Introduction to Physical Acoustics

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1 Introduction to Physical Acoustics

2 Class webpage CMSC 828D: Algorithms and systems for capture and playback of spatial audio. Send me a test message with the subject cmsc828d

3 Goals Physical Acoustics is the branch of physics studying propagation of sound Our goals: understand some background material about sound propagation

4 Fluid Mechanics 101 Properties of Matter Density ρ Pressure p Compressibility (dp/dρ) viscosity Conservation Laws Mass is conserved (in the absence of sources) Momentum is conserved (F=Ma) Energy is conserved Three Conservation Laws describe how imposed changes affect a fluid Treat the fluid as a continuum subject to the equations of continuum mechanics Equations governing acoustics will be a special (simpler) case of these equations

5 Mathematical Modeling One of the extraordinary successes of the 19 th and 20 th centuries is the development of mathematical models to predict the behavior of fluid and solid media Aircraft, automobile, buildings, mechanical design of all products, engines etc. based on this understanding

6 Conservation of Mass Derivation Consider a box of size x y z through which fluid flows It has a density ρ (x) and the flow vector u=(u,v,w)

7 Fluid element and properties The behavior of the fluid is described in terms of macroscopic properties: Velocity u. Pressure p. Density ρ. Temperature T. Energy E. Typically ignore (x,y,z,t) in the notation. Properties are averages of a sufficiently large number of molecules. A fluid element can be thought of as the smallest volume for which the continuum assumption is valid. Fluid element for conservation laws y y z (x,y,z) x Faces are labeled North, East, West, South, Top and Bottom Properties at faces are expressed as first two terms of a Taylor series expansion, p 1 p 1 e.g. for p : pw = p x and pe = p+ x x 2 x 2 x z

8 Mass balance Rate of increase of mass in fluid element equals the net rate of flow of mass into element. Rate of increase is: The inflows (positive) and outflows (negative) are shown here: z y x t z y x t ρ ρ = ) ( x y z ( ) 1. 2 w w z x y z ρ ρ + ( ) 1. 2 v v y x z y ρ ρ + z y x x u u ρ ρ 2 1. ) ( z x y y v v ρ ρ 2 1. ) ( y x z z w w ρ ρ 2 1. ) ( ( ) 1. 2 u u x y z x ρ ρ +

9 Mass Conservation ( Continuity ) equation Summing all terms in the previous slide and dividing by the volume xyz results in: ρ + t ( ρu) x + ( ρv) y + ( ρw) z = 0 In vector notation: ρ + u ρ + ρ u = 0 t Creation of mass Change in density Convective term: flow of mass out For incompressible constant property fluids ρ / t =0, and ρ = 0 the equation becomes: div u = 0. u Alternative ways to write this: i = 0 x i

10 Rate of change for a stationary fluid element In most cases we are interested in the changes of a flow property for a fluid element, or fluid volume, that is stationary in space. However, some equations are easier derived for fluid particles. For a moving fluid particle, the total derivative per unit volume of this property φ is given by: (for moving fluid particle) Dφ φ ρ = ρ + u. grad φ Dt t (for given location in space) For a fluid element, for an arbitrary conserved property φ: ρ + div ( ρ u) = 0 t Continuity equation ( ρφ) + div ( ρφ u) = t Arbitrary property 0

11 Relevant entries for Φ x-momentum y-momentum z-momentum Energy u v w E Du ( ρu) ρ + div( ρuu) Dt t Dv ( ρ v) ρ + div( ρvu) Dt t Dw ( ρw) ρ + div( ρwu) Dt t DE ( ρe) ρ + div( ρeu) Dt t

12 Conservation Laws Equations in a gas (like air) Variables ρ density p pressure u velocity T temperature Constants C p is heat capacity Mass ρ + u ρ + ρ u = 0, t Momentum for an inviscid uid ρ u + ρu (u) + p = 0 t Energy (neglecting heat conduction) µ T ρc p t + u (ρt ) Dp Dt = 0. Equation of state relates three of the quantities p ρ = RT

13 Conservation Laws Eliminate ρ Introduce the short hand notation Yields the system of equations ρ t = D Dt = t µ p RT t + u γ is the ratio of specific heats for the gas 1 Dp p Dt 1 T p RT DT Dt + u = 0 Du Dt + p = 0 1 T DT Dt γ 1 γp Dp Dt = 0

14 Equations of Acoustics Acoustics govern the propagation of small perturbations through the system Let the system be in equilibrium with pressure p 0, Temperature T 0, and zero velocity Then a small disturbance p will upset the equilibrium Using conservation laws, we can derive the equations of acoustics that govern the propagation of sound waves in the medium. Key assumption ( acoustic approximation ) All perturbations are much smaller than equilibrium values

15 Acoustic equations Equations under these assumptions are\ Can eliminate T from the system 1 p γp 0 t + u = 0 p 0 RT 0 u t + p = 0 1 p p 0 t 1 T T 0 t + u = 0 p 0 u RT 0 t + p = 0 1 T 0 T t γ 1 γp 0 p t = 0

16 The wave equation Almost there Differentiate first equation with respect to time Substitute for u / t from second equation 1 2 p γp 0 t 2 + u t = 0 u t = 1 p ρ p c 2 p = 0 t2

17 Laplace operator = 2 is the Laplace operator (Divergence of gradient) Extremely common in partial differential equations So the wave equation can be written as 1 2 p c 2 t 2 2 p =0

18 Wave equation for the Velocity potential Let φ be the velocity potential so that u= φ Then the conservation equations become From these we can eliminate p to arrive at the wave equation for the velocity potential 1 p γp 0 t + 2 φ =0 Ã! p0 φ RT 0 t + p =0 1 c 2 2 p t φ =0

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