Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 018
Properties of Sound Sound Waves Requires medium for propagation Mainly longitudinal (displacement along propagation direction) Wavelength much longer than interatomic spacing so can treat medium as continuous Fundamental functions Mass density Velocity field x, y, z, t v x, y, z, t Two fundamental equations Continuity equation (Conservation of mass) Velocity equation (Conservation of momentum) Newton s Law in disguise Theoretical Mechanics Fall 018
Fundamental Functions Density ρ(x,y,z), mass per unit volume M x, y, z, t lim V 0 V dm x, y, z, t dxdydz Velocity field v x, y, z, t o x, y, z v x, y, z, t Theoretical Mechanics Fall 018
Continuity Equation Consider mass entering differential volume element dy x, y, z dx dz Mass entering box in a short time Δt v x x, y, z, t vxx dx, y, z, tdydz t Take limit Δt 0,,,,,, vy x y z t vy x y dy z t dzdxt,,,,,, v z x y z t vz x y z dz t dxdy t x, y, z, t t x, y, z, tdxdydz t v dv Theoretical Mechanics Fall 018
dv dxdydz v dxdydz t t dv By Stoke s Theorem. Because true for all dv Mass current density (flux) (kg/(sec m )) Jm v Sometimes rendered in terms of the total time derivative (moving along with the flow) Incompressible flow v 0 and ρ constant dv t v 0 d v v 0 v t dt Theoretical Mechanics Fall 018
Pressure Scalar Displace material from a small volume dv with sides given by da. The pressure p is defined to the force acting on the area element df da Pressure is normal to the area element Doesn t depend on orientation of volume External forces (e.g., gravitational force) must be balanced by a pressure gradient to get a stationary fluid in equilibrium Pressure force (per unit volume) F pr p p x Theoretical Mechanics Fall 018
Fluid at rest Fluid in motion Hydrostatic Equilibrium f app p dv 0 f app p dv dv Fnet p fapp dv m dv dt dt As with density use total derivative (sometimes called material derivative or convective derivative) dv dt v v t v Theoretical Mechanics Fall 018
Fluid Dynamic Equations dv v p v v fapp dt t Manipulate with vector identity vv v v v v Final velocity equation v v v v v f p app t One more thing: equation of state relating p and ρ Theoretical Mechanics Fall 018
Energy Conservation For energy in a fixed volume 3 v Etot d x V ε internal energy per unit mass Work done (first law in co-moving frame) Mp Md pdv d p s, d Isentropic process (s constant, no heat transfer in) p t t Theoretical Mechanics Fall 018
1 1 t v v v v p v fapp p p v pv v t p p pv v p t 1 1 t 1 je v p v v p v fapp v Theoretical Mechanics Fall 018
Bernoulli s Theorem Exact first integral of velocity equation when Irrotational motion v 0 v External force conservative f U Flow incompressible with fixed ρ Bernouli s Theorem If flow compressible but isentropic app 0 p U t 0 p U t Theoretical Mechanics Fall 018
Kelvin s Theorem on Circulation Already discussed this in the Arnold material dv v v p v U dt t t To linear order t ds v C t C s, t t C s, t t v C s, t, t 1 v C C s, t t t t v C s, t t, t t ds s C s, t t v C s, t, t ds s Theoretical Mechanics Fall 018
d dv C,,,, C s t t ds v C s t t v C s, t, t ds dt dt s s p C v C U ds ds s s 0 (the integrand is exact!) The circulation is constant about any closed curve that moves with the fluid. If a fluid is stationary and acted on by a conservative force, the flow in a simply connected region necessarily remains irrotational. Theoretical Mechanics Fall 018
Lagrangian for Isentropic Flow Two independent field variables: ρ and Φ Lagrangian density t p U t L U t Canonical momenta L P / t P 0 L / t 0 0 Theoretical Mechanics Fall 018
Euler Lagrange Equations L L P 0 0 t t p Hamiltonian Density L L p P U t t H P U t P t L internal energy plus potential energy plus kinetic energy 0 Theoretical Mechanics Fall 018
Sound Waves Linearize about a uniform stationary state,, 0, x t v x t v p x t p p Continuity equation 1 0v 0 0 v 0 t 0 t Velocity equation v 1 p t Eisentropic equation of state 0 0 0 p p0 p ps, 0 p0 p c s Theoretical Mechanics Fall 018
Flow Irrotational Take curl of velocity equation. Conclude flow irrotational v p v t t t 1 p c t 0 t 0 t Scalar wave equation 1 c t c t Boundary conditions nˆ nˆ V 0 for a fixed boundary 0 free surface t 0 Theoretical Mechanics Fall 018
3-D Plane Wave Solutions Ansatz, Re e i k xt k c v 0 0 ik 0 c i iv ik 0 c Energy flux 0 0 j E ik i k c k t 1 * 1 Re ˆ 0 0 0 0 0 Theoretical Mechanics Fall 018
Helmholz Equation and Organ Pipes Theoretical Mechanics Fall 018
Theoretical Mechanics Fall 018
Green Function for Wave Equation Green Function in 3-D Apply Fourier Transforms u r u r f r 3 ipr f p d re f r 3 ipr f r d pe f p 3 1 Fourier transform equation to solve and integrate by parts twice p u p u p f p Theoretical Mechanics Fall 018
Green Function Solution The Fourier transform of the solution is The solution is u p 1 f p ipx u r 3 e d p p The Green function is p f 1 1 p p 3 ipr ipr 3 3 u r e e d pf r d r 3 1 1 p ipr ipr 3 G r r e e d p 3 Theoretical Mechanics Fall 018
Alternate equation for Green function 1 ipr ipr 3 Gr r e e d p 3 r r Simplify iprcos iprsin 1 e 3 1 e sin 3 p 0 0 G R d p p dp d p R 1 psin pr 1 psin pr e dp 4 R dp 0 R p R p Yukawa potential (Green function) rr e Gr r 4 r r Theoretical Mechanics Fall 018
Helmholtz Equation Driven (Inhomogeneous) Wave Equation 1 c t Time Fourier Transform r, t f r, t Wave Equation Fourier Transformed r, f r, c 1 f r t 1 d e f r it r, t de r, it,, Theoretical Mechanics Fall 018
c Green function satisfies Green Function 3 r, t d r dtg r r, t t f r, t,, k k f k, ik r t 1 f k 3 r, t 4 d kd e k c 1 3 3 1 ik r t ik rt r, t d r dt d k d e e f 4 r, t k c Theoretical Mechanics Fall 018
Green function is ik r t ik rt 1 3 e e G r r, t t d k d 4 k c Satisfies 1 c t G r r t t r r t t, Also, with causal boundary conditions is G r r, i r r / c e 4 r r Theoretical Mechanics Fall 018
Causal Boundary Conditions Can get causal B. C. by correct pole choice ω k plane kc i kc i i / c i / c Gives so-called retarded Green function Green function evaluated 3 ik R, d k e G R 3 k i / c 1 e e e kdk k i / c R ikr ikr ir/ c 8 ir 4 Theoretical Mechanics Fall 018