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11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:

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Presentation on theme: "11/11/2013PHY 711 Fall 2013 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29 -- Chap. 9:"— Presentation transcript:

1 11/11/2013PHY 711 Fall Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture Chap. 9: Wave equation for sound 1.Standing waves 2.Green’s function for wave equation; wave scattering

2 11/11/2013PHY 711 Fall Lecture 292

3 11/11/2013PHY 711 Fall Lecture 293

4 11/11/2013PHY 711 Fall Lecture 294

5 11/11/2013PHY 711 Fall Lecture 295 Comments about exam Note that this figure is misleading;  >  for physical case

6 11/11/2013PHY 711 Fall Lecture 296 Comment about exam – continued

7 11/11/2013PHY 711 Fall Lecture 297 Comment about exam – continued

8 11/11/2013PHY 711 Fall Lecture 298 Linearization of the fluid dynamics relations:

9 11/11/2013PHY 711 Fall Lecture 299

10 11/11/2013PHY 711 Fall Lecture 2910

11 11/11/2013PHY 711 Fall Lecture 2911 Time harmonic standing waves in a pipe L a

12 11/11/2013PHY 711 Fall Lecture 2912

13 11/11/2013PHY 711 Fall Lecture 2913

14 11/11/2013PHY 711 Fall Lecture 2914

15 11/11/2013PHY 711 Fall Lecture 2915 m=0 m=1 m=2

16 11/11/2013PHY 711 Fall Lecture 2916 m=0 m=1 m=2 Zeros of derivatives: m=0: , , m=1: , , m=2: , ,

17 11/11/2013PHY 711 Fall Lecture 2917 Boundary condition for z=0, z=L:

18 11/11/2013PHY 711 Fall Lecture 2918 Example

19 11/11/2013PHY 711 Fall Lecture 2919 Alternate boundary condition for z=0, z=L:

20 11/11/2013PHY 711 Fall Lecture 2920 Other solutions to wave equation:

21 11/11/2013PHY 711 Fall Lecture 2921 Wave equation with source:

22 11/11/2013PHY 711 Fall Lecture 2922 Wave equation with source -- continued:

23 11/11/2013PHY 711 Fall Lecture 2923 Derivation of Green’s function for wave equation

24 11/11/2013PHY 711 Fall Lecture 2924 Derivation of Green’s function for wave equation -- continued

25 11/11/2013PHY 711 Fall Lecture 2925 Derivation of Green’s function for wave equation -- continued

26 11/11/2013PHY 711 Fall Lecture 2926 Derivation of Green’s function for wave equation -- continued

27 11/11/2013PHY 711 Fall Lecture 2927 Derivation of Green’s function for wave equation – continued need to find A and B.

28 11/11/2013PHY 711 Fall Lecture 2928 Derivation of Green’s function for wave equation – continued

29 11/11/2013PHY 711 Fall Lecture 2929 Derivation of Green’s function for wave equation – continued

30 11/11/2013PHY 711 Fall Lecture 2930 For time harmonic forcing term we can use the corresponding Green’s function: In fact, this Green’s function is appropriate for boundary conditions at infinity. For surface boundary conditions where we know the boundary values or their gradients, the Green’s function must be modified.

31 11/11/2013PHY 711 Fall Lecture 2931

32 11/11/2013PHY 711 Fall Lecture 2932

33 11/11/2013PHY 711 Fall Lecture 2933 Wave equation with source: z y x

34 11/11/2013PHY 711 Fall Lecture 2934 Treatment of boundary values for time-harmonic force:

35 11/11/2013PHY 711 Fall Lecture 2935

36 11/11/2013PHY 711 Fall Lecture 2936

37 11/11/2013PHY 711 Fall Lecture 2937

38 11/11/2013PHY 711 Fall Lecture 2938


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