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Waves. Chapter 16: Traveling waves Chapter 18: Standing waves, interference Chapter 37 & 38: Interference and diffraction of electromagnetic waves. Wave Motion. time t. t +Δ t. A wave is a moving pattern that moves along a medium . For example, a wave on a stretched string:. v.
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Waves Chapter 16: Traveling waves Chapter 18: Standing waves, interference Chapter 37 & 38: Interference and diffraction of electromagnetic waves
Wave Motion time t t +Δt A wave is a moving pattern that moves along a medium. For example, a wave on a stretched string: v 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) AB) BC) CD) All move with the same speed
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 .
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). y = f (x) at time t = 0: y v vt at time t: y = f (x - vt) y Note: y = y(x,t), a function of two variables;
For non-dispersive waves: 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)
y Quiz v x 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: • y= - f(x-vt) • y= - f(x+vt) • y= - f(- x-vt) • y= - f(- x+vt) • something else
Travelling Waves The most general form of a traveling sine wave (harmonic waves) isy(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
y A x -A For sinusoidal waves, the shape is a sine function, eg., f(x) = y(x,0) = A sin(kx) (A and k are constants) One wavelength Then y (x,t) = f(x – vt) = A sin[k(x – vt)] ory (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+)
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
