9.3.2 Stationary Waves in Air Column

A stationary wave (or standing wave) is formed by the superposition of two identical waves traveling in opposite directions. This interference pattern results in points of zero displacement called nodes and points of maximum displacement called antinodes.

• Transverse Stationary Waves: Can be formed on a string fixed at both ends. The particles vibrate perpendicular to the string's length.
• Longitudinal Stationary Waves: Formed in air columns (like organ pipes). Air particles oscillate parallel to the length of the pipe, creating nodes (points of no displacement/high pressure variation) and antinodes (points of maximum displacement/low pressure variation).

Closed-End Organ Pipe

In an organ pipe that is closed at one end and open at the other:

• A displacement node is always formed at the closed end.
• A displacement antinode is always formed at the open end.

Because of these boundary conditions, a closed-end pipe can support only odd harmonics (1st, 3rd, 5th, ...).

1st Harmonic (Fundamental Frequency)

• This is the simplest standing wave pattern, with one node and one antinode.
• The length of the pipe ($L$) is equal to one-quarter of a wavelength (${\lambda }_1$): $L=\frac{{\lambda }_1}{4}⟹{\lambda }_1=4L$
• Using the wave speed equation $v=f_1{\lambda }_1$, the fundamental frequency is: $f_1=\frac{v}{4L}$

3rd Harmonic (First Overtone)

• The next possible standing wave has an additional node and antinode.
• The length of the pipe contains three-quarters of a wavelength (${\lambda }_3$): $L=\frac{3{\lambda }_3}{4}⟹{\lambda }_3=\frac{4L}{3}$
• The frequency for this harmonic is: $f_3=\frac{v}{{\lambda }_3}=\frac{v}{4L/3}=\frac{3v}{4L}$
• This frequency is three times the fundamental frequency: $f_3=3f_1$

5th Harmonic (Second Overtone)

• The next pattern contains five-quarters of a wavelength (${\lambda }_5$): $L=\frac{5{\lambda }_5}{4}⟹{\lambda }_5=\frac{4L}{5}$
• The frequency for the 5th harmonic is: $f_5=\frac{v}{{\lambda }_5}=\frac{v}{4L/5}=\frac{5v}{4L}$
• This frequency is five times the fundamental frequency: $f_5=5f_1$

Open-End Organ Pipe

In an organ pipe that is open at both ends:

• A displacement antinode is formed at both open ends.

This configuration allows an open-end pipe to support all harmonics (both odd and even).

1st Harmonic (Fundamental Frequency)

• The simplest pattern has an antinode at each end and one node in the middle.
• The length of the pipe is equal to half a wavelength (${\lambda }_1$): $L=\frac{{\lambda }_1}{2}⟹{\lambda }_1=2L$
• The fundamental frequency is: $f_1=\frac{v}{2L}$

2nd Harmonic (First Overtone)

• The next standing wave has two nodes.
• The length of the pipe is equal to one full wavelength (${\lambda }_2$): $L={\lambda }_2$
• The frequency for the 2nd harmonic is: $f_2=\frac{v}{L}=\frac{v}{2L}\times 2$
• This frequency is twice the fundamental frequency: $f_2=2f_1$

3rd Harmonic (Second Overtone)

• This pattern has three nodes.
• The length of the pipe is equal to three-halves of a wavelength (${\lambda }_3$): $L=\frac{3{\lambda }_3}{2}⟹{\lambda }_3=\frac{2L}{3}$
• The frequency for the 3rd harmonic is: $f_3=\frac{v}{{\lambda }_3}=\frac{v}{2L/3}=\frac{3v}{2L}$
• This frequency is three times the fundamental frequency: $f_3=3f_1$

End Correction

In practical scenarios, the antinode does not form exactly at the open end but slightly outside it. This is known as End Correction.

Summary Table