$$v_o = 10 	imes \frac{5}{18} = \frac{25}{9} 	ext{ m/s}$$

Now, let's substitute all our known values into the Doppler equation:
$$f' = 660 \left( \frac{330 + \frac{25}{9}}{330} ight)$$

To solve this elegantly without a calculator, let's distribute the denominator:
$$f' = 660 \left( 1 + \frac{25}{9 	imes 330} ight)$$
$$f' = 660 + 660 	imes \left( \frac{25}{2970} ight)$$

Notice how the $660$ and $2970$ can be simplified. Dividing both by $330$ gives $2$ and $9$ respectively:
$$f' = 660 + 2 	imes \left( \frac{25}{9} ight)$$
$$f' = 660 + \frac{50}{9}$$

Since $\frac{50}{9}$ is approximately $5.55$, we get:
$$f' = 660 + 5.55 = 665.55 	ext{ Hz}$$

Rounding to the nearest integer, the frequency you hear as you run towards the musician is **$666 	ext{ Hz}$**.

Through a beautiful combination of standing wave mechanics and relative kinematics, we've successfully decoded the physics of the running listener!

Animated Solution for Physics - Waves: A musician produce the sound of second harmonics from open end flute of 50 cm. The other person moves toward the musician with speed 10 km/h from the second end of room. If the speed of sound 330 m/s, the frequency heard by running person will be

Select Answer:

Visualized Solution

Visualizing the Setup

Formula for $f_{2}$

Substituting Values for $f_{s}$

Calculating $f_{s}$

Doppler Effect Formula

Substituting Values for $f^{'}$

Simplifying the Expression

Final Apparent Frequency $f^{'}$

The Sigma Insight: Doppler Effect

The Symphony of Physics

The Flute's Melody

The Doppler Effect

The Final Calculation

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