physics / waves.mdon commit [spec] additions to complex graphs and exp identities (4de6207)
   1# Waves
   2
   3## Mechanical waves
   4- need a medium to travel through (fields for electromagnetic waves)
   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![](graphics/longitudinal-waves.png)
  19
  20### Transverse waves
  21**Direction of motion is perpendicular to wave**
  22- rarefactions (expansions)
  23- compressions
  24![](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
  52- some energy is *reflected*
  53- some energy is *absorbed* by new medium
  54- some energy is *transmitted*
  55
  56**Superposition** - stimuli add together at a given point (vector addition)  
  57**Standing wave** - constructive interference at resonant frequency
  58
  59### Reflection
  60
  61**Diffuse** reflection - irregular surface reflects each ray in a different angle.
  62
  63#### Rays
  64Two- or three-dimensional *wave fronts* can be reflected, e.g. waves at a beach.
  65
  66Direction of motion of wave fronts can be shown by arrows, called *rays*, which are perpendicular to the wave front:
  67
  68![](graphics/rays.png)
  69
  70Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
  71
  72- Normal: $\perp$ to wall
  73- Incident wave front: $\perp$ to incident ray
  74- Incident ray: $\theta_i$
  75
  76#### Transverse
  77- sign of reflected transverse wave is inverted when endpoint is fixed in y-axis (equivalent to $180^\circ=\pi={\lambda \over 2}$ phase change)
  78- phase is constant if endpoint is free to move in y-axis (**reflected** is same as **incident**)
  79
  80## Harmonics
  81
  82**Harmonic** - fundamental (lowest) frequency to produce a certain number of wavelengths
  83**Overtone** - a multiple of the fundamental harmonic which produces the same no. of wavelengths at a different frequency (due to constructive interference)
  84
  85#### Wave has antinodes at both ends:
  86$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)  
  87$f = {nv \div 2l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
  88
  89#### Wave has antinode at one end:
  90$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)  
  91$f = {nv \div 4l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
  92
  93## Light
  94
  95Newton - light as a particle
  96- light speeds up as it travels through a solid medium
  97
  98Hooke - light as a wave
  99- light slows down through solid medium
 100
 101### Huygen's principle
 102**Each point on a wavefront can be considered a source of secondary wavelets**
 103![](graphics/huygen.png)
 104
 105### Refraction
 106**Change in direction caused by change in speed** e.g. prism
 107$\Delta v$ depends on $\lambda$, so wavelengths become "split"
 108![](graphics/refraction.png)
 109
 110Refractive index of a medium depends $\Delta v$ from $c$
 111$n={c \over v}\quad$ (refractive index of medium)
 112$n_1v_1=n_2v_2$ (equivalence between media)
 113
 114### Snell's law
 115$n$ can be used to determine how much a ray will refract going between two media.
 116
 117$$n_1 \sin \theta_1=n_2 \sin \theta_2$$
 118
 119### Total internal reflection
 120When $n_1 < n_2$, light is refracted *towards* normal ($90^\circ$ to medium border - "vertical" line in case of air/water).
 121When $n_1 > n_2$, light is reflected *away* from normal.
 122**Critical angle $\theta_c$** - angle of incidence $\theta_1$ at which $\theta_2 \gt 90^\circ$ to normal
 123$n_1 sin \theta_c = n_2 \sin 90^\circ$
 124$\therefore \theta_c = {n_2 \over n_1}$
 125
 126### Dispersion
 127
 128### Double Slit
 129
 130![](graphics/double-slit.png)
 131**(a) Double slit as theorised by particle model** - "streams" of photons are concentrated in bright spots  
 132**(b) Double slit as theorised by wave model** - waves disperse onto screen (overlapping)
 133
 134Young's double slit experiment supports wave model:
 135- parallel slits of thickness comparable to $\lambda$
 136- multiple wave fronts combine to form constructive / destructive interference
 137- fringes - points of constructive interference (bright)
 138- constructive interference when waves are **coherent** (in phase)
 139- fringe in centre of slits
 140- solve path difference using pythag
 141
 142![](graphics/double-slit-interference.png)
 143
 144Path difference $pd = |S_1P-S_2P|$ for point $p$ on screen
 145
 146Constructive interference when $pd = n\lambda$ where $n \in [0, 1, 2, ...]$  
 147Destructive interference when $pd = (n-{1 \over 2})\lambda$ where $n \in [1, 2, 3, ...]$
 148
 149Fringe separation:
 150$$\Delta x = {{\lambda l }\over d}$$
 151
 152where
 153$\Delta x$ is distance between fringes  
 154$l$ is distance from slits to screen  
 155$d$ is separation between sluts ($=S_1-S_2$)
 156
 157## Electromagnetic waves
 158
 159![](graphics/em-waves.png)
 160
 161- electric waves and magnetic waves are perpendicular to each other due to Faraday's law
 162
 163Wave equation:
 164
 165$$c = f \lambda$$
 166
 167where
 168$c$ is velocity (speed of light in this case)
 169$f$ is frequency (Hz)
 170$\lambda$ is wavelength (m)