1. Wave Equations: EM Waves

2. Electromagnetic waves for E field for B field

3. In general, electromagnetic waves Where  represents E or B or their components

4. # A plane wave satisfies wave equation in Cartesian coordinates # A spherical wave satisfies wave equation in spherical polar coordinates # A cylindrical wave satisfies wave equation in cylindrical coordinates

5. Solution of 3D wave equation In Cartesian coordinates Separation of variables

6. Substituting for  we obtain Variables are separated out Each variable-term independent And must be a constant

7. So we may write where we use

8. Solutions are then Total Solution is plane wave

9. Traveling 3D plane wave

10. spherical coordinates

11. spherical waves

14. Alternatively The wave equation becomes

15. Put Then  Hence

16. Therefore Wave equation transforms to 

17. Which follows that Separation of variables Solutions are Total solution is

18. outgoingwaves incomingwaves Final form of solution General solution spherical wave

19. Cylindrical waves

20. with angular and azimuthal symmetry, the Laplacian simplifies and the wave equation

21. The solutions are Bessel functions. r For large r, they are approximated as

22. A plane wave satisfies one-dimensional wave equation in Cartesian coordinates The position vector must remain perpendicular to the given plane

23. The wave then satisfies the generalization of the one-dimensional wave equation

24. Plane EM waves in vacuum

25. Wave vector k is perpendicular to E Wave vector k is perpendicular to B

26. B is perpendicular to E

27. B, k and E make a right handed Cartesian co-ordinate system

28. Plane EM waves in vacuum