Physics 451/551 Theoretical Mechanics

1. Physics 451/551Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18

2. Sound Waves • Properties of Sound • 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 • Two fundamental equations • Continuity equation (Conservation of mass) • Velocity equation (Conservation of momentum) • Newton’s Law in disguise

3. Fundamental Functions • Density ρ(x,y,z),mass per unit volume • Velocity field

4. Continuity Equation • Consider mass entering differential volume element • Mass entering box in a short time Δt • Take limit Δt→0

5. By Stoke’s Theorem. Because true for all dV • Mass current density (flux) (kg/(sec m2)) • Sometimes rendered in terms of the total time derivative (moving along with the flow) • Incompressible flow and ρ constant

6. 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 • 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)

7. Hydrostatic Equilibrium • Fluid at rest • Fluid in motion • As with density use total derivative (sometimes called material derivative or convective derivative)

8. Fluid Dynamic Equations • Manipulate with vector identity • Final velocity equation • One more thing: equation of state relating p and ρ

9. Energy Conservation • For energy in a fixed volume ε internal energy per unit mass • Work done (first law in co-moving frame) • Isentropic process (s constant, no heat transfer in)

11. Bernoulli’s Theorem • Exact first integral of velocity equation when • Irrotational motion • External force conservative • Flow incompressible with fixed ρ • Bernouli’s Theorem • If flow compressible but isentropic

12. Kelvin’s Theorem on Circulation • Already discussed this in the Arnold material • To linear order

13. 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.

14. Lagrangian for Isentropic Flow • Two independent field variables: ρand Φ • Lagrangian density • Canonical momenta

15. Euler Lagrange Equations • Hamiltonian Density internal energy plus potential energy plus kinetic energy

16. Sound Waves • Linearize about a uniform stationary state • Continuity equation • Velocity equation • Isentropic equation of state

17. Flow Irrotational • Take curl of velocity equation. Conclude flow irrotational • Scalar wave equation • Boundary conditions

18. 3-D Plane Wave Solutions • Ansatz • Energy flux

19. Helmholz Equation and Organ Pipes • Velocity potential solves Helmholtz equation • BCs • Cylindrical Solutions

20. Bessel Function Solutions • Bessel Functions solve • Eigenfunctions • Fundamental • Open ended

21. Green Function for Wave Equation • Green Function in 3-D • Apply Fourier Transforms • Fourier transform equation to solve and integrate by parts twice

22. Green Function Solution • The Fourier transform of the solution is • The solution is • The Green function is

23. Alternate equation for Green function • Simplify • Yukawa potential (Green function)

24. Helmholtz Equation • Driven (Inhomogeneous) Wave Equation • Time Fourier Transform • Wave Equation Fourier Transformed

25. Green Function • Green function satisfies

26. Green function is • Satisfies • Also, with causal boundary conditions is

27. Causal Boundary Conditions • Can get causal B. C. by correct pole choice • Gives so-called retarded Green function • Green function evaluated ω k plane

28. Method of Images • Suppose have homogeneous boundary conditions on the x-y half plane. The can solve the problem by making an image source and making a combined Green function. The rigid boundary solution has • To satisfy the boundary condition so that the solution vanishes on the boundary

29. Kirchhoff’s Approximation • We all know sound waves diffract (easily pass around corners). Standard approximation “schema” • Zeroth solution the Image GF • Boundary condition not correct at hole

30. In RHP • Exact relation • For short wavelengths, evaluate RHS as if screen not there! Huygens’ Principle

31. Babinet’s Principle • Apply Green’s identity

32. Diffracted Amplitude • Fresnel diffraction: phase shifts across the aperture important. Full integral must be completed • Fraunhofer diffraction Pattern is the transverse Fourier Transform!

33. Two Cases • Rectangular aperture • Destructive interference at qxa=π • Circular aperture • Airy disk (angle of first zero)

34. Equation for Heat Conduction • Field variable: temperature scalar • Additional inputs: heat capacity (at constant pressure) cp, thermal conductivity kth • Thermal diffusivity • Heat Equation

35. Boundary Conditions • Closed boundary surface held at constant Tex • Insulating surface • Separate variables • Helmholtz again

36. Long Rectangular Rod • Long ends held at temperature T0 • Eigensolutions

37. General Solution • Find expansion coefficients with the orthogonality relations • Long term solution dominated by slowest decaying mode

38. Thermal Waves • Put periodic boundary condition on plane z = 0 • 1-D problem

39. Penetration Depth • Exponential falloff length (for amplitude) • Solution for thermal wave • On earth, 3.2 m with a one year period!

40. Green Function for Heat Equation • Fourier Transform spatial dependence • Solve using initial condition

41. Complete the square