A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at

Q: Why does the density of the gas affect the speed of sound?

According to Laplace's formula, the speed of sound is inversely proportional to the square root of the gas density. A denser gas has more inertia, which slows down the propagation of the sound wave.

Q: Why do we assume pressure and gamma are constant?

The problem states the gas is at STP (Standard Temperature and Pressure), so the pressure is the same as air. We also assume it's a diatomic gas like air, so the adiabatic index ($\gamma$) remains the same.

Q: What is the difference between an open and closed organ pipe?

An open pipe has antinodes at both ends, allowing all harmonics (1st, 2nd, 3rd...). A closed pipe has a node at the closed end and an antinode at the open end, allowing only odd harmonics (1st, 3rd, 5th...).

Q: Why is the frequency difference exactly equal to the fundamental frequency?

For an open pipe, the harmonics are integer multiples of the fundamental ($f_1, 2f_1, 3f_1$). Therefore, the difference between any two consecutive harmonics is always equal to the fundamental frequency ($f_1$).

Q: How do I quickly calculate $75\sqrt{2}$ without a calculator?

Memorize the approximate value of $\sqrt{2} \approx 1.414$. Then, $75 \times 1.414 = 75 \times (1 + 0.4 + 0.014) = 75 + 30 + 1.05 = 106.05$, which is very close to the exact answer.

Question

A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is $300	ext{ m/s}$, the frequency difference between the fundamental and second harmonic of this pipe is ...... Hz.
[2020, 8 Jan Shift-I]

Sigma Percentile · Accepted Answer

### The Setup: A Pipe in a Heavy Gas

Imagine a $1	ext{ m}$ long organ pipe, open at both ends. But here is the twist—it is not filled with normal air. It is kept in a gas that is twice as dense as air at standard temperature and pressure (STP).

Before we can even think about the frequencies of the sound waves resonating inside this pipe, we must first figure out how fast sound travels in this new, denser environment.

### The Speed of Sound: Laplace's Insight

Remember Laplace's formula for the speed of sound in a gas? It tells us that the speed $v$ is given by:

$$v = \sqrt{\frac{\gamma P}{ho}}$$

Since the gas is at STP, the pressure $P$ is the same as that of air. Assuming the gas is diatomic like air, the adiabatic index $\gamma$ also remains constant. This means the speed of sound is inversely proportional to the square root of the gas density $ho$.

We can set up a beautiful ratio comparing the speed of sound in the gas to the speed of sound in air:

$$\frac{v_{gas}}{v_{air}} = \sqrt{\frac{ho_{air}}{ho_{gas}}}$$

We are given that the density of the gas is twice that of air, so $ho_{gas} = 2ho_{air}$. Substituting this into our ratio, along with the given speed of sound in air ($300	ext{ m/s}$), we get:

$$\frac{v_{gas}}{300} = \sqrt{\frac{1}{2}}$$

Solving for $v_{gas}$, we find:

$$v_{gas} = \frac{300}{\sqrt{2}} = 150\sqrt{2}	ext{ m/s}$$

### The Harmonics: Standing Waves in an Open Pipe

Now, let's look at the standing waves forming inside the pipe. For a pipe open at both ends, antinodes form at the open ends. The fundamental frequency (the first harmonic) is given by:

$$f_1 = \frac{v_{gas}}{2L}$$

The second harmonic is simply twice the fundamental frequency:

$$f_2 = \frac{v_{gas}}{L}$$

The question asks for the frequency difference between the second harmonic and the fundamental. Let's subtract them:

$$\Delta f = f_2 - f_1 = \frac{v_{gas}}{L} - \frac{v_{gas}}{2L} = \frac{v_{gas}}{2L}$$

Notice how the difference between the second harmonic and the fundamental is exactly equal to the fundamental frequency itself! This is a classic property of open organ pipes.

### The Final Crescendo: Calculating the Difference

We have our formula, and we have our values. Let's substitute $v_{gas} = 150\sqrt{2}	ext{ m/s}$ and $L = 1	ext{ m}$ into our expression for $\Delta f$:

$$\Delta f = \frac{150\sqrt{2}}{2 	imes 1} = 75\sqrt{2}$$

Using the approximate value of $\sqrt{2} \approx 1.414$, we can calculate the final numerical answer:

$$\Delta f = 75 	imes 1.414 \approx 106.06	ext{ Hz}$$

**Final Answer:** The frequency difference is **$106.06	ext{ Hz}$**.

Enter Numerical Value:

Visualized Solution

The Open Organ Pipe

Speed of Sound in a Gas

Ratio of Speeds

Calculating $v_{g a s}$

Harmonics of an Open Pipe

Frequency Difference

Final Calculation

What if the pipe was closed?

The Sigma Insight: Standing Waves in Strings and Organ Pipes

The Setup

The Speed of Sound

The Harmonics

The Final Crescendo

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