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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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Particle Velocity of a Gas