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Acoustic Vector Sensing |
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Undergraduate Research by Ian Beil and Evan Nixon |
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Physical Properties of Acoustic Fields |
In order to determine the direction of arrival of an acoustic source, we first need to develop a model of the sound waves which the AVS will detect. For the purposes of our research, we assumed that each acoustic source propagated as a uniform plane wave. The model of an acoustic wave in three dimensions is derived from the following three equations:
· Conservation of mass in a fluid · Acceleration towards decreasing pressure in a fluid · Pascal’s Law
The following discusses the derivation and physical meaning of each equation in detail. In the analysis, ρ(x,t) denotes the density at location x and time t, υ(x,t) denotes the fluid velocity at location x and time t, and n(x) denotes the normal vector at location x.
Conservation of mass in a fluid
Starting with the definition of conservation of mass:
Using Gauss’s Equation:
Rearranging terms, we get the conservation of mass in a fluid:
Acceleration towards decreasing pressure in a fluid
Starting with the fact that the time derivative of momentum is force:
Realizing that the gradient of pressure is force divided by volume:
By substituting these equations, we obtain Reynolds’s Transport Theorem:
After defining the Dv/Dt operator, we prove that the acceleration of a fluid is always in the decreasing of greatest decrease in pressure:
Pascal’s Equation
Pascal’s Equation provides an estimate for the relationship between pressure and density of a fluid, where m is dependent on the fluid and the temperature:
For air at room temperature, m is roughly equal to 1, so we obtain the equation:
Wave Equation Combining these equations and using Taylor approximation of first order we obtain the following relationship:
p(r,t) – pressure at position r and time t v(r,t) – particle velocity at position r and time t u - [cos(ϕ) cos(Ψ) sin(ϕ) cos (Ψ), sin(Ψ)]T c – speed of sound ρ0 – ambient pressure |
