Waves I

(yeet.)

Chapter 2 of PC5131 is Waves... I. This is effectively singular waves, rather than interfering waves. This includes normal wave properties and equations, wave intensity, string waves, polarisation, sound waves (longitudinal waves) and the Doppler Effect. This is relatively easier than the next topic (Waves II) so uh yay thank god only this much is tested in the Common Test on Monday. Anyways yeah good luck, and let's see what comes out of this, right? Aight. See yah.

Basics

Wave Quantities

\[\begin{align*}\omega &=2\pi f = \frac{2\pi}{T}\\k &= \frac{2\pi}{\lambda}\\v &= f\lambda = \frac{f}{T} = \frac{\omega}{k}\end{align*}\]

Wave Equation

\[\begin{align*}\psi(x, t)&=A\sin(\phi_x -\phi_t + \phi_0)\\&=A\sin(kx-\omega t+\phi_0)\end{align*}\]

For wave moving backwards:

\[\psi(x, t) = A\sin(kx+\omega t+\phi_0)\]

Phase Difference

\[\begin{align*}\Delta\phi &= \phi_2 - \phi_1 \\ &= (kx_2 - \omega t + \phi_0) - (kx_1-\omega t+\phi_0) \\ &= k(x_2 - x_1) \\ &= k\Delta x \\ &= 2\pi \frac{\Delta x}\lambda \\ &= -\omega \Delta t \\ &=-2\pi \frac{\Delta t}{T} \end{align*}\]

This applies to both \(\Delta t\) and \(\Delta x\), which is great. If the wave is moving backwards, the phase difference in terms of \(\Delta t\) is positive, not negative.

Intensity of Wave

\[\begin{align*} I &= \frac{P}{A}\\&=\frac{P}{4\pi r^2} \\ I &\propto \frac{1}{r^2} \\ \frac{I_1}{I_2} &= \left(\frac{r_2}{r_1}\right)^2 \end{align*}\]

String Wave

\[v = \sqrt{\frac{T}{\mu}}\]

Polarisation

Polarising Filter

\[\begin{align*}I_1 &= \frac{I_0}{2}\\A_{n+1} &= A_n \cos\phi_{n+1,n}\\I_{n+1} &= I_n \cos^2\phi_{n+1,n}\end{align*}\]

Brewster's Angle

\[\begin{align*}n_a \sin\theta_i &= n_a \sin\theta_{reflect} \\ n_a \sin\theta_i &= n_b \sin\theta_{refract} \\ n_a \sin\theta_{reflect} &= n_b \sin\theta_{refract}\end{align*}\]

From here, we note the following for Brewster's Angle, which is defined as \(\theta_b\).

\[\begin{align*}\theta_{reflect} &= \theta_{brewster} = \theta_b \\ \theta_{refract} &= \theta_{polarised} = \theta_p \\ \theta_{b} + \theta_{p} &= \frac{\pi}2 \\ \sin\theta_p &= \cos\theta_b \\ n_a \sin\theta_{b} &= n_b \sin\theta_{p} = n_b\cos\theta_b \\ \frac{\sin\theta_b}{\cos\theta_b} &= \frac{n_b}{n_a} \\ \tan\theta_b &= \frac{n_b}{n_a} \\ \theta_b &= tan^{-1}\left(\frac{n_b}{n_a} \right)\end{align*}\]

Sound

Speeds of Sound

Condition	Speed
0°C (Standard Temperature and Pressure)	330 m/s
30°C (Room Temperature and Pressure)	343 m/s
T°C (Any Temperature)	\(331.5 \times \sqrt{1 + \frac{T}{273.15}}\)

Doppler Effect

\[f' = f \frac{c \pm v_o}{c \mp v_s}\]

Source	Observer	Numerator	Denominator	Remarks
stationary	stationary	c	c	-
approaching	stationary	\(c\)	\(c-v_s\)	-
moving away	stationary	\(c\)	\(c+v_s\)	-
stationary	approaching	\(c+v_o\)	\(c\)	-
stationary,	moving away	\(c-v_o\)	\(c\)	-
approaching	approaching	\(c+v_o\)	\(c-v_s\)	maximum
moving away	approaching	\(c+v_o\)	\(c+v_s\)	-
approaching	moving away	\(c-v_o\)	\(c-v_s\)	-
moving away	moving away	\(c-v_o\)	\(c+v_s\)	minimum

Just draw a damn diagram, for god's sake.

How to pick the correct signs?

Look at how the frequency scales. For example, the frequency should increase if the target is moving towards the source, so make numerator big so numerator has a positive sign. The frequency should increase if the source is moving in the same direction as the sound so make denominator small so denominator has a negative sign.
Look at just the speed of the source, target and sound. The numerator is the "speed at which the target and source are approaching each other." The denominator is the "relative speed of the signal relative to the source."