1. Chapter 2 Wave motion August 22,24 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 small.
Mathematical description of a wave: For a wave that does not change its shape:
Disturbance is a function of position and time: y (x, t) =f (x, t)
Example: Displacement D(x,t) and pressure P(x,t) of sound wave. E(x, t) and B(x, t) of light
y (x, 0) = f (x, 0)= f (x) (wave profile, snapshot)
y (x, t) =f (x-vt) (General form of a wave)
Example: y (x, t) = exp[-a(x-vt)2] x
t = 0 t vt

2. 2.1.1 The differential wave equation
Specifying a wave: Amplitude and wavelength  Second order differential equation
*A partial, linear, second order, homogeneous differential equation

3. 2.2 Harmonic waves Harmonic waves: y (x, t) = f (x-vt) =A sin k(x-vt)
Parameters:
Amplitude: A
Wavelength: l
Wave vector (propagation number): k = 2p/l
Period: t = l/v
Frequency: n = 1/t (distinguish n and v)
Speed of wave: v=n l
Angular frequency: w =2pn =2p/t
Wave number : k =1/ l
Real waves:
Monochromatic waves 
Band of frequencies:
Quasi-monochromatic waves
Example 2.3, 2.4
Remember all of them by heart.

4. 2.3 Phase and phase velocity
General harmonic wave functions:
y (x, t) =A sin(kx – wt+e)
Phase: (x, t)= kx – wt + e
Initial phase: (x, t)|x=0, t=0= e
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: 5,9,17,22,24,36,41,42
Due: September 2

6. August 26 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:
Algebraic representation:
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. A 
2.6 Phasors and the addition of waves
Harmonic wave: A1 A2 A 2 1 
Re(y)
Im(y)
Phasor representation: Using a rotating arrow (vector) to represents the wave.
The addition of waves = the addition of vectors.

8. 2.7 Plane waves x y z k r
Wavefront: The surface composed by the points of equal phase of a wave at a given time.
Plane wave: Waves whose wavefronts are planes.
Equation for a plane perpendicular to
 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).
Example 2.6

10. Read: Ch2: 4-7
No homework

11. August 29 Spherical waves
2.8 The three-dimensional differential wave equation
Plane wave:
Laplacian operator:
General solution:

12. 2.9 Spherical waves
Spherical waves: Waves whose wavefronts are spheres.
Spherical coordinates: (r, q, f) x z y r q f
Laplacian operator in spherical coordinates:
Spherical symmetry: 
Differential wave equation:
Solution:
General solution:
The inverse square law: Intensity of a spherical wave  1/r2.

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

14. Read: Ch2: 8-10
Homework: Ch2: 49,54
Due: September 9