theme C flashies

simple harmonic motion, wave model, wave phenomena, standing waves, and the Doppler effect

amplitude

amplitude

the maximum displacement of the oscillator away from its equilibrium position

simple harmonic motion

• motion never ends (no energy losses)

• the magnitude of the restoring force is directly proportional to the displacement of the object

• the direction of the force is always towards the equilibrium position

acceleration and velocity in SHM

• amplitude: max a, v = 0

• equilibrium position: a = 0, max v

energy maxima

• equilibrium position: max Ek

• amplitude: max Eh/Ep

• Ek → Eh/Ep transfer f is 2x oscillator f

transverse waves

direction of energy transfer is perpendicular to direction of propagation

longitudinal waves

direction of energy transfer is parallel to direction of propagation

displacement-distance graph

• compressions and rarefactions occur at x=0

• positive tangent: rarefaction

• negative tangent: compression

mechanical waves in states

• solid: can sustain both

• liquid: can sustain longitudinal, but transverse only at the surface

• gas: can sustain longitudinal, but no transverse waves (no restoring force)

sound waves

• longitudinal

• require a medium

• air pressure and air displacement π/2 rad out of phase

• difference in air pressure is greatest at compressions and rarefactions

electromagnetic waves

• transverse waves

• do not need a medium to travel through

• travel at the speed of light in vacuum

• consist of fluctuations in perpendicular magnetic and electric fields

where do electromagnetic waves come from

the motion of charged particles, or release of photons due to energy changes

wavefront

surfaces that connect particles in phase and move with the wave. perpendicular to the direction of motion

ray

lines that show the direction of travel

reflection

the incident ray is identical to the reflected ray in a plane mirror

refraction

waves in a more (optically) dense medium move slower, causing a change in wavelength and a change in direction. frequency does not change!

relative refractive index

_med1n_med2 = n_med2/n_med1

total internal reflection

occurs when incident ray is at an angle equal or larger to critical angle

snell’s law at critical angle

sinθ_c = n₂/n₁

principle of superposition

when two waves of the same type meet at a point in a medium, their individual displacements add

constructive interference

waves that are displaced in the same direction will superpose to give a wave with a larger amplitude

destructive interference

waves that are displaced in opposite directions will superpose to cancel out or give a lower amplitude

diffraction

occurs when a wave must move around an obstacle or through an aperture

diffraction effects

• no wave properties change, except amplitude decreases

• a change in direction

• smaller slits → more diffraction

coherence

when two waves of the same type have identical phases and equal frequencies

minima

areas of destructive interference when diffracted waves π rad out of phase superpose

maxima

areas of constructive interference when diffracted waves completely phase superpose

fringe width (s)

distance between two maxima or two minima

standing wave properties

• nodes where amplitudes cancel out

• antinodes where amplitudes add

• all particles move in phase between adjacent nodes/antinodes

• phase difference between nodal points = π rad

• standing wave frequency = frequency of travelling wave that form it

wave inversion

reflects, flips, π rad out of phase

fixed end

cannot move, displacement = 0

• opposite direction

• opposite displacement

• π rad out of phase

free end

always at maximum, displacement = 2x amplitude

• opposite direction

• equal displacement

• completely in phase

change in length per harmonic

+ λ/2

two fixed ends or two free ends

L = nλ/2, f_n=nf₁

one fixed end, one free end

L = (2n-1)λ/4, f_2n-1=(2n-1)f₁

sounding frequencies

the sounding frequency for a pipe closed at one end occurs at half the sounding frequency of a pipe of the same length, open at both ends

natural frequency (f₀)

the frequency at which an oscillating system oscillates when there is little to no friction

damping

frictional/resistive forces that transfer energy away from an oscillation. acts in the opposite direction of the restoring force

as damping increases…

• amplitude half life decreases

• oscillatory behaviour decreases

• time period increases

critical damping

the oscillator takes the least possible time to come to rest, therefore there is no oscillatory behaviour

forced vibrations

when an oscillating system is driven by another oscillator

driving frequency and natural frequency

• very different: amplitude low

• getting closer: amplitude increases

• very close/identical: amplitude very high, possibly enough to break

effect of damping on maximum amplitude

as damping increases, the oscillating system will show a maximum amplitude at lower values of f

when is amplitude constant

when the driving frequency supplies energy at the same rate that damping removes it, the amplitude of the oscillating body remains constant

moving source, stationary observer

• moving closer: difference in successive wavefronts is smaller, λ decreases, f increases, c constant

• moving away: difference in successive wavefronts is larger, λ increases, f decreases, c constant

stationary source, moving observer

• moving closer: difference in successive wavefronts in smaller, f increases, c increases, λ constant

• moving away: difference in successive wavefronts is larger, f decreases, c decreases, λ constant