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

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

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.

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

Read: Ch2: 1-3 Homework: Ch2: 5,9,17,22,24,36,41,42 Due: September 2

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

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.

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:

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

Read: Ch2: 4-7 No homework

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

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.

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,

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