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

where, when time and/or position arguments have been dropped, as in the last line above, they are all understood to be and , respectively. To apply Newton's second law equating net force to mass times acceleration, we need the mass of the volume element

where denotes air density.

The center-of-mass acceleration of the volume element can be written
as
where
is particle velocity.^{C.15} Applying Newton's second law
, we
obtain

or, dividing through by ,

In terms of the logarithmic derivative of , this can be written

Note that denotes

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University