# Waves
## Mechanical waves
-- need a medium to travel through
+- need a medium to travel through (fields for electromagnetic waves)
- cannot transfer energy through vacuum
- individual particles have little movement regardless of the distance of wave
- **transfer of energy without net transfer of matter**
-**Nodes** - point of no motion (fixed point on graph)
+**Nodes** - point of no motion (fixed point on graph)
**Antinodes** - point of maximum motion (peaks)
**Crests** (peaks) & **troughs** (azimuths)
**Direction of motion is parallel to wave**
-
+
### Transverse waves
**Direction of motion is perpendicular to wave**
- rarefactions (expansions)
- compressions
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+
### Measuring mechanical waves
-**Amplitude $A$** - max displacement from rest position (0)
-**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)
+**Amplitude $A$** - max displacement from rest position (0)
+**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)
**Frequency $f$** - number of cycles (wavelengths) per second
-$T={1 \over f}\quad$(period: time for one cycle)
+$T={1 \over f}\quad$(period: time for one cycle)
$v=f \lambda \quad$(speed: displacement per second)
### Doppler effect
- applies to all types of wave
- only affects apparent $f$; actual $f$ is constant
-When $P_1$ 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.
+When $P_1$ 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.
## Interference patterns
When a medium changes character:
+
- some energy is *reflected*
- some energy is *absorbed* by new medium
- some energy is *transmitted*
-**Superposition** - stimuli add together at a given point (vector addition)
+**Superposition** - stimuli add together at a given point (vector addition)
**Standing wave** - constructive interference at resonant frequency
### Reflection
Direction of motion of wave fronts can be shown by arrows, called *rays*, which are perpendicular to the wave front:
-
+
Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
+
- Normal: $\perp$ to wall
- Incident wave front: $\perp$ to incident ray
-- Incident ray: $
+- Incident ray: $\theta_i$
#### Transverse
- sign of reflected transverse wave is inverted when endpoint is fixed in y-axis (equivalent to $180^\circ=\pi={\lambda \over 2}$ phase change)
**Overtone** - a multiple of the fundamental harmonic which produces the same no. of wavelengths at a different frequency (due to constructive interference)
#### Wave has antinodes at both ends:
-$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
$f = {nv \div 2l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
#### Wave has antinode at one end:
-$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
$f = {nv \div 4l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
## Light
-
-
Newton - light as a particle
- light speeds up as it travels through a solid medium
### Huygen's principle
**Each point on a wavefront can be considered a source of secondary wavelets**
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+
### Refraction
**Change in direction caused by change in speed** e.g. prism
$\Delta v$ depends on $\lambda$, so wavelengths become "split"
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+
Refractive index of a medium depends $\Delta v$ from $c$
-$n={c \over v}\quad$ (refractive index of a medium)
+$n={c \over v}\quad$ (refractive index of medium)
$n_1v_1=n_2v_2$ (equivalence between media)
### Snell's law
$\therefore \theta_c = {n_2 \over n_1}$
### Dispersion
+
+### Double Slit
+
+
+**(a) Double slit as theorised by particle model** - "streams" of photons are concentrated in bright spots
+**(b) Double slit as theorised by wave model** - waves disperse onto screen (overlapping)
+
+Young's double slit experiment supports wave model:
+- parallel slits of thickness comparable to $\lambda$
+- multiple wave fronts combine to form constructive / destructive interference
+- fringes - points of constructive interference (bright)
+- constructive interference when waves are **coherent** (in phase)
+- fringe in centre of slits
+- solve path difference using pythag
+
+
+
+Path difference $pd = |S_1P-S_2P|$ for point $p$ on screen
+
+Constructive interference when $pd = n\lambda$ where $n \in [0, 1, 2, ...]$
+Destructive interference when $pd = (n-{1 \over 2})\lambda$ where $n \in [1, 2, 3, ...]$
+
+Fringe separation:
+$$\Delta x = {{\lambda l }\over d}$$
+
+where
+$\Delta x$ is distance between fringes
+$l$ is distance from slits to screen
+$d$ is separation between sluts ($=S_1-S_2$)
+
+## Electromagnetic waves
+
+
+
+- electric waves and magnetic waves are perpendicular to each other due to Faraday's law
+
+Wave equation:
+
+$$c = f \lambda$$
+
+where
+$c$ is velocity (speed of light in this case)
+$f$ is frequency (Hz)
+$\lambda$ is wavelength (m)