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

Transcription

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

2 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 18

3 Fundamental Functions Density ρ(x,y,z), mass per unit volume M x, y, z, t lim V 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 18

4 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,,,,,, 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 18

5 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 and ρ constant dv t v d v v v t dt Theoretical Mechanics Fall 18

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 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 18

7 Fluid at rest Fluid in motion Hydrostatic Equilibrium f app p dv 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 18

8 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 18

9 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 18

10 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 18

11 Bernoulli s Theorem Exact first integral of velocity equation when Irrotational motion v v External force conservative f U Flow incompressible with fixed ρ Bernouli s Theorem If flow compressible but isentropic app p U t p U t Theoretical Mechanics Fall 18

12 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 18

13 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 (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 18

14 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 L / t Theoretical Mechanics Fall 18

15 Euler Lagrange Equations L L P 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 Theoretical Mechanics Fall 18

16 Sound Waves Linearize about a uniform stationary state,,, x t v x t v p x t p p Continuity equation 1 v v t t Velocity equation v 1 p t Isentropic equation of state p p p ps, p p c s Theoretical Mechanics Fall 18

17 Flow Irrotational Take curl of velocity equation. Conclude flow irrotational v p v t t t 1 p c t t t Scalar wave equation 1 c t c t Boundary conditions nˆ nˆ V for a fixed boundary free surface t Theoretical Mechanics Fall 18

18 3-D Plane Wave Solutions Ansatz, Re e i k xt k c v ik c i iv ik c Energy flux j E ik i k c k t 1 * 1 Re ˆ Theoretical Mechanics Fall 18

19 Helmholz Equation and Organ Pipes Velocity potential solves Helmholtz equation r k r BCs vr r a vz r z z, L Cylindrical Solutions 1 1 r k r r r r z r,, z Rr F Z z im p F e Z z cos z zero possible L Theoretical Mechanics Fall 18

20 Bessel Function Solutions Bessel Functions solve 1 d d r m r J mr Jmr k r dr dr r Eigenfunctions mn p mnp r, t ReJm r cos z expim it a L mnp Fundamental Open ended mn p c kmnp c a L Theoretical Mechanics Fall 18 ck c L ck 1 1 c L

21 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 18

22 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 p ipr ipr 3 G r r e e d p 3 Theoretical Mechanics Fall 18

23 Alternate equation for Green function 1 ipr ipr 3 Gr r e e d p 3 r r Simplify iprcos iprcos 1 e 3 1 e sin 3 p G R d p p dp d p R 1 psin pr 1 psin pr e dp 4 R dp R p R p Yukawa potential (Green function) rr e Gr r 4 r r Theoretical Mechanics Fall 18

24 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 18

25 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 ik r t ik rt r, t d r dt d k d e e f 4 r, t k c Theoretical Mechanics Fall 18

26 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 18

27 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 18

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 i r r / c i r r / c e e G r r r r r r 4 r r 4 r r,, x y z To satisfy the boundary condition so that the solution vanishes on the boundary i r r / c i r r / c e e G r r r r r r 4 r r 4 r r,, x y z Theoretical Mechanics Fall 18

29 V Kirchhoff s Approximation We all know sound waves diffract (easily pass around corners). Standard approximation schema r k r r r Zeroth solution the Image GF i r r / c i r r / c e e G r r r r x, r y, r z 4 r r 4 r r Boundary condition not correct at hole d r da 3 A da G G lim da G 3 3 d r G G d r k G r r Gk r R R H Theoretical Mechanics Fall 18

30 Exact relation In RHP For short wavelengths, evaluate RHS as if screen not there! Huygens Principle r dag da H H ik e e ik r r e r r z ik r r ik r r r da G da 8 H H r r r r r r z Theoretical Mechanics Fall 18

31 Babinet s Principle r da G r, r H r da G r, r PH ik r r e z 4 r r r r da G r, r P ik r r e z 4 r r ik r r e z 4 r r Apply Green s identity ik r r e inc 4 r r r r r G r, r r r r diff diff inc ik rr e 4 r r Theoretical Mechanics Fall 18

32 r r r rˆ r Diffracted Amplitude ˆ r r r r ik r kˆ r ikr r r r ik e ikr k ˆ kˆ r r dae cos exp 8 rr ik H r kˆ r r Fresnel diffraction: phase shifts across the aperture important. Full integral must be completed Fraunhofer diffraction ka / r 1 ka / r 1 Pattern is the transverse Fourier Transform! ik r r ik r r irq ik e ikr k ˆ kˆ ik e r dae dae 8 r r 8 r r H H Theoretical Mechanics Fall 18

33 Rectangular aperture Two Cases sin qa sin qb x y I r I qxa qyb Destructive interference at q x a=π Circular aperture J1 sin q a I r I qa Airy disk (angle of first zero) sin.61 a Theoretical Mechanics Fall 18

34 Equation for Heat Conduction Field variable: temperature scalar Additional inputs: heat capacity (at constant pressure) c p, thermal conductivity k th dt c de Thermal diffusivity Heat Equation H p j k T k th th c p T T t q c p Theoretical Mechanics Fall 18

35 Boundary Conditions Closed boundary surface held at constant T ex Insulating surface nt Separate variables, T r t T r e t Helmholtz again T r k T r q c p Theoretical Mechanics Fall 18

36 Long Rectangular Rod Long ends held at temperature T Eigensolutions T r X xy y Z z X x Y y Z z k mnp m x sin a m 1,,3 n y cos b n,1,,3 p z cos c p,1,,3 m n p a b c Theoretical Mechanics Fall 18

37 General Solution m x n y p z T r, t Cmnp sin cos cos e a b c mnp m x n y p z T r, t Cmnp sin cos cos a b c mnp Find expansion coefficients with the orthogonality relations mnp t Long term solution dominated by slowest decaying mode x T r t T C1 e a 1, sin t Theoretical Mechanics Fall 18

38 Thermal Waves Put periodic boundary condition on plane z = T z, t T cost 1-D problem T z T z t, Re d T dz T z 1 T t T z e i T z e it i 1 i Theoretical Mechanics Fall 18

39 Penetration Depth Exponential falloff length (for amplitude) 1/ 1/ T Solution for thermal wave /, z z T z t Te cos t On earth, 3. m with a one year period! Theoretical Mechanics Fall 18

40 Green Function for Heat Equation Fourier Transform spatial dependence T T t Solve using initial condition T k, t t, T k t Theoretical Mechanics Fall 18 k T k, t A k e ik r 3,, T r t e d r T k t A k 1 ik r 3 k t ik r 3 T r, t T r, t e d r e e d k 3 1 kt ik r r 3 G r r, t e e d k 3 kt

41 Complete the square 1 kt ik r r cos G r r, t e e k dkd cosd 3 1 cos k t ik r r r r k t ik r r ik r r cos i 1 e e k dkd e e e kdk 1 1 kt ik r r e e kdk e i r r r r i l t k i r r /t t k ik r r /t r r /4 t r r /4t 1 r r /4 t l 1 r r /4t / 3/ i t t r r dl G e e l i r r t e 4 kdk e Theoretical Mechanics Fall 18