1. 1 Propagation of waves Friday October 18, 2002

2. 2 Propagation of waves in 3D Imagine a disturbane that results in waves propagating equally in all directions  E.g. sound wave source in air or water, light source in a dielectric medium etc.. The generalization of the wave equation to 3-dimensions is straight forward if the medium is homogeneous Let  = amplitude of disturbance (could be amplitude of E-field also)

3. 3 Propagation of waves in 3D  depends on x, y and z such that it satisfies the wave equation or, where in cartesian co-ordinates,

4. 4 1. Special Case: Plane Waves along x Suppose  (x, y, z, t)=  (x, t) (depends only on x) Then  = f (kx-ωt) + g (kx+ωt) Then for a given position x o,  has the same value for all y, z at any time t o. i.e. the disturbance has the same value in the y-z plane that intersects the x-axis at x o. This is a surface of constant phase

5. 5 Plane waves along x Planes perpendicular to the x-axis are wave fronts – by definition

6. 6 2. Plane waves along an arbitrary direction (n) of propagation Now  will be constant in plane perpendicular to n – if wave is plane For all points P’ in plane Ozx y d P’ P

7. 7 2. Plane waves along an arbitrary direction (n) of propagation For all points P’ in plane or, for the disturbance at P

8. 8 2. Plane waves along an arbitrary direction (n) of propagation Ozx y d P’ P If wave is plane,  must be the same everywhere in plane  to n This plane is defined by is equation of a plane  to n, a distance d from the origin

9. 9 2. Plane waves along an arbitrary direction (n) of propagation is the equation of a plane wave propagating in k-direction

10. 10 3. Spherical Waves Assume has spherical symmetry about origin (where source is located) In spherical polar co-ordinates θ φ x y z r

11. 11 3. Spherical Waves Given spherical symmetry,  depends only on r, not φ or θ Consequently, the wave equation can be written,

12. 12 3. Spherical Waves Now note that,

13. 13 3. Spherical Waves But, is just the wave equation, whose solution is, i.e. amplitude decreases as 1/ r !! Wave fronts are spheres

14. 14 4. Cylindrical Waves (e.g. line source) The corresponding expression is, for a cylindrical wave traveling along positive 

15. 15 Electromagnetic waves Consider propagation in a homogeneous medium (no absorption) characterized by a dielectric constant  o = permittivity of free space

16. 16 Electromagnetic waves Maxwell’s equations are, in a region of no free charges, Gauss’ law – electric field from a charge distribution No magnetic monopoles Electromagnetic induction (time varying magnetic field producing an electric field) Magnetic fields being induced By currents and a time-varying electric fields µ o = permeability of free space (medium is diamagnetic)

17. 17 Electromagnetic waves or, For the electric field E, i.e. wave equation with v 2 = 1/µ o 

18. 18 Electromagnetic waves Similarly for the magnetic field i.e. wave equation with v 2 = 1/µ o  In free space,  =   o =  o (  = 1) c = 3.0 X 10 8 m/s

19. 19 Electromagnetic waves In a dielectric medium,  = n 2 and  =   o = n 2  o

20. 20 Electromagnetic waves: Phase relations The solutions to the wave equations, can be plane waves,