Sources of Musical Sound — Question 1

Question 1

Pressure boundary conditions select the organ-pipe spectrum

Consider ideal tubes of length $L=0.85\,\mathrm m$ with sound speed $c=340\,\mathrm{m/s}$. An open end is a pressure node; a closed end is a particle-velocity node and therefore $p_x=0$. Neglect end corrections.

1. Derive the allowed frequencies for an open-open tube and for a closed-open tube.

2. List the first three frequencies for each and identify the first overtone of the closed-open tube.

Question 1 - Solution

Pressure boundary conditions select the organ-pipe spectrum

(a) Method and calculation

For open-open ends, a pressure mode $p\propto\sin kx$ obeys $\sin kL=0$, giving $k_n=n\pi/L$ and $f_n=nc/(2L)$. For a closed left end, $p\propto\cos kx$ automatically has $p_x(0)=0$. The open right end requires$$\cos kL=0,\quad k_m=\frac{(2m+1)\pi}{2L},\quad f_m=\frac{(2m+1)c}{4L}.$$

(b) Interpretation and further calculation

The open-open fundamental is $340/(2\times0.85)=200\,\mathrm{Hz}$, giving $200,400,600\,\mathrm{Hz}$. The closed-open fundamental is $100\,\mathrm{Hz}$, giving $100,300,500\,\mathrm{Hz}$. Its first overtone is $300\,\mathrm{Hz}$, the third harmonic; overtone order counts allowed modes, not every integer multiple.

Physical check

The ideal closed-open tube lacks even harmonics. Pressure and displacement nodes exchange roles at either type of end.