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 19 20### Transverse waves 21**Direction of motion is perpendicular to wave** 22- rarefactions (expansions) 23- compressions 24 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 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** 101 102 103### Refraction 104**Change in direction caused by change in speed** e.g. prism 105$\Delta v$ depends on $\lambda$, so wavelengths become "split" 106 107 108Refractive index of a medium depends $\Delta v$ from $c$ 109$n={c \over v}\quad$ (refractive index of 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 125 126### Double Slit 127 128 129**(a) Double slit as theorised by particle model** - "streams" of photons are concentrated in bright spots 130**(b) Double slit as theorised by wave model** - waves disperse onto screen (overlapping) 131 132Young's double slit experiment supports wave model: 133- parallel slits of thickness comparable to $\lambda$ 134- multiple wave fronts combine to form constructive / destructive interference 135- fringes - points of constructive interference (bright) 136- constructive interference when waves are **coherent** (in phase) 137- fringe in centre of slits 138- solve path difference using pythag 139 140 141 142Path difference $pd = |S_1P-S_2P|$ for point $p$ on screen 143 144Constructive interference when $pd = n\lambda$ where $n \in [0, 1, 2, ...]$ 145Destructive interference when $pd = (n-{1 \over 2})\lambda$ where $n \in [1, 2, 3, ...]$ 146 147Fringe separation: 148$$\Delta x = {{\lambda l }\over d}$$ 149 150where 151$\Delta x$ is distance between fringes 152$l$ is distance from slits to screen 153$d$ is separation between sluts ($=S_1-S_2$) 154 155## Electromagnetic waves 156 157 158 159- electric waves and magnetic waves are perpendicular to each other due to Faraday's law 160 161Wave equation: 162 163$$c = f \lambda$$ 164 165where 166$c$ is velocity (speed of light in this case) 167$f$ is frequency (Hz) 168$\lambda$ is wavelength (m)