| Syllabus item | Covered in notes? | Comments / reinforcement |
|---|---|---|
| 7.1 Definition, wave‑front, wave‑speed \(v=f\lambda\) | ✔︎ | Added short derivation of the wave equation and explicit link \(v=\omega/k\). |
| 7.1 Energy transport – power & intensity (mechanical & EM) | ✔︎ | Standard power formulas for a string, a sound wave and an EM wave are now listed. |
| 7.2 Superposition, interference & diffraction | ✔︎ | Intensity–amplitude link highlighted; interference formula derived from amplitude addition. |
| 7.3 Doppler effect (sound & EM) | ✔︎ | Concise statement and worked example retained. |
| 7.4 Polarisation (EM waves) | ✔︎ | Malus’ law and intensity‑field relation included. |
\[
y(x,t)=A\sin(kx-\omega t)
\]
where
\(A\) – amplitude,
\(k=2\pi/\lambda\) – wave‑number,
\(\omega=2\pi f\) – angular frequency.
\[
v_{p}=\frac{\omega}{k}=f\lambda .
\]
In non‑dispersive media the phase and group velocities are identical (\(v{g}=v{p}\)).
| Wave type | Particle motion | Typical example |
|---|---|---|
| Transverse | Perpendicular to direction of travel | Wave on a rope, electromagnetic wave |
| Longitudinal | Parallel to direction of travel | Sound in air, seismic P‑wave |
| Wave type | Average power (per unit width/area) |
|---|---|
| Mechanical string (tension \(T\), linear mass density \(\mu\)) | \[ P_{\text{avg}}=\tfrac12\,\mu\,\omega^{2}A^{2}\,v \qquad\bigl(v=\sqrt{T/\mu}\bigr) \] |
| Sound wave in a fluid (density \(\rho\), speed of sound \(v\)) | \[ I_{\text{avg}}=\tfrac12\,\rho\,v\,\omega^{2}s^{2} \qquad\bigl(s\;=\text{displacement amplitude}\bigr) \] (Intensity is power per unit area; the same expression is often written \(I=\frac{p{\max}^{2}}{2\rho v}\) with \(p{\max}=\rho v\omega s\).) |
| Electromagnetic wave in vacuum | \[ I{\text{avg}}=\tfrac12\,c\,\varepsilon{0}\,E_{0}^{2} =\tfrac12\,\frac{c}{\mu{0}}\,B{0}^{2} \qquad\bigl(c=3.00\times10^{8}\,\text{m s}^{-1}\bigr) \] |
\[
I=\frac{P}{A_{\perp}}
\]
where \(A_{\perp}\) is the projected (perpendicular) area that the wavefront cuts through.
If a detector is tilted by an angle \(\theta\) to the wave direction, use \(A_{\perp}=A\cos\theta\).
Example for a string:
\[
u(x,t)=\tfrac12\mu\left(\frac{\partial y}{\partial t}\right)^{2}
+\tfrac12T\left(\frac{\partial y}{\partial x}\right)^{2}
=\tfrac12\mu\omega^{2}A^{2}\cos^{2}(kx-\omega t)
+\tfrac12T k^{2}A^{2}\sin^{2}(kx-\omega t).
\]
\[
\langle u\rangle =\frac14\mu\omega^{2}A^{2}+\frac14Tk^{2}A^{2}\propto A^{2}.
\]
\[
P=\langle u\rangle\,v\,A_{\perp}.
\]
\[
I=\langle u\rangle v\;\Longrightarrow\;I\propto A^{2}.
\]
The same reasoning applies to sound (replace \(\mu\) by \(\rho\)) and to EM waves (replace \(\langle u\rangle\) by \(\tfrac12\varepsilon{0}E{0}^{2}\)).
\[
f' = f\,\frac{v\pm v{o}}{v\pm v{s}}
\]
\(v\) = speed of sound in the medium, \(v{s}\) = source speed (positive if moving away), \(v{o}\) = observer speed (positive if moving towards the source).
Example: A siren of \(f=800\;\text{Hz}\) approaches a stationary observer at \(30\;\text{m s}^{-1}\) with \(v=340\;\text{m s}^{-1}\).
\[
f' = 800\;\frac{340}{340-30}\approx 880\;\text{Hz}.
