Vibrations, Waves & Optics

1. The Wave Equation

Derivation from a vibrating string. Consider a string of linear mass density \( \mu \) under tension T. An infinitesimal element of length dx at position x is displaced vertically by \( u(x,t) \). The net vertical force on the element comes from the difference in the vertical components of tension at its two ends. For small displacements (\( \sin\theta \approx \tan\theta = \partial u/\partial x \)):

Applying Newton's second law (\( F = ma \)) to the element of mass \( dm = \mu\,dx \):\( \mu\,dx\,\frac{\partial^2 u}{\partial t^2} = T\frac{\partial^2 u}{\partial x^2}\,dx \). Dividing through by \( \mu\,dx \):

d'Alembert's general solution. Substituting the change of variables\( \xi = x - vt \), \( \eta = x + vt \) transforms the wave equation into\( \frac{\partial^2 u}{\partial\xi\,\partial\eta} = 0 \), which integrates to:

Any shape propagates without distortion at speed v: f travels right, g travels left. This is the most general solution and shows that wave propagation is a fundamental consequence of Newton's laws applied to elastic media.

2. Sinusoidal Traveling Waves

The most important special case of d'Alembert's solution is the harmonic (sinusoidal) traveling wave, which forms a complete basis for arbitrary waveforms via Fourier analysis:

Parameter definitions and relations:

• A: Amplitude (maximum displacement)
• k = 2π/λ: Wave number (spatial frequency)
• ω = 2πf: Angular frequency
• φ: Initial phase

Substitution into the wave equation requires \( \omega^2 = v^2 k^2 \), giving the fundamental relations:

Complex notation \( u = A\,e^{i(kx - \omega t)} \) (taking the real part) greatly simplifies calculations involving superposition, interference, and impedance matching.

3. Maxwell's Equations and the EM Wave Equation

The four Maxwell equations in differential form (SI units):

Derivation of EM wave equation. In free space (\( \rho = 0, \vec{J} = 0 \)), take the curl of Faraday's law:\( \nabla\times(\nabla\times\vec{E}) = -\frac{\partial}{\partial t}(\nabla\times\vec{B}) \). Using the vector identity \( \nabla\times(\nabla\times\vec{E}) = \nabla(\nabla\cdot\vec{E}) - \nabla^2\vec{E} \), with \( \nabla\cdot\vec{E} = 0 \) in free space, and substituting Ampere-Maxwell:

This is a wave equation with speed \( c = 1/\sqrt{\mu_0\epsilon_0} \approx 3\times 10^8 \) m/s. An identical equation holds for \( \vec{B} \). Maxwell's profound insight: light is an electromagnetic wave.

4. Electromagnetic Plane Waves

A monochromatic plane wave propagating in the x-direction has the form:

Faraday's law applied to this solution requires \( E_0 = cB_0 \). The E and B fields are perpendicular to each other and to the direction of propagation (transverse wave), and oscillate in phase.

Energy transport: Poynting vector. The energy flux (power per unit area) carried by the wave is:

The time-averaged intensity is \( \langle S \rangle = \frac{1}{2}\frac{E_0^2}{\mu_0 c} = \frac{1}{2}c\epsilon_0 E_0^2 \). The Poynting vector points in the direction of wave propagation and its magnitude gives the intensity (power per unit area) -- essential for understanding radiation pressure, antenna radiation patterns, and energy balance in optical systems.

5. Dispersion Relations

The dispersion relation \( \omega(k) \) encodes all propagation properties of a medium. It is obtained by substituting a plane-wave ansatz\( e^{i(kx - \omega t)} \) into the governing wave equation.

Phase and group velocity. These two distinct velocities characterize wave propagation:

The phase velocity is the speed of individual wave crests. The group velocity is the speed at which the envelope of a wave packet (and energy/information) propagates. When \( v_p \neq v_g \) the medium is dispersive: wave packets spread over time. Normal dispersion: \( dv_p/dk < 0 \) (\( v_g < v_p \)). Anomalous dispersion: \( dv_p/dk > 0 \) (\( v_g > v_p \)).

