Resonant Air Columns Resonance occurs when the natural

Resonant Air Columns

• Resonance occurs when the natural vibration rates of two objects are the same or when the vibration rate of one of them is equal to one of the harmonics of the other.

• Resonance in Closed Tubes • During resonance, a compression of the reflected wave unites with a compression of the direct wave, and a rarefaction of the reflected wave unites with a rarefaction of the direct wave. The constructive superposition of the two waves in phase amplifies the sound. • The resonant air column is simply a standing longitudinal wave system.

• The water surface closes the lower end of the tube and prevents longitudinal displacement of the molecules of the air immediately adjacent to it. • This termination effectively clamps the air column at the closed end and gives rise to a displacement node. Because the upper end of the tube is open, it provides a free-ended termination for the air column and gives rise to a displacement antinode, or loop.

• The fundamental frequency of the resonant column corresponds approximately to a displacement node at the closed end an adjacent displacement loop at the open end. • Since the distance separating a node and an adjacent loop of a standing wave is one-fourth wavelength, the length of the closed tube is approximately one fourth the wavelength of its fundamental resonant frequency.

• By applying a small empirical correction proportional to the diameter of the tube (because the motion of molecules of the air at the open end of the tube is not strictly in one dimension), we can state the relationship more precisely as follows: • λ = 4 L + 1. 6 d

Resonance in Open Tubes • What if, instead we have a tube open at both ends? • Compressions are reflected as rarefactions and vice versa.

• The fundamental frequency of a resonant air column in an open tube corresponds approximately to displacement loops at opposite ends and a displacement node in the middle.

• The relationship between wavelength, the length of the open tube, and the diameter of the open tube is given by the formula: • λ = 2 L + 1. 6 d

• For a closed tube, resonance only occurs for odd harmonics. • For an open tube, resonance occurs for all harmonics.

See the following website • Read section 11. 15 at… • http: //boomeria. org/physicstextbook/ch 11. html