1. THE WAVE EQUATION Alemayehu Adugna Arara Supervisor : Dr. J.H.M. ten Thije Boonkkamp November 04, 2009

2. Outline Occurence of the Wave Equation 1D Waves Spherical Waves Cylinderical Waves Supersonic Flow Past a body Revolution Initial value Problems in Two and Three Dimensions

3. 1. Occurrence of the Wave Equation Acoustics Electromagnetism Elasticity Hyperpolic wave equation.

4. Acoustics Linearized Small disturbance about an equilibrium state. Body forces are neglected The initial disturbance has a uniform entropy

5. From conservation of mass we have Acoustics From balance momentum we also have Then from these equation we have

6. Acoustics Introducing Hence we have,

7. Linearized Supersonic Flow Disturbance is to remain small, –the motion should have to be very small or – the body must be very slender. Slender body moving with arbitrary constant velocity relates acoustics with aerodynamics. The velocity components in frame are

8. Linearized Supersonic Flow Hence the equation wave for acoustics become The velocity

9. Electromagnetic Waves Maxwell‘s equation And E satisfies the same equation. Where B is the magnetic induction and E is the electric field. Therefore

10. 2. 1D Wave Equation The functions f and g follows from initial and boundary conditions For the intial value problem,

11. 3. Spherical Waves For waves symmetric about the origin This can be written as The general solution will be

12. 3. Spherical Waves For outgoing waves, the solution is Standard form is to prescribe source is In acoustics, ∂φ/∂R is radial velocity and Q(t) is the flux of volume.

13. 3. Spherical Waves IVP, “balloon problem“ in acoustics: No source at the origin

14. 3. Spherical Waves

16. t R0R0 R+ a 0 t=R 0 R 0 /2a 0 R 0 /a 0 R 0 /20 R R- a 0 t=R 0 R- a 0 t=0 A C D E B

17. 3. Spherical Waves Region A: Region B:

18. 3. Spherical Waves Region C: Region D: Region E:

19. 3. Spherical Waves Time Evolution of pressure difference R P p-p 0 p1p1 p2p2 0R 0 -a 0 tR 0 +a 0 tR0R0

20. 3. Spherical Waves R P p-p 0 p1p1 p2p2 0R 0 -a 0 tR 0 +a 0 tR0R0

21. 3. Spherical Waves R p-p 0 p2p2 0R 0 +a 0 tR0R0

22. 4. Cylinderical Waves The sources are uniformily distrubeted on the z axis with a uniform strength q(t) per unit length. Total disturbance is r R q(t)dz- z

23. 4.Cylinderical Waves Various forms of this solution are valuable. If q‘(t) → 0 suficiently fast as t →- ∞.

24. 4. Cylinderical Waves

25. 5. Supersonic Flow Past a body of Revolution

26. Linearization Body must be slender R‘(x) is small and Φ x, are both small.

27. 5. Supersonic Flow Past a body of Revolution The components of the velocity perturbation are obtained by suitable modification

28. 6. Initial Value Problem in Two and Three Dimensions We know from the spherical wave solution is We consider

29. 6. Initial Value Problem in Two and Three Dimensions The initial source strength determining f acts only for an instant,

30. 6. Initial Value Problem in Two and Three Dimensions

31. The general initial value therefore is

32. 6. Initial Value Problem in Two and Three Dimensions Two Dimesional Problem

33. Conclusion The wave equations occurs in many problems. The solution strongly depends on the initial value and boundary conditions. The wave equations are easier to solve in odd dimensions. One can easily solve the IVP in two and three dimensions.

34. THANK YOU SO MUCH!! !!!