04 Wave Phenomena 🌊
Simple Harmonic Motion
Polarisation
1⃣ Acceleration = -Displacement
2⃣ Acceleration ∝ Displacement
Questions
May 2016 Q4
Nov 2015
Longtitudinal
Direction of wave propagation is parallel to the force causing it
May 2017 Q7e
Nov 2017
Q3 (SHM)
Q8 Pt 2
Questions
Nov 15 Q6 Pt 2
Q2f (SHM)
Q5a (Light)
Nov 2018
Q4 (pipe)
due to restoring force that
- always acts towards centre
- proportional to displacement
Mass-Spring
Simple Pendulum
Force 🔨
\(kx\)
Force 🔨
\(mg\sin\theta\)
Acceleration 🏁
\(ma=-kx\)
Acceleration 🏁
\(ma=-mg\sin\theta\)
\(a=-\frac{k}{m}x\)
\(a=-g\sin\frac{x}{L}\)
\(a=-\omega^2x\)
Period ⏱
\(\omega^2=\frac{k}{m}\)
\(\omega=\sqrt{\frac{k}{m}}\)
\(T=\frac{2\pi}{\omega}\)
\(\theta=\frac{x}{L}\)
Period ⏱
\(\omega^2=\frac{g}{L}\)
\(a=-\frac{g}{L}x\)
only proportional if \(\sin\theta=\theta\)
\(\omega=\sqrt{\frac{g}{L}}\)
\(T=\frac{2\pi}{\omega}\)
Energy
- \(x=x_0\cos(\omega t)\)
- \(v=-\omega x_0\sin(\omega t)\)
- \(a=-\omega^2 x_0\cos(\omega t)=-\omega^2x\)
- \(x=x_0\sin(\omega t)\)
- \(v=\omega x_0\cos(\omega t)\)
- \(a=-\omega^2x_0\sin(\omega t)=-\omega^2x\)
Kinetic Energy
\(\frac{1}{2}mv^2\)
Total Energy
\(\frac{1}{2}m(-\omega x_0\sin(\omega t))^2\)
\(\frac{1}{2}m\omega^2x_0^2\sin^2(\omega t)\)
Equivalent to
max KE
\(\frac{1}{2}m\omega^2x_0^2\)
Potential Energy
\(\frac{1}{2}m\omega^2x_0^2-\frac{1}{2}m\omega^2x_0^2\sin^2(\omega t)\)
\(\frac{1}{2}m\omega^2x_0^2(1-\sin^2(\omega t))\)
\(\frac{1}{2}m\omega^2x^2\)
Phase Difference
Travelling wave
click to edit
Standing Waves
Boundary
Conditions
Pipe closed
at one end
Refraction
Refractive
Index \(n\)
Light travels fastest in a vacuum
\(n_{m}=\frac{c}{c_m}\)
Angle of Incidence
or Refraction
- \(n\) is 1⃣ for vaccum
- \(n\) is MORE than 1⃣ for optically denser materials
Snell's Law
\(\frac{\sin\theta_1}{\sin\theta_2}=\frac{n_2}{n_1}=\frac{v_1}{v_2}\)
DENSE ➡ less dense
less dense ➡ DENSE
Refractive Index
small \(n\) ➡ BIG \(n\)
Wave Speed
Fast 🐎 ➡ slow 🐌
- Decrease in speed
- Increase in refractive index
Angle is proportional
to wave speed
\(\theta\propto v\)
Angle is inversely proportional to refractive index
\(\theta\propto\frac{1}{n}\)
Decrease in \(\theta\)
Refractive Index
BIG \(n\) ➡ small \(n\)
Wave Speed
Slow 🐌 ➡ fast 🐎
- Increase in speed
- Decrease in refractive index
Increase in \(\theta\)
air ☁
water 💧:
water 💧:
air ☁
formed when two identical waves travelling in opposite directions superimpose
\(E=E_0\cos\theta\)
Since \(I=kA^2\)
\(I=k(E_0\cos\theta)^2\)
\(I=k(E_0)^2\cos^2\theta\)
\(I=I_0\cos^2\theta\)
\(\theta\) is angle to
the VERTICAL
Pipe open at both ends
1st harmonic
\(L=\frac{1}{4}\lambda_1\)
\(\lambda_1=4L\)
3rd harmonic
\(L=\frac{3}{4}\lambda_3\)
\(\lambda_3=\frac{4}{3}L\)
nth harmonic
\(\lambda_n=\frac{4L}{n};\)
\(n=1,3,5\dots\)
1st harmonic
\(L=\frac{1}{2}\lambda_1\)
\(\lambda_1=2L\)
2nd harmonic
\(L=\lambda_2\)
\(\lambda_2=L\)
nth harmonic
\(\lambda_n=\frac{2L}{n}\)
\(n=1,2,3,4,5\dots\)
node
antinode
\(\frac{shift}{period}\times 360º\)
Standing wave
- Points in same half wavelength IN PHASE
- Points in adjacent half wavelengths ANTIPHASE