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Volume Velocity of a Gas

The volume velocity $ U$ of a gas flow is defined as particle velocity $ u$ times the cross-sectional area $ A$ of the flow, or

$\displaystyle U(t,x) = u(t,x) A(x)
$

where $ x$ denotes position along the flow, and $ t$ denotes time in seconds. Volume velocity is thus in physical units of volume per second (m$ \null^3$ /s).

When a flow is confined within an enclosed channel, as it is in an acoustic tube, volume velocity is conserved when the tube changes cross-sectional area, assuming the density $ \rho$ remains constant. This follows directly from conservation of mass in a flow: The total mass passing a given point $ x$ along the flow is given by the mass density $ \rho$ times the integral of the volume volume velocity at that point, or

$\displaystyle M(t_0:t_f,x) = \rho\int_{t_0}^{t_f} U(t,x) dt.
$

As a simple example, consider a constant flow through two cylindrical acoustic tube sections having cross-sectional areas $ A_1$ and $ A_2$ , respectively. If the particle velocity in cylinder 1 is $ u_1$ , then the particle velocity in cylinder 2 may be found by solving

$\displaystyle u_1 A_1 = u_2 A_2
$

for $ u_2$ .

It is common in the field of acoustics to denote volume velocity by an upper-case $ U$ . Thus, for the two-cylinder acoustic tube example above, we would define $ U_1\isdeftext u_1A_1$ and $ U_2\isdeftext u_2A_2$ , so that

$\displaystyle U_1 = U_2
$

would express the conservation of volume velocity from one tube segment to the next.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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