Physics 451551 Theoretical Mechanics G A Krafft Old

Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 2018

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 Theoretical Mechanics Fall 2018

Fundamental Functions • Density ρ(x, y, z), mass per unit volume • Velocity field Theoretical Mechanics Fall 2018

Continuity Equation • Consider mass entering differential volume element • Mass entering box in a short time Δt • Take limit Δt→ 0 Theoretical Mechanics Fall 2018

• By Stoke’s Theorem. Because true for all d. V • Mass current density (flux) (kg/(sec m 2)) • Sometimes rendered in terms of the total time derivative (moving along with the flow) • Incompressible flow and ρ constant Theoretical Mechanics Fall 2018

Pressure Scalar • Displace material from a small volume d. V with sides given by d. A. 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) Theoretical Mechanics Fall 2018

Hydrostatic Equilibrium • Fluid at rest • Fluid in motion • As with density use total derivative (sometimes called material derivative or convective derivative) Theoretical Mechanics Fall 2018

Fluid Dynamic Equations • Manipulate with vector identity • Final velocity equation • One more thing: equation of state relating p and ρ Theoretical Mechanics Fall 2018

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) Theoretical Mechanics Fall 2018

Theoretical Mechanics Fall 2018

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 Theoretical Mechanics Fall 2018

Kelvin’s Theorem on Circulation • Already discussed this in the Arnold material • To linear order Theoretical Mechanics Fall 2018

• 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 2018

Lagrangian for Isentropic Flow • Two independent field variables: ρ and Φ • Lagrangian density • Canonical momenta Theoretical Mechanics Fall 2018

• Euler Lagrange Equations • Hamiltonian Density internal energy plus potential energy plus kinetic energy Theoretical Mechanics Fall 2018

Sound Waves • Linearize about a uniform stationary state • Continuity equation • Velocity equation • Eisentropic equation of state Theoretical Mechanics Fall 2018

Flow Irrotational • Take curl of velocity equation. Conclude flow irrotational • Scalar wave equation • Boundary conditions Theoretical Mechanics Fall 2018

3 -D Plane Wave Solutions • Ansatz • Energy flux Theoretical Mechanics Fall 2018

Helmholz Equation and Organ Pipes Theoretical Mechanics Fall 2018

Theoretical Mechanics Fall 2018

Green Function for Wave Equation • Green Function in 3 -D • Apply Fourier Transforms • Fourier transform equation to solve and integrate by parts twice Theoretical Mechanics Fall 2018

Green Function Solution • The Fourier transform of the solution is • The Green function is Theoretical Mechanics Fall 2018

• Alternate equation for Green function • Simplify • Yukawa potential (Green function) Theoretical Mechanics Fall 2018

Helmholtz Equation • Driven (Inhomogeneous) Wave Equation • Time Fourier Transform • Wave Equation Fourier Transformed Theoretical Mechanics Fall 2018

Green Function • Green function satisfies Theoretical Mechanics Fall 2018

• Green function is • Satisfies • Also, with causal boundary conditions is Theoretical Mechanics Fall 2018

Causal Boundary Conditions • Can get causal B. C. by correct pole choice ω k plane • Gives so-called retarded Green function • Green function evaluated Theoretical Mechanics Fall 2018