Wave Properties and Types

A progressive wave carries energy from one point to another. Each point on the wave oscillates about its equilibrium position.

Properties

• Amplitude ($A$): Maximum displacement from equilibrium
• Wavelength ($\lambda$): Distance between consecutive points in phase
• Frequency ($f$): Number of complete oscillations per second (Hz)
• Period ($T$): Time for one complete oscillation; $T = 1/f$
• Wave speed: $v = f\lambda$
• Phase difference ($\Delta\phi$): Measured in radians or degrees. Points one wavelength apart have $\Delta\phi = 2\pi$

$\Delta\phi = \frac{2\pi x}{\lambda}$

where $x$ is the path difference.

Wave Equation

$y = A\sin(\omega t - kx)$

Where:

• $\omega = 2\pi f$ = angular frequency
• $k = 2\pi/\lambda$ = wave number
• The minus sign indicates the wave travels in the positive x-direction

Transverse: Oscillations perpendicular to direction of propagation. Can be polarised.

• EM waves, water surface waves, waves on strings, S-waves

Longitudinal: Oscillations parallel to direction of propagation. Cannot be polarised.

• Sound, ultrasound, P-waves, compression waves in springs

Intensity is the power transmitted per unit area:

$\boxed{I = \frac{P}{A_{\text{area}}}}$

Units: W/m²

For a wave: intensity is proportional to amplitude squared:

$I \propto A^2$

If amplitude doubles, intensity quadruples.

For a point source radiating equally in all directions: $I = \frac{P}{4\pi r^2}$

This gives the inverse square law: $I \propto 1/r^2$.

Polarisation restricts the oscillations of a transverse wave to a single plane.

• Only transverse waves can be polarised
• This proves light is transverse (since it can be polarised)
• A polariser (Polaroid filter) only transmits oscillations in one plane
• Two polarisers at 90° block all light

Malus's Law

When polarised light of intensity $I_0$ passes through a second polariser (analyser) at angle $\theta$:

$I = I_0 \cos^2\theta$

Applications

• Polarising sunglasses (reduce glare from reflected light)
• LCD screens
• Stress analysis in engineering
• Radio/TV antenna alignment

All EM waves:

• Travel at $c = 3 \times 10^8$ m/s in vacuum
• Are transverse
• Can be polarised
• Consist of oscillating electric and magnetic fields
• $c = f\lambda$

Spectrum: Radio → Microwave → IR → Visible → UV → X-ray → Gamma

• Progressive waves carry energy; described by $y = A\sin(\omega t - kx)$
• $v = f\lambda$; $\omega = 2\pi f$; $k = 2\pi/\lambda$
• $I \propto A^2$; for point source $I \propto 1/r^2$
• Only transverse waves can be polarised
• Malus's Law: $I = I_0\cos^2\theta$