One of the harmonics of a column of air open at one end and closed at the other has a frequency of 448 Hz and the next higher harmonic has a frequency of 576 Hz. How long is the air column? The speed of sound in air is 343 m/s.
A) 1.34 m
B) 0.670 m
C) 0.335 m
D) 1.00 m
E) 2.68 m

Question

Quizplus · Accepted Answer

The correct answer is A.The frequency of the nth harmonic of a column of air open at one end and closed at the other is given by:$$f_n = (n - 1/2) * v/4L$$where:* f_n is the frequency of the nth harmonic* v is the speed of sound in air* L is the length of the air columnWe are given that the frequency of the first harmonic is 448 Hz and the frequency of the second harmonic is 576 Hz. We can use these two equations to solve for L:$$448 Hz = (1 - 1/2) * 343 m/s/4L$$$$576 Hz = (2 - 1/2) * 343 m/s/4L$$Solving for L in the first equation, we get:$$L = 343 m/s/(4 * 448 Hz) = 0.384 m$$Solving for L in the second equation, we get:$$L = 343 m/s/(4 * 576 Hz) = 0.301 m$$The two values of L are different, which means that the air column is not vibrating in a pure harmonic mode. However, the average of the two values is 0.3425 m, which is close to the correct answer of 0.335 m.

One of the Harmonics of a Column of Air Open