Momentum Conservation in Nonuniform Tubes
Newton's second law ``force equals mass times acceleration'' implies that
the pressure gradient in a gas is proportional to the acceleration of a
differential volume element in the gas. Let denote the area of the
surface of constant phase at radial coordinate
in the tube. Then the
total force acting on the surface due to pressure is
, as
shown in Fig.C.45.
The net force to the right across the volume element
between
and
is then
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where, when time and/or position arguments have been dropped, as in the last line above, they are all understood to be
![$ t$](http://www.dsprelated.com/josimages_new/pasp/img122.png)
![$ x$](http://www.dsprelated.com/josimages_new/pasp/img179.png)
![\begin{eqnarray*}
dM(t,x) &=& \int_x^{x+dx} \rho(t,\xi) A(\xi)\,d\xi \\ [5pt]
&\...
...\rho A' \right)\frac{(dx)^2}{2}\\ [5pt]
&\approx& \rho\, A\,dx,
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4273.png)
where denotes air density.
The center-of-mass acceleration of the volume element can be written
as
where
is particle velocity.C.16 Applying Newton's second law
, we
obtain
or, dividing through by
![$ -A\,dx$](http://www.dsprelated.com/josimages_new/pasp/img4280.png)
In terms of the logarithmic derivative of
![$ A$](http://www.dsprelated.com/josimages_new/pasp/img251.png)
Note that
![$ p$](http://www.dsprelated.com/josimages_new/pasp/img290.png)
![$ \rho$](http://www.dsprelated.com/josimages_new/pasp/img1197.png)
![$ \rho$](http://www.dsprelated.com/josimages_new/pasp/img1197.png)
Cylindrical Tubes
In the case of cylindrical tubes, the logarithmic derivative of the
area variation,
ln, is zero, and Eq.
(C.148)
reduces to the usual momentum conservation equation
encountered when deriving the wave equation for plane waves
[318,349,47]. The present case reduces to the
cylindrical case when
![$\displaystyle \frac{A'}{A} \;\ll\; \frac{p'}{p}
$](http://www.dsprelated.com/josimages_new/pasp/img4286.png)
If we look at sinusoidal spatial waves,
and
, then
and
, and the condition
for cylindrical-wave behavior becomes
, i.e., the spatial
frequency of the wall variation must be much less than that of the
wave. Another way to say this is that the wall must be approximately
flat across a wavelength. This is true for smooth horns/bores at
sufficiently high wave frequencies.
Next Section:
Wave Impedance in a Cone
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Conical Acoustic Tubes