physics / waves.mdon commit add midyear cheatsheet, add ch8-9 notes for waves (e57f9b6)
   1# Waves
   2
   3## Mechanical waves
   4- need a medium to travel through
   5- cannot transfer energy through vacuum
   6- individual particles have little movement regardless of the distance of wave
   7- **transfer of energy without net transfer of matter**
   8
   9**Nodes** - point of no motion (fixed point on graph)
  10**Antinodes** - point of maximum motion (peaks)
  11
  12**Crests** (peaks) & **troughs** (azimuths)
  13
  14### Longitudinal waves
  15
  16**Direction of motion is parallel to wave**
  17
  18![](/mnt/andrew/graphics/longitudinal-waves.png)
  19
  20### Transverse waves
  21**Direction of motion is perpendicular to wave**
  22- rarefactions (expansions)
  23- compressions
  24![](/mnt/andrew/graphics/transverse-waves.png)
  25
  26### Measuring mechanical waves
  27
  28**Amplitude $A$** - max displacement from rest position (0)
  29**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)
  30**Frequency $f$** - number of cycles (wavelengths) per second
  31
  32$T={1 \over f}\quad$(period: time for one cycle)
  33$v=f \lambda \quad$(speed: displacement per second)
  34
  35### Doppler effect
  36- occurs when there is relative movement between source and observer
  37- inverse relationship between frequency and distance: $f \propto {1 \over d}$
  38- applies to all types of wave
  39- only affects apparent $f$; actual $f$ is constant
  40
  41When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to travel than $w_{n-1}. Hence, $w_n$ reaches the observer sooner than $w_{n-1}, increasing "apparent" wavelength.
  42
  43
  44
  45
  46
  47
  48## Interference patterns
  49
  50When a medium changes character:
  51- some energy is *reflected*
  52- some energy is *absorbed* by new medium
  53- some energy is *transmitted*
  54
  55**Superposition** - stimuli add together at a given point (vector addition)
  56**Standing wave** - constructive interference at resonant frequency
  57
  58### Reflection
  59
  60**Diffuse** reflection - irregular surface reflects each ray in a different angle.
  61
  62#### Rays
  63Two- or three-dimensional *wave fronts* can be reflected, e.g. waves at a beach.
  64
  65Direction of motion of wave fronts can be shown by arrows, called *rays*, which are perpendicular to the wave front:
  66
  67![](/mnt/andrew/graphics/rays.png)
  68
  69Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
  70- Normal: $\perp$ to wall
  71- Incident wave front: $\perp$ to incident ray
  72- Incident ray: $
  73
  74#### Transverse
  75- sign of reflected transverse wave is inverted when endpoint is fixed in y-axis (equivalent to $180^\circ=\pi={\lambda \over 2}$ phase change)
  76- phase is constant if endpoint is free to move in y-axis (**reflected** is same as **incident**)
  77
  78## Harmonics
  79
  80**Harmonic** - fundamental (lowest) frequency to produce a certain number of wavelengths
  81**Overtone** - a multiple of the fundamental harmonic which produces the same no. of wavelengths at a different frequency (due to constructive interference)
  82
  83#### Wave has antinodes at both ends:
  84$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
  85$f = {nv \div 2l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
  86
  87#### Wave has antinode at one end:
  88$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
  89$f = {nv \div 4l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
  90
  91## Light
  92
  93Newton - light as a particle
  94- light speeds up as it travels through a solid medium
  95
  96Hooke - light as a wave
  97- light slows down through solid medium
  98
  99### Huygen's principle
 100**Each point on a wavefront can be considered a source of secondary wavelets**
 101![](/mnt/andrew/graphics/huygen.png)
 102
 103### Refraction
 104**Change in direction caused by change in speed** e.g. prism
 105$\Delta v$ depends on $\lambda$, so wavelengths become "split"
 106![](/mnt/andrew/graphics/refraction.png)
 107
 108Refractive index of a medium depends $\Delta v$ from $c$
 109$n={c \over v}\quad$ (refractive index of a medium)
 110$n_1v_1=n_2v_2$ (equivalence between media)
 111
 112### Snell's law
 113$n$ can be used to determine how much a ray will refract going between two media.
 114
 115$$n_1 \sin \theta_1=n_2 \sin \theta_2$$
 116
 117### Total internal reflection
 118When $n_1 < n_2$, light is refracted *towards* normal ($90^\circ$ to medium border - "vertical" line in case of air/water).
 119When $n_1 > n_2$, light is reflected *away* from normal.
 120**Critical angle $\theta_c$** - angle of incidence $\theta_1$ at which $\theta_2 \gt 90^\circ$ to normal
 121$n_1 sin \theta_c = n_2 \sin 90^\circ$
 122$\therefore \theta_c = {n_2 \over n_1}$
 123
 124### Dispersion