recall and use intensity = power/area and intensity ∝ (amplitude )2 for a progressive wave

Progressive (Travelling) Waves – Cambridge AS/A‑Level Physics (9702)

1. Syllabus mapping (7 – Waves)

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.

2. What is a progressive wave?

• A disturbance that travels through a medium (or vacuum) carrying energy and momentum while the medium’s particles undergo only temporary displacements.
• Wave‑front: a surface of constant phase; it is always perpendicular to the direction of propagation.
• Mathematical form (harmonic wave) \[ y(x,t)=A\sin(kx-\omega t) \] where \(A\) – amplitude, \(k=2\pi/\lambda\) – wave‑number, \(\omega=2\pi f\) – angular frequency.
• From the argument of the sine function, a point of constant phase satisfies \(kx-\omega t=\text{constant}\); differentiating gives the **phase velocity** \[ v_{p}=\frac{\omega}{k}=f\lambda . \] In non‑dispersive media the phase and group velocities are identical (\(v_{g}=v_{p}\)).

3. Types of progressive waves

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

4. Quantitative descriptors (common to all waves)

• Amplitude \(A\) – maximum displacement from equilibrium (or maximum field strength for EM waves).
• Wavelength \(\lambda\) – distance between successive points of equal phase.
• Frequency \(f\) and angular frequency \(\omega = 2\pi f\).
• Wave‑number \(k = 2\pi/\lambda\).
• Wave speed \(v\) – \(v = f\lambda = \omega/k\).

5. Energy transport

5.1 Power carried by a harmonic 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	\[ I_{\text{avg}}=\tfrac12\,c\,\varepsilon_{0}\,E_{0}^{2} =\tfrac12\,\frac{c}{\mu_{0}}\,B_{0}^{2} \qquad\bigl(c=3.00\times10^{8}\,\text{m s}^{-1}\bigr) \]

5.2 Definition of intensity

\[ I=\frac{P}{A_{\perp}} \] 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 \(A_{\perp}=A\cos\theta\).

5.3 Deriving the \(I\propto A^{2}\) relationship

1. For any harmonic wave the instantaneous energy density (kinetic + potential) is proportional to the square of the local amplitude. Example for a string: \[ u(x,t)=\tfrac12\mu\left(\frac{\partial y}{\partial t}\right)^{2} +\tfrac12T\left(\frac{\partial y}{\partial x}\right)^{2} =\tfrac12\mu\omega^{2}A^{2}\cos^{2}(kx-\omega t) +\tfrac12T k^{2}A^{2}\sin^{2}(kx-\omega t). \]
2. Time‑averaging over a full cycle replaces \(\cos^{2}\) and \(\sin^{2}\) by their mean value \(1/2\), giving \[ \langle u\rangle =\frac14\mu\omega^{2}A^{2}+\frac14Tk^{2}A^{2}\propto A^{2}. \]
3. Average power crossing a perpendicular area is \[ P=\langle u\rangle\,v\,A_{\perp}. \]
4. Dividing by \(A_{\perp}\) yields the intensity \[ I=\langle u\rangle v\;\Longrightarrow\;I\propto A^{2}. \] The same reasoning applies to sound (replace \(\mu\) by \(\rho\)) and to EM waves (replace \(\langle u\rangle\) by \(\tfrac12\varepsilon_{0}E_{0}^{2}\)).

5.4 Practical consequences

• Doubling the amplitude increases the intensity by a factor of four.
• When two coherent waves interfere, the resultant intensity depends on the **square of the sum of amplitudes** – the basis of interference, diffraction and the \(\cos^{2}\) law for polarisation.

6. Related wave phenomena (exam‑relevant)

6.1 Doppler effect (sound)

\[ f' = f\,\frac{v\pm v_{o}}{v\pm v_{s}} \] \(v\) = 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 \(v=340\;\text{m s}^{-1}\). \[ f' = 800\;\frac{340}{340-30}\approx 880\;\text{Hz}. \]

6.2 Electromagnetic spectrum (quick reference)

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

6.3 Polarisation (plane EM waves)

• The electric field oscillates in a plane perpendicular to propagation; the direction of this oscillation defines the polarisation.
• Malus’ law for an ideal polariser: \[ I = I_{0}\cos^{2}\theta, \] where \(\theta\) is the angle between the incident polarisation and the transmission axis.

6.4 Interference & diffraction – intensity from amplitude addition

\[ I_{\text{tot}} = \big\langle\bigl(A_{1}\cos\phi_{1}+A_{2}\cos\phi_{2}\bigr)^{2}\big\rangle = I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos\Delta\phi . \] This formula underpins:

• Young’s double‑slit experiment
• Thin‑film fringes
• Single‑slit and grating diffraction patterns

7. Worked examples

7.1 Intensity from a loudspeaker (sound)

Given: Average power \(P=2.0\;\text{W}\); spherical wave approximated as planar at distance \(r=0.10\;\text{m}\).

1. Projected area: \(A_{\perp}= \pi r^{2}= \pi(0.10)^{2}=3.14\times10^{-2}\;\text{m}^{2}\).
2. Intensity: \(I = P/A_{\perp}= 2.0/3.14\times10^{-2}\approx 6.4\times10^{1}\;\text{W m}^{-2}\).

7.2 Power in a vibrating string

Given: String tension \(T=50\;\text{N}\), linear density \(\mu=0.01\;\text{kg m}^{-1}\), frequency \(f=100\;\text{Hz}\), amplitude \(A=2\;\text{mm}\).

1. Wave speed: \(v=\sqrt{T/\mu}= \sqrt{50/0.01}=70.7\;\text{m s}^{-1}\).
2. \(\omega = 2\pi f = 628\;\text{rad s}^{-1}\).
3. Average power: \(P_{\text{avg}}=\tfrac12\mu\omega^{2}A^{2}v =\tfrac12(0.01)(628)^{2}(2\times10^{-3})^{2}(70.7) \approx 0.28\;\text{W}.\)

8. Common misconceptions

• Intensity is not the same as amplitude. Intensity varies with the **square** of the amplitude (or field strength).
• Area must be perpendicular to the direction of travel. Use the projected area \(A_{\perp}=A\cos\theta\) for inclined detectors.
• Intensity is a scalar quantity. It has magnitude only; the direction is carried by the power (energy‑flux) vector.
• All waves obey \(I\propto A^{2}\). The relation holds for linear, harmonic waves in non‑absorbing media. Strongly absorbing or non‑linear media require additional factors.

9. Summary of key formulas

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	\(I_{\text{tot}}\)	\(I_{\text{tot}} = I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos\Delta\phi\)

10. Quick checklist for exam questions

1. Identify the wave type and the relevant amplitude (displacement, pressure, \(E\)‑field).
2. Confirm that the area used is the **projected** area perpendicular to propagation.
3. If power is given, convert to intensity with \(I=P/A_{\perp}\); if intensity is given, obtain power by \(P=IA_{\perp}\).
4. When the problem involves a change of amplitude, apply \(I\propto A^{2}\) (or \(I\propto E_{0}^{2}\) for EM).
5. For Doppler or polarisation questions, write the appropriate formula first, then substitute the numbers.
6. In interference or diffraction problems, start from amplitude addition, square the result and take the time average to obtain intensity.
7. Check units carefully (W, W m⁻², Hz, m, etc.) and remember that intensity is a scalar.