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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

 (C.143)

or, dividing through by ,

 (C.144)

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

 ln (C.145)

Note that denotes small-signal acoustic pressure, while denotes the full gas density (not just an acoustic perturbation in the density). We may therefore treat as a constant.

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