Show that in case of a closed organ pipe the ratio of frequencies of the harmonics is 1:3:5:7

To understand why the frequencies of the harmonics in a closed organ pipe are in the ratio of 1:3:5:7, let's start by examining how sound waves behave in such a pipe. A closed organ pipe is one that is closed at one end and open at the other. This configuration creates a specific pattern of standing waves, which directly influences the frequencies emitted by the pipe.

The Basics of Standing Waves

In a closed organ pipe, standing waves form due to the reflection of sound waves at the closed end. Since the closed end must be a node (where there is no movement of air), and the open end must be an antinode (where the maximum movement occurs), the pipe supports only certain wavelengths of sound. The fundamental frequency corresponds to the simplest standing wave, which has one quarter of a wavelength fitting into the length of the pipe.

Wavelengths and Frequencies

The fundamental frequency, or first harmonic, can be expressed as follows:

• First Harmonic (n=1): The wavelength (λ) is 4L, where L is the length of the pipe. The frequency (f) is given by the formula:
• f₁ = v / λ = v / (4L), where v is the speed of sound in air.

For harmonics in a closed pipe, only odd harmonics are present. This means that the second harmonic (n=2) does not exist, and we jump to the third harmonic:

• Third Harmonic (n=3): The wavelength for the third harmonic is 4L/3. Its frequency is:
• f₃ = v / (4L/3) = (3v) / (4L).

Continuing this pattern, we find:

• Fifth Harmonic (n=5): The wavelength is 4L/5, leading to:
• f₅ = v / (4L/5) = (5v) / (4L).

• Seventh Harmonic (n=7): The wavelength is 4L/7, yielding:
• f₇ = v / (4L/7) = (7v) / (4L).

Frequency Ratios

Now, let’s summarize the frequencies of these harmonics:

• f₁ = v / (4L)
• f₃ = (3v) / (4L)
• f₅ = (5v) / (4L)
• f₇ = (7v) / (4L)

To find the ratios of the harmonics, we can express the frequencies as ratios of f₁:

• f₁ : f₃ : f₅ : f₇ = (v / (4L)) : ((3v) / (4L)) : ((5v) / (4L)) : ((7v) / (4L))

This simplifies to:

• 1 : 3 : 5 : 7

Visualizing the Concept

Think of it like a musical instrument. Each harmonic can be seen as a different note that the pipe can play, similar to how a guitar string vibrates at different modes. In our case, a closed pipe only supports odd-numbered modes, which gives rise to this unique frequency pattern.

Final Thoughts

This characteristic of closed organ pipes not only dictates the sound they produce but also influences the design of various musical instruments. Understanding these principles helps musicians and acousticians create and tune instruments effectively. The ratio of 1:3:5:7 thus reveals the underlying harmony and structure in the world of sound produced by closed organ pipes.