Introduction to Physical Acoustics Class webpage

• CMSC 828D: Algorithms and systems for capture and playback of spatial audio. www.umiacs.umd.edu/~ramani/cmsc828d_audio

• Send me a test email message with the subject cmsc828d Goals • Physical Acoustics is the branch of physics studying propagation of sound • Our goals: understand some background material about sound propagation 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 Mathematical Modeling

• One of the extraordinary successes of the 19th and 20th 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 Conservation of Mass

• Derivation • Consider a box of size δx × δy × δz through which fluid flows • It has a density ρ (x) and the flow vector u=(u,v,w) Fluid element and properties Fluid element for conservation laws • The behavior of the fluid is described in terms of macroscopic properties: – Velocity u. – Pressure p. (x,y,z) δz – Density ρ.

– Temperature T. δy – Energy E. δx z • Typically ignore (x,y,z,t) in the notation. y • Properties are averages of a sufficiently x large number of molecules. Faces are labeled • A fluid element can be thought of as the North, East, West, smallest volume for which the continuum South, Top and Bottom assumption is valid. Properties at faces are expressed as first two terms of a Taylor series expansion, ∂∂pp11 e.g. for p :p =−pxppxδ and =+ δ WE∂∂xx22 Mass balance

• Rate of increase of mass in fluid element equals the net rate of flow of mass into element. ρδ ∂ρ δ • Rate of increase is: ∂ ( xδyδz) = xδyδz ∂t ∂t • The inflows (positive) and outflows (negative) are shown here:

⎛⎞∂()1ρw ⎜⎟ρwzx+ . δδδy ⎝⎠∂z 2 ⎛⎞∂()ρv 1 ⎜⎟ρvyxz+ . δδδ ⎝⎠∂y 2

⎛⎞∂()ρu 1 ⎛ ∂(ρu) 1δ ⎞ ⎜⎟ρux+ . δδδy z ⎜ ρu − . x⎟ δyδz ⎝⎠∂x 2 ⎝ ∂x 2 ⎠ ρ ρ δ ⎛ ∂( v) 1 ⎞ δ ⎜ v − . y⎟ xδz z ⎝ ∂y 2 ⎠ y ⎛ ∂(ρw) 1δ ⎞ x ⎜ ρw − . z⎟ δxδy ⎝ ∂z 2 ⎠ Mass Conservation (“Continuity”) equation

• Summing all terms in the previous slide and dividing by the volume δxδyδz results in:

∂ρ + ∂(ρu) + ∂(ρv) + ∂(ρw) = 0 ∂t ∂x ∂y ∂z

• In vector notation: ∂ρ + uu⋅∇ρρ + ∇⋅ = 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 ∂xi 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: Dφ ⎛ ∂φ ⎞ (for moving fluid particle) ρ = ρ ⎜ + u.grad φ ⎟ (for given location in space) Dt ⎝ ∂t ⎠

• For a fluid element, for an arbitrary conserved property φ:

∂ρ ∂(ρφ) + div (ρ u) = 0 + div (ρφ u) = 0 ∂t ∂t Continuity equation Arbitrary property Relevant entries for Φ

Du ∂(ρu) x-momentum u ρ + div(ρuu) Dt ∂t

Dv ∂(ρv) y-momentum v ρ + div(ρvu) Dt ∂t

Dw ∂(ρw) z-momentum w ρ + div(ρwu) Dt ∂t

DE ∂(ρE) Energy E ρ + div(ρEu) Dt ∂t

Conservation Laws Mass • Equations in a ∂ρ + u ρ + ρ u =0, gas (like air) ∂t · ∇ ∇ · • Variables Momentum for an inviscid ° uid – ρ –density ∂u ρ + ρu (u)+ p =0 – p – pressure ∂t · ∇ ∇ – u – velocity Energy (neglecting heat conduction) – T – ∂T Dp temperature ρCp + u (ρT ) =0. ∂t · ∇ − Dt • Constants µ ¶ Equation of state relates three of the quanti- – Cp is “heat capacity” ties p ρ = RT Conservation Laws ∂ρ ∂ p • Eliminate ρ = ∂t ∂t RT • Introduce the short hand notation µ ¶ • D ∂ = + u Dt ∂t · ∇ • Yields the system of equations

1 Dp 1 DT + u =0 • γ is the ratio of specific p Dt − T Dt ∇ · heats for the gas p Du + p =0 RT Dt ∇ 1 DT γ 1 Dp − =0 T Dt − γp Dt Equations of Acoustics

• Acoustics govern the propagation of small perturbations through the system

• Let the system be in equilibrium with pressure p0, Temperature T0, 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 Acoustic equations

• Equations under these assumptions are\ 1 ∂p 1 ∂T + u = 0 • Can eliminate T from the p ∂t − T ∂t ∇ · system 0 0 p ∂u 1 ∂p 0 + p =0 + u =0 RT0 ∂t ∇ γp0 ∂t ∇ · 1 ∂T γ 1 ∂p − =0 p ∂u T0 ∂t − γp0 ∂t 0 + p =0 RT0 ∂t ∇ The wave equation

• Almost there … • Differentiate first equation with respect to time 1 ∂2p ∂u 2 + =0 γp0 ∂t ∇ · ∂t • Substitute for ∂ u /∂ t from second equation ∂u 1 = p ∂t −ρ0 ∇ 1 ∂2p p =0 c2 ∂t2 −∇· ∇ 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 ∂2p 2p =0 c2 ∂t2 −∇ Wave equation for the Velocity potential 1 ∂p • Let φ be the velocity + 2φ =0 potential so that u=∇ φ γp0 ∂t ∇ • Then the conservation p ∂φ 0 + p =0 equations become ∇ ÃRT0 ∂t ! • From these we can eliminate p to arrive at the wave equation for the velocity potential 1 ∂2p + 2φ =0 −c2 ∂t2 ∇