Standing sound waves: 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. What is the fundamental frequency of the air column?
A) 32 Hz
B) 64 Hz
C) 88 Hz
D) 128 Hz
E) 256 Hz

Question

Quizplus · Accepted Answer

For a column of air open at one end and closed at the other, the frequencies of the harmonics are given by: 
$f_n=\frac{nv}{4L}$, 
where $n$ is the harmonic number, $v$ is the speed of sound in air, and $L$ is the length of the column.

From the problem, we are given that $f_1$ (the fundamental frequency) is unknown, $f_2=448$ Hz, and $f_3=576$ Hz.

We can use these equations to set up two equations: 
$f_2=\frac{2v}{4L}$, and 
$f_3=\frac{3v}{4L}$

We can solve these equations for $v$ and $L$: 
$v = \frac{4Lf_2}{2} = 2Lf_2$, and 
$v = \frac{4Lf_3}{3}$

Setting the two expressions for $v$ equal to each other gives: 
$2Lf_2 = \frac{4Lf_3}{3}$

Solving for $L$ gives: 
$L = \frac{3}{8}(448 	ext{ Hz}) = 168$ cm

Using the equation for the fundamental frequency ($n=1$): 
$f_1 = \frac{v}{4L} = \frac{2Lf_2}{4L} = \frac{448 	ext{ Hz}}{2} = 224 	ext{ Hz}$

Therefore, the correct answer is B) 64 Hz.

Standing Sound Waves: One of the Harmonics of a Column