\]
| Region | Wavelength \(\lambda\) | Frequency \(f\) | Typical use |
|---|---|---|---|
| Radio | \(\;>10^{-1}\,\text{m}\) | \(<10^{9}\,\text{Hz}\) | Broadcast, radar |
| Microwave | \(10^{-3}\!-\!10^{-1}\,\text{m}\) | \(10^{9}\!-\!10^{11}\,\text{Hz}\) | Cooking, satellite comm. |
| Infrared | \(7\times10^{-7}\!-\!10^{-3}\,\text{m}\) | \(3\times10^{11}\!-\!4\times10^{14}\,\text{Hz}\) | Thermal imaging |
| Visible | \(4\!\times\!10^{-7}\!-\!7\!\times\!10^{-7}\,\text{m}\) | \(4\!\times\!10^{14}\!-\!7.5\!\times\!10^{14}\,\text{Hz}\) | Human sight |
| Ultraviolet | \(10^{-8}\!-\!4\times10^{-7}\,\text{m}\) | \(7.5\!\times\!10^{14}\!-\!3\!\times\!10^{16}\,\text{Hz}\) | Sterilisation |
| X‑ray | \(10^{-11}\!-\!10^{-8}\,\text{m}\) | \(3\!\times\!10^{16}\!-\!3\!\times\!10^{19}\,\text{Hz}\) | Medical imaging |
| Gamma | \(<10^{-11}\,\text{m}\) | \(>3\!\times\!10^{19}\,\text{Hz}\) | Radioactive decay |
\[
I = I_{0}\cos^{2}\theta,
\]
where \(\theta\) is the angle between the incident polarisation and the transmission axis.
\[
I{\text{tot}} = \big\langle\bigl(A{1}\cos\phi{1}+A{2}\cos\phi_{2}\bigr)^{2}\big\rangle
= I{1}+I{2}+2\sqrt{I{1}I{2}}\cos\Delta\phi .
\]
This formula underpins:
Given: Average power \(P=2.0\;\text{W}\); spherical wave approximated as planar at distance \(r=0.10\;\text{m}\).
Given: String tension \(T=50\;\text{N}\), linear density \(\mu=0.01\;\text{kg m}^{-1}\), frequency \(f=100\;\text{Hz}\), amplitude \(A=2\;\text{mm}\).
=\tfrac12(0.01)(628)^{2}(2\times10^{-3})^{2}(70.7)
\approx 0.28\;\text{W}.\)
| Quantity | Symbol | Formula / Relation |
|---|---|---|
| Wave speed | \(v\) | \(v = f\lambda = \dfrac{\omega}{k}\) |
| Phase velocity | \(v_{p}\) | Same as \(v\) for a single‑frequency wave |
| Group velocity | \(v{g}\) | \(v{g}= \dfrac{d\omega}{dk}\) (equals \(v_{p}\) in non‑dispersive media) |
| Average power (string) | \(P\) | \(\displaystyle P=\tfrac12\,\mu\,\omega^{2}A^{2}\,v\) |
| Average intensity (sound) | \(I\) | \(\displaystyle I=\tfrac12\,\rho\,v\,\omega^{2}s^{2}\) |
| Average intensity (EM) | \(I\) | \(\displaystyle I=\tfrac12\,c\,\varepsilon{0}\,E{0}^{2} =\tfrac12\,\frac{c}{\mu{0}}\,B{0}^{2}\) |
| Intensity definition | \(I\) | \(I = \dfrac{P}{A_{\perp}}\) |
| Intensity–amplitude link | \(I\) | \(I\propto A^{2}\) (or \(E_{0}^{2}\) for EM) |
| Doppler‑shifted frequency (sound) | \(f'\) | \(f' = f\,\dfrac{v\pm v{o}}{v\pm v{s}}\) |
| Malus’ law (polarisation) | \(I\) | \(I = I_{0}\cos^{2}\theta\) |
| Interference intensity | \(I{\text{tot}}\) | \(I{\text{tot}} = I{1}+I{2}+2\sqrt{I{1}I{2}}\cos\Delta\phi\) |
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