Superposition of Waves - Stationary Waves in Organ Pipe

Introduction

The superposition of waves is a fundamental concept in physics that describes the behavior of waves as they interact with each other. When two or more waves meet, they combine to form a new wave pattern. This principle applies to all types of waves, including sound waves, which are of particular interest when discussing organ pipes.

In an organ pipe, sound waves reflect back and forth, creating standing waves or stationary waves. These standing waves are the result of the superposition of two waves traveling in opposite directions with the same frequency.

Superposition Principle

The superposition principle states that when two or more waves overlap, the resultant wave displacement at any point is the algebraic sum of the displacements of the individual waves at that point.

Mathematically, if wave 1 has a displacement $y_1(x,t)$ and wave 2 has a displacement $y_2(x,t)$, the resultant wave $y(x,t)$ is given by:

$$ y(x,t) = y_1(x,t) + y_2(x,t) $$

Stationary Waves in Organ Pipes

When a sound wave is produced in an organ pipe, it reflects off the ends of the pipe and interferes with incoming waves. Depending on the boundary conditions (open or closed ends), stationary waves with specific patterns, known as modes or harmonics, are formed.

Open Pipe

An open pipe has both ends open to the air. The boundary condition here is that the pressure at both ends must be equal to the atmospheric pressure, leading to antinodes (points of maximum displacement) at both ends.

Closed Pipe

A closed pipe has one end closed and the other open. The closed end is a displacement node (point of zero displacement) because the air cannot move at the rigid boundary, while the open end is an antinode.

Formulas for Stationary Waves in Organ Pipes

The frequency of the nth harmonic in an organ pipe can be calculated using the following formulas:

Open Pipe

For an open pipe, the wavelength $\lambda_n$ of the nth harmonic is given by:

$$ \lambda_n = \frac{2L}{n} $$

where $L$ is the length of the pipe and $n$ is an integer representing the harmonic number (1, 2, 3, ...). The frequency $f_n$ is then:

$$ f_n = \frac{v}{\lambda_n} = \frac{nv}{2L} $$

where $v$ is the speed of sound in air.

Closed Pipe

For a closed pipe, only odd harmonics are present. The wavelength $\lambda_n$ of the nth harmonic (n = 1, 3, 5, ...) is:

$$ \lambda_n = \frac{4L}{n} $$

The frequency $f_n$ is:

$$ f_n = \frac{v}{\lambda_n} = \frac{nv}{4L} $$

Differences Between Open and Closed Pipes

Feature	Open Pipe	Closed Pipe
Ends	Both open	One closed, one open
Harmonics	All harmonics (1, 2, 3, ...)	Only odd harmonics (1, 3, 5, ...)
Fundamental Frequency	$f_1 = \frac{v}{2L}$	$f_1 = \frac{v}{4L}$
Node/Antinode at Ends	Antinode/Antinode	Node/Antinode

Examples

Example 1: Fundamental Frequency of an Open Pipe

An open organ pipe is 0.5 meters long. If the speed of sound is 340 m/s, what is the fundamental frequency?

Using the formula for an open pipe:

$$ f_1 = \frac{v}{2L} = \frac{340 \text{ m/s}}{2 \times 0.5 \text{ m}} = 340 \text{ Hz} $$

Example 2: Third Harmonic of a Closed Pipe

A closed organ pipe is 0.25 meters long. What is the frequency of the third harmonic?

Since only odd harmonics are present in a closed pipe, the third harmonic (n=3) frequency is:

$$ f_3 = \frac{3v}{4L} = \frac{3 \times 340 \text{ m/s}}{4 \times 0.25 \text{ m}} = 1020 \text{ Hz} $$

Conclusion

Understanding the superposition of waves and the formation of stationary waves in organ pipes is crucial for comprehending the physics of sound. The behavior of sound waves in open and closed pipes leads to the creation of musical notes with specific frequencies, which are determined by the length of the pipe and the speed of sound. By applying the principles of wave superposition and the formulas for stationary waves, one can predict the harmonic frequencies produced by organ pipes.