# Waves
-## Longitudinal and Transverse Waves
+## Mechanical waves
+- need a medium to travel through
+- cannot transfer energy through vacuum
+- individual particles have little movement regardless of the distance of wave
+- **transfer of energy without net transfer of matter**
-Longitudinal - direction of motion parallel to wave
-Transverse - direction of motion perpendicular to wave
+**Nodes** - point of no motion (fixed point on graph)
+**Antinodes** - point of maximum motion (peaks)
+**Crests** (peaks) & **troughs** (azimuths)
-## Wavelength & frequency
+### Longitudinal waves
+**Direction of motion is parallel to wave**
-$$v=f \times \lambda$$
+
+
+### Transverse waves
+**Direction of motion is perpendicular to wave**
+- rarefactions (expansions)
+- compressions
+
+
+### 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)
+**Frequency $f$** - number of cycles (wavelengths) per second
+
+$T={1 \over f}\quad$(period: time for one cycle)
+$v=f \lambda \quad$(speed: displacement per second)
+
+### Doppler effect
+- occurs when there is relative movement between source and observer
+- inverse relationship between frequency and distance: $f \propto {1 \over d}$
+- 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.
-Nodes - point of no motion
-Antinodes - point of maximum motion (peaks)
-Crests (peaks) & troughs (azimuths)
## Interference patterns
-Standing wave - constructive interference at resonant frequency
+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)
+**Standing wave** - constructive interference at resonant frequency
+
+### Reflection
+
+**Diffuse** reflection - irregular surface reflects each ray in a different angle.
+
+#### Rays
+Two- or three-dimensional *wave fronts* can be reflected, e.g. waves at a beach.
+
+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: $
+
+#### 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)
+- phase is constant if endpoint is free to move in y-axis (**reflected** is same as **incident**)
+
+## Harmonics
+
+**Harmonic** - fundamental (lowest) frequency to produce a certain number of wavelengths
+**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)
+$f = {nv \div 2l}\quad$ (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)
+$f = {nv \div 4l}\quad$ (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
+
+Hooke - light as a wave
+- light slows down through solid medium
+
+### Huygen's principle
+**Each point on a wavefront can be considered a source of secondary wavelets**
+
+
+### Refraction
+**Change in direction caused by change in speed** e.g. prism
+$\Delta v$ depends on $\lambda$, so wavelengths become "split"
+
+
+Refractive index of a medium depends $\Delta v$ from $c$
+$n={c \over v}\quad$ (refractive index of a medium)
+$n_1v_1=n_2v_2$ (equivalence between media)
+
+### Snell's law
+$n$ can be used to determine how much a ray will refract going between two media.
+
+$$n_1 \sin \theta_1=n_2 \sin \theta_2$$
+
+### Total internal reflection
+When $n_1 < n_2$, light is refracted *towards* normal ($90^\circ$ to medium border - "vertical" line in case of air/water).
+When $n_1 > n_2$, light is reflected *away* from normal.
+**Critical angle $\theta_c$** - angle of incidence $\theta_1$ at which $\theta_2 \gt 90^\circ$ to normal
+$n_1 sin \theta_c = n_2 \sin 90^\circ$
+$\therefore \theta_c = {n_2 \over n_1}$
+
+### Dispersion