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.

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.

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

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

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.



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.

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

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.

$$U=\tilde{u}.S $$

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