# Engineering Acoustics/Primary variables of interest

## Basic Assumptions

Consider a piston moving in a tube. The piston starts moving at time t=0
with a velocity u=$u_p$. The piston fits inside the tube smoothly
without any friction or gap. The motion of the piston creates a planar
sound wave or acoustic disturbance traveling down the tube at a constant
speed c\>\>$u_p$. In a case where the tube is very small, one can
neglect the time it takes for acoustic disturbance to travel from the
piston to the end of the tube. Hence, one can assume that the acoustic
disturbance is uniform throughout the tube domain.

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![](Acousticplanewave1.gif "Acousticplanewave1.gif")

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### Assumptions

1\. Although sound can exist in solids or fluid, we will first consider
the medium to be a fluid at rest. The ambient, undisturbed state of the
fluid will be designated using subscript zero. Recall that a fluid is a
substance that deforms continuously under the application of any shear
(tangential) stress.

2\. Disturbance is a compressional one (as opposed to transverse).

3\. Fluid is a continuum: infinitely divisible substance. Each fluid
property assumed to have definite value at each point.

4\. The disturbance created by the motion of the piston travels at a
constant speed. It is a function of the properties of the ambient fluid.
Since the properties are assumed to be uniform (the same at every
location in the tube) then the speed of the disturbance has to be
constant. The speed of the disturbance is the speed of sound, denoted by
letter $c_0$ with subscript zero to denote ambient property.

5\. The piston is perfectly flat, and there is no leakage flow between
the piston and the tube inner wall. Both the piston and the tube walls
are perfectly rigid. Tube is infinitely long, and has a constant area of
cross section, A.

6\. The disturbance is uniform. All deviations in fluid properties are
the same across the tube for any location x. Therefore the instantaneous
fluid properties are only a function of the Cartesian coordinate x (see
sketch). Deviations from the ambient will be denoted by primed
variables.

## Variables of interest

### Pressure (force / unit area)

Pressure is defined as the normal force per unit area acting on any
control surface within the fluid.

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![](Acousticcontrolsurface.gif "Acousticcontrolsurface.gif")

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$p = \frac {\tilde{F}.\tilde{n}}{dS}$

For the present case,inside a tube filled with a working fluid, pressure
is the ratio of the surface force acting onto the fluid in the control
region and the tube area. The pressure is decomposed into two
components - a constant equilibrium component, $p_0$, superimposed with
a varying disturbance $p^'(x)$. The deviation $p^'$is also called the
acoustic pressure. Note that $p^'$ can be positive or negative. Unit:
$kg/ms^2$. Acoustical pressure can be measured using a microphone.

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![](Acousticpressure1.gif "Acousticpressure1.gif")

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### Density

Density is mass of fluid per unit volume. The density, ρ, is also
decomposed into the sum of ambient value (usually around ρ0= 1.15 kg/m3)
and a disturbance ρ'(x). The disturbance can be positive or negative, as
for the pressure. Unit: $kg/m^3$

### Acoustic volume velocity

Rate of change of fluid particles position as a function of time. Its
the well known fluid mechanics term, flow rate.

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$U=\int_{s}\tilde{u}.\tilde{n}\, dS$

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In most cases, the velocity is assumed constant over the entire cross
section (plug flow), which gives acoustic volume velocity as a product
of fluid velocity $\tilde{u}$ and cross section S.

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$U=\tilde{u}.S$

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