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# Waves

<!-- **Nodes** are fixed on graph -->
<!-- **Crests** (peaks) & **troughs** (azimuths) -->
## Longitudinal (motion $||$ wave)

**rarefactions** (expansions) and **compressions**

![](graphics/longitudinal-waves.png)

## Transverse waves (motion $\perp$ wave)

**nodes** are fixed on graph

![](graphics/transverse-waves.png)

## Measuring mechanical waves

**Amplitude $A$** - max displacement from rest position  
**Wavelength $\lambda$** - $x$ distance between $y_1=y_2$  
**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
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

When a medium changes character, energy is _reflected_, _absorbed_, and _transmitted_

**Standing waves** - constructive int. at resonant freq

## Polarisation

![](graphics/polarisation.png){#id .class width=20%}

## Refraction

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![](graphics/refraction.png)

Angle of incidence $\theta_i =$ angle of reflection $\theta_r$

Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$

Snell's law - $n_1 \sin \theta_1=n_2 \sin \theta_2$

## Harmonics

where $a=2$ for antinodes at both ends, $a=4$ for antinodes at one end:

$\lambda = {{al} \div n}\quad$ (wavelength for $n^{th}$ harmonic)  
$f = {nv \div al}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)


## Double split

Path difference $pd = |S_1P-S_2P|$ for point $p$ on screen

Constructive: $pd = n\lambda$ where $n \in [0, 1, 2, ...]$  
Destructive: $pd = (n-{1 \over 2})\lambda$ 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 ($=S_1-S_2$)

![](graphics/em-spectrum.png){#id .class height=100px}