1. Waves Chapter 16:Traveling waves Chapter 18:Standing waves, interference Chapter 37 & 38:Interference and diffraction of electromagnetic waves

2. A wave is a moving pattern that moves along a medium. For example, a wave on a stretched string: The wave speed v is the speed of the pattern. Waves carry energy and momentum. v time t t +Δt Wave Motion Δx = v Δt

3. The wave moves this way The particles move up and down If the particle motion is perpendicular to the direction the wave travels, the wave is called a “transverse wave”. Transverse waves Examples: Waves on a string; light & other electromagnetic waves.

4. Longitudinal waves Example: sound waves in gases Even in longitudinal waves, the particle velocities are quite different from the wave velocity. The particles move back and forth. The wave moves this way

5. Quiz wave motion A B C Which particle is moving at the highest speed? A) A B) B C) C D) All move with the same speed

6. For these waves, the wave speed is constant for all sizes & shapes of waves. lengthmass/unit tension  v Eg. stretched string: Non-dispersive waves: the wave always keeps the same shape as it moves. Here the wave speed depends only on tension and mass density .

7. Reflections Waves (partially) reflect from any boundary in the medium: 1) “Soft” boundary: Reflection is upright light string, or free end Transmitted is upright

8. Reflection is inverted Reflections 2) “Hard” boundary: heavy rope, or fixed end. Transmitted is upright

9. The math: Suppose the shape of the wave at t = 0, is given by some function y = f(x). y y = f (x) v at time t : y = f (x - vt) y vt at time t = 0 : Note: y = y(x,t), a function of two variables;

10. y (x,t) = f (x ± vt) + sign: wave travels in negative x direction - sign: wave travels in positive x direction f is any (smooth) function of one variable. eg. f(x) = A sin (kx) For non-dispersive waves:

11. Quiz v y x A) y=  f(x  vt) B) y=  f(x+vt) C) y=  f(  x  vt) D) y=  f(  x+vt) E)something else A wave y=f(x  vt) travels in the positive x direction. It reflects from a fixed end at the origin. The reflected wave is described by:

12. y(x,t) = A sin (kx ±  t +  ) Wavenumber angular frequency ω = 2πf amplitude “phase” The most general form of a traveling sine wave (harmonic waves) is y(x,t)=A sin(kx ± ωt +  ) The wave speed is phase constant  Travelling Waves This is NOT the speed of the particles on the wave

13. f(x) = y(x,0) = A sin(kx) For sinusoidal waves, the shape is a sine function, eg., Then y (x,t) = f(x – vt) = A sin[k(x – vt)] or y (x,t) = A sin (kx –  t) with  = kv A -A y x ( A and k are constants) One wavelength

14. f(x) = y(x,0) = A sin(kx+  ) A -A y x Phase constant: To solve for the phase constant , use y(0,0): eg: y(0,0)=5, y(x,t)=10sin(kx-ωt+  ) <- snapshot at t=0

15. Principle of Superposition When two waves meet, the displacements add algebraically: So, waves can pass through each other: v v

16. Light Waves (extra) Light (electromagnetic) waves are produced by oscillations in the electric and magnetic fields: These are ‘linear wave equations’, with For a string it can be shown that (with F=T):