Sound Waves: Resonance

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Sound Waves: Resonance

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Resonance is a special case of forced oscillation in a system, wherein the first object's driving frequency matches the second object's natural frequency, forcing the second object to vibrate at the same frequency with a significantly higher amplitude.

Consider a tuning fork and a hollow cylindrical tube closed at one end. When the tines of the tuning fork vibrate at their natural frequency, a sound wave is generated, which impinges upon the entrance of the tube, forcing the air inside the tube to vibrate at the same frequency. If the tuning fork's natural frequency matches the air column's normal modes, resonance results in a louder sound.

A similar thing happens for a tube open at both ends. The air columns in the tube have maximum air displacements at both ends, and thus, the reflections of the sound at each end produce a large-amplitude wave for particular frequencies.

The resonant frequencies for a tube open at both ends follow the same equation as that of a standing wave on a string fixed at both ends.

Sound Waves: Resonance

Resonance is produced depending on the boundary conditions imposed on a wave. Resonance can be produced in a string under tension with symmetrical boundary conditions (i.e., has a node at each end). A node is defined as a fixed point where the string does not move. The symmetrical boundary conditions result in some frequencies resonating and producing standing waves, while other frequencies interfere destructively. Sound waves can resonate in a hollow tube, and the frequencies of the sound waves that resonate depend on the boundary conditions.

For instance, consider a tube that is closed at one end and open at the other. If a vibrating tuning fork is placed near the open end of the tube, an incident sound wave travels through the tube and reflects off the closed end. The reflected sound has the same frequency and wavelength as the incident sound wave but travels in the opposite direction. At the closed end of the tube, the air molecules have very little freedom to oscillate, and a node arises. At the open end, the molecules are free to move, and at the right frequency, an antinode occurs.

Unlike the symmetrical boundary conditions for standing waves on a string, the boundary conditions for a tube open at one end and closed at the other end are anti-symmetrical; there is a node at the closed end and an antinode at the open end. If the tuning fork has the right frequency, the air column in the tube resonates loudly. Thus, it vibrates very little at most frequencies, suggesting that the air column has only specific natural frequencies.

In another instance, if the standing waves travel through a tube that is open at both ends, the boundary conditions are symmetrical—an antinode at each end. The resonance in tubes open at both ends can be analyzed similarly to those for tubes closed at one end. The air columns in tubes open at both ends have maximum air displacements at both ends.

This text is adapted from Openstax, University Physics Volume 1, Section 17.4: Normal Modes of a Standing Sound Wave.