Standing waves in closed tubes | Mechanical waves and sound | Physics | Khan Academy

Open-Open Tube

  • An open-open pipe has anti-nodes at both ends.

  • Specific wavelengths are allowed based on the formula derived from the tube length (L) and the harmonic number (N).

    • Fundamental frequency (first harmonic, N=1)

    • Second harmonic (N=2)

    • Third harmonic (N=3)

  • Formula for wavelength:

    • ( \lambda = \frac{2L}{N} )

Open-Closed Tube

  • The scenario changes significantly when one end of the tube is closed.

    • Closed end: no displacement, represents a node.

    • Open end: free to oscillate, represents an anti-node.

Fundamental Frequency (N=1)

  • First possible wavelength configuration:

    • Waveform travels from anti-node (open end) to node (closed end).

    • Identified as one-fourth of a wavelength ( ( \lambda )).

    • Relation to tube length: ( L = \frac{1}{4} \lambda ) → ( \lambda = 4L ).

Second Harmonic (N=3)

  • Configuration with one node in the middle:

    • Starting at the anti-node, oscillates to the first node and back to the open end.

    • This wave is three-fourths of a wavelength.

    • Relation: ( L = \frac{3}{4} \lambda ) → ( \lambda = \frac{4L}{3} ).

Third Harmonic (N=5)

  • Configuration with two nodes in the middle:

    • Starting at the anti-node, oscillating to a node, an anti-node, and back to the closed end.

    • Total length for L corresponds to five-fourths of a wavelength.

    • Relation: ( L = \frac{5}{4} \lambda ) → ( \lambda = \frac{4L}{5} ).

General Wave Formula for Open-Closed Tube

  • Pattern observed for wavelengths: ( \lambda = \frac{4L}{N} ) where N is the harmonic number.

  • Allowed harmonic numbers (N): only odd integers (1, 3, 5, ...)

  • Distinction from open-open tube:

    • Open-open uses all integers (N)

    • Open-closed restricts to odd integers.

Real-life Application

  • Practical experiment: blowing over an open tube (like a soda bottle).

    • As soda level decreases, the effective length (L) of the tube increases.

    • Resulting effect: decrease in frequency leads to a lower pitch sound.

    • Demonstrates the principle of wavelength relation to tube length and frequency: ( V = \lambda , F ).

Summary of Wavelengths

  • ( \lambda_{1} = 4L ) (fundamental)

  • ( \lambda_{2} = \frac{4L}{3} ) (third harmonic)

  • Fifth harmonic of an open-closed tube is calculated by taking four times the tube's length and then dividing that value by five.

  • Pattern represents that wavelengths are dictated by odd harmonics only.