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

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: ρ and 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