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

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

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

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

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

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

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

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

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

y (x,t) = f (x ± vt) For non-dispersive waves: + 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)

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

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

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

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

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