Chapter 20: Problem 39

Question:        S in fig. 22 (chap 20) is a small loudspeaker driven by an audio oscillator and amplifier, adjustable in frequency from $1000\ {\rm Hz}$ to $2000\ {\rm Hz}$ only. D is a piece of cylindrical sheet-metal pipe $L=0.457\ {\rm m}$ long and open at both ends.

1.

At what frequencies will resonance occur when the frequency emitted by the speaker is varies between $1000\ {\rm Hz}$ to $2000\ {\rm Hz}$.

2.

Sketch the displacement modes for each resonance.

Neglect end effects.

Solution:        

The resonance condition for the pipe is (open end boundary conditions) $k L = n\pi$, where n=1,2,3... is an integer. We will take the speed of sound to be $c=343\ {\rm m/s}$The resonant frequencies are

$$\nu_n n\ {c\over 2 L}$$ =

$$\rightarrow \ \ \nu_3 = 1126.0\ {\rm Hz}\ \ \nu_4 = 1501.2\ {\rm Hz}\ \ \nu_5 = 1876.5\ {\rm Hz}\ \ .$$ (7)