1 Chapter 2 Wave motion August 25,27 Harmonic waves 2.1 One-dimensional waves Wave: A disturbance of the medium, which propagates through the space, transporting energy and momentum. Types of waves: Mechanical waves, electromagnetic (EM) waves. Longitudinal waves, Transverse waves. Question: The type of wave in the corn field in Macomb, IL. Suppose the wind is weak. Mathematical description of a wave: For a wave that does not change its shape: x 0 t = 0 x 0 t vt Disturbance is a function of position and time:  (x, t) =f (x, t) Example: E(x, t) and B(x, t) of light  (x, 0) =f (x, 0)=f (x) (wave profile, snapshot)  (x, t) =f (x-vt) (General form of a wave) Example:  (x, t) = exp[-a(x-vt) 2 ]

The differential wave equation *A partial, linear, second order, homogeneous differential equation Specifying a wave: Amplitude and wavelength  Second order differential equation

3 2.2 Harmonic waves Harmonic waves:  (x, t) = f (x-vt) =A sin k(x-vt) Parameters: Amplitude: A Wavelength: Wave vector (propagation number): k = 2  / Period:  = /v Frequency:  = 1/  Speed of wave: v=  Angular frequency:  =2  2  /  Wave number :  =1/  Real waves: Monochromatic waves  Band of frequencies: Quasi-monochromatic waves Remember all of them by heart.

4 2.3 Phase and phase velocity General harmonic wave functions:  (x, t) =A sin(kx-  t+  ) Phase:  (x, t)= kx –  t +  Initial phase:  (x, t)| x=0, t=0 =  Rate-of-change of phase with time: Rate-of-change of phase with space: Phase velocity: The speed of propagation of the condition of constant phase. In general

5 Read: Ch2: 1-3 Homework: Ch2: 4,17,18,29,34,35 Due: September 5

6 August 29 Addition of waves 2.4 The superposition principle Superposition principle: The total disturbance from two waves at each point is the algebraic sum of the individual waves at that point. Superposition of harmonic waves  Interference: in-phase, out-of-phase

7 2.5 The complex representation Real harmonic wave:  Complex representation: The actual wave is the real part. Easy to manipulate mathematically, especially in the addition of waves. Use with care when perform multiplication of waves. 2.6 Phasors and the addition of waves Harmonic wave: Phasor: A rotating arrow (vector) that represents the wave  The addition of waves = the addition of vectors. A1A1 A2A2 A 22 11  Re(  ) Im(  ) A 

8 2.7 Plane waves Equation for a plane perpendicular to Wavefront: The surface composed by the points of equal phase at a given time. Plane wave: Waves whose wavefronts are planes. x y z k r  Description of a plane wave: k: propagation vector (wave vector). Including time variable: In Cartesian coordinates:

9 Significance of plane waves: Easy to generate (harmonic oscillator). Any 3-dimensional wave can be expressed as a combination of plane waves (Fourier analysis).

10 Read: Ch2: 4-7 No homework

11 September 3 Spherical waves 2.8 The three-dimensional differential wave equation Plane wave: Laplacian operator: General solution:

Spherical waves Spherical waves: Waves whose wavefronts are spheres. Spherical coordinates: (r, ,  ) x z y r   Laplacian operator in spherical coordinates: Spherical symmetry:  Differential wave equation: Solution: General solution: The inverse square law: Intensity of a spherical wave  1/r 2.

13 Harmonic spherical wave: 2.10 Cylindrical waves Cylindrical waves: Waves whose wavefronts are cylinders. Cylindrical coordinates: (r, , z) Laplacian operator in cylindrical coordinates: Cylindrical symmetry: x z y r  z Differential wave equation: Solution: When r is sufficiently large, A is the source strength.

14 Read: Ch2: 8-10 Homework: Ch2: 40,43 Due: September 12