PHYSICS HELP - resonant lengths, air columns, etc.?

Question

Find the second resonant length of an air column that resonates with a sound of frequency 1.0 kHz. at 15.0 degrees C under each of the following conditions:

a) The air column is closed at both ends.

b) The air column is open at both ends.

c) The air column is open at one end and closed at another.

Answer 1

PHYSICS HELP - resonant lengths, air columns, etc.?
Find the second resonant length of an air column that resonates with a sound of frequency 1.0 kHz. at 15.0 degrees C under each of the following conditions:

a) The air column is closed at both ends.

**Here, there both ends have maximum pressure fluxuations, so both are "anti-nodes." The first frequency that can fit is 1/2 a wavelength, this is the resonant frequency. The second frequency that can fit where both ends are anti-nodes is 1 wavelength.

Speed of sound at 15 deg C = 340.3 m/sec . . . ref: http://en.wikipedia.org/wiki/Speed_of_sound

Wavelength = v/frequ

=340.3 m/sec / 1000/sec
=.340 m


b) The air column is open at both ends.

**Here, there is no pressure at either end, so both are "nodes." The first frequency that can fit is 1/2 a wavelength, this is the resonant frequency. The second frequency that can fit where both ends are nodes is 1 wavelength. Same answer as "a," above.

c) The air column is open at one end and closed at another.

Here, you have one node and one anti-node. This can resonate at a length of 1/4 the wave length. The second resonance would be at 1/2 the wave length.

Since we know the wavelength from "a," above, we divide by 2 to get our answer:

1/2 Wavelegth = .340m/2
=.170 m