Find the Length of a Tube with a 128 Hz Fundamental

The length $L$ can be found from the relationship in $f_n=n\frac{v_{\mathrm{w}}}{4L}$, but we will first need to find the speed of sound $v_{\mathrm{w}}$.

(1) Identify knowns:

• the fundamental frequency is 128 Hz
• the air temperature is $\mathrm{22.0ºC}$

(2) Use $f_n=n\frac{v_{\mathrm{w}}}{4L}$ to find the fundamental frequency ($n=1$).

(3) Solve this equation for length.

(4) Find the speed of sound using $v_{\mathrm{w}}=\left(\text{331 m/s}\right)\sqrt{\frac{T}{\text{273 K}}}$ .

(5) Enter the values of the speed of sound and frequency into the expression for $L$.

Many wind instruments are modified tubes that have finger holes, valves, and other devices for changing the length of the resonating air column and hence, the frequency of the note played. Horns producing very low frequencies, such as tubas, require tubes so long that they are coiled into loops.

(1) Identify knowns:

• the first overtone has $n=3$
• the second overtone has $n=5$
• the third overtone has $n=7$
• the fourth overtone has $n=9$

(2) Enter the value for the fourth overtone into $f_n=n\frac{v_{\mathrm{w}}}{4L}$.

Whether this overtone occurs in a simple tube or a musical instrument depends on how it is stimulated to vibrate and the details of its shape. The trombone, for example, does not produce its fundamental frequency and only makes overtones.