Sources of Musical Sound — Question 9

Question 9

A stepped bore changes resonance without changing total length

A closed-open tube has two equal-length segments, each $L/2$, with areas $A_1$ and $A_2$. The fluid has the same $\rho,c$ throughout. Use pressure shapes $p_1=C\cos kx$ and $p_2=D\sin[k(L-x)]$ and harmonic time dependence. At the junction, pressure and volume velocity are continuous; volume velocity is proportional to area times $p_x$.

1. Derive the equation for $k$ at the junction.

2. For $A_2=4A_1$, find the fundamental frequency relative to that of a uniform closed-open tube of length $L$.

Question 9 - Solution

A stepped bore changes resonance without changing total length

(a) Method and calculation

Write $z=kL/2$. Pressure continuity gives $C\cos z=D\sin z$. Both spatial derivatives have a negative sign at the junction, so volume-velocity continuity gives$$A_1C\sin z=A_2D\cos z.$$Eliminating the coefficients yields$$\tan^2z=A_2/A_1.$$

(b) Interpretation and further calculation

The smallest positive solution for the area ratio four is $z=\tan^{-1}2$. Thus$$f=\frac{ck}{2\pi}=\frac c{\pi L}\tan^{-1}2,\qquad\frac f{c/(4L)}=\frac4\pi\tan^{-1}2=1.4097.$$The enlarged open-end segment raises the fundamental in this model. Equal total length and sound speed do not guarantee the same spectrum when bore impedance changes.

Physical check

For $A_1=A_2$, the smallest root is $z=\pi/4$, recovering the ordinary quarter-wave fundamental.