6. Superposition and Interference

Principle of superposition. The wave equation is linear, so any sum of solutions is also a solution. When two coherent waves of equal amplitude meet with a phase difference \( \delta \), their combined intensity follows from adding the amplitudes:

Two-source interference (Young's experiment). Two slits separated by distance d illuminate a distant screen. The path difference is \( \Delta = d\sin\theta \) and the phase difference is \( \delta = 2\pi d\sin\theta/\lambda \). The resulting intensity pattern:

Constructive interference (\( I = 4I_0 \)) when \( d\sin\theta = m\lambda \) (integer m). Destructive interference (\( I = 0 \)) when \( d\sin\theta = (m + \frac{1}{2})\lambda \). This fringe pattern provided the first definitive evidence for the wave nature of light (Thomas Young, 1801).

7. Diffraction

Single-slit Fraunhofer diffraction. When a plane wave passes through a slit of width a, each point in the aperture acts as a secondary source (Huygens' principle). In the far field (Fraunhofer limit), integrating contributions across the slit with path difference accounting:

Minima occur when \( \beta = m\pi \) (\( m = \pm 1, \pm 2, \ldots \)), i.e.,\( a\sin\theta = m\lambda \). The central maximum has angular half-width\( \theta_1 \approx \lambda/a \) -- smaller apertures produce wider diffraction patterns.

Fraunhofer vs Fresnel diffraction. Fraunhofer (far-field) diffraction applies when both source and observation point are effectively at infinity (\( a^2/\lambda L \ll 1 \), Fresnel number ≪ 1). Fresnel (near-field) diffraction includes the curvature of the wavefront and involves more complex Fresnel integrals.

8. Snell's Law of Refraction

Derivation from Fermat's principle. Light travels between two points along the path that minimizes the optical path length \( \int n\,ds \). For a ray crossing an interface between media with indices \( n_1 \) and \( n_2 \), minimizing the total transit time yields:

Alternative derivation from wavefront matching. At the boundary, the tangential component of the wave vector must be continuous (the wavefronts must match along the interface). Since \( k = n\omega/c \) and \( k_{\parallel} = k\sin\theta \):\( k_1\sin\theta_1 = k_2\sin\theta_2 \), which gives Snell's law.

Total internal reflection occurs when \( n_1 > n_2 \) and\( \theta_1 > \theta_c = \arcsin(n_2/n_1) \). The refracted angle would exceed 90 degrees, so all light is reflected. This is the basis for optical fibers and prism-based reflectors.

9. Fresnel Equations

Reflection and transmission at an interface. Applying the electromagnetic boundary conditions (continuity of tangential E and H) at a planar interface between media, we obtain the Fresnel amplitude coefficients for s-polarization (E perpendicular to plane of incidence) and p-polarization (E in the plane of incidence):

Brewster's angle. The p-polarized reflection coefficient vanishes (\( r_p = 0 \)) when \( n_2\cos\theta_i = n_1\cos\theta_t \). Combined with Snell's law, this gives:

At Brewster's angle, only s-polarized light is reflected. This is used in polarizing beam splitters and explains why sunlight reflected from water/glass at glancing angles is partially polarized (polarized sunglasses exploit this effect).

10. Fourier Transform and Wave Packets

Any waveform can be decomposed into sinusoidal components. The Fourier transform pair relates a function in position space to its spectral content in wave-number space:

Uncertainty principle. A fundamental property of Fourier transform pairs: the widths of a function and its transform are inversely related. For a wave packet with spatial width\( \Delta x \) and spectral width \( \Delta k \):

A well-localized wave packet (small \( \Delta x \)) requires a broad spectrum (large\( \Delta k \)), and vice versa. Using de Broglie's relation \( p = \hbar k \), this becomes the Heisenberg uncertainty principle \( \Delta x\,\Delta p \geq \hbar/2 \) -- the quantum mechanical uncertainty principle is purely a property of waves.

11. Thin Lens and Lensmaker's Equation

Derivation from refraction at two surfaces. Apply Snell's law (in the paraxial/small-angle approximation \( \sin\theta \approx \theta \)) to refraction at a single spherical surface of radius R separating media \( n_1 \) and \( n_2 \):\( \frac{n_1}{s} + \frac{n_2}{s'} = \frac{n_2 - n_1}{R} \). Applying this twice (front and back surfaces of the lens) and taking the thin-lens limit (lens thickness ≈ 0) gives the Lensmaker's equation:

where n is the lens refractive index (in air), and \( R_1, R_2 \) are the radii of curvature of the two surfaces. The thin lens equation relating object and image distances:

The magnification is \( M = -d_i/d_o \). A negative \( d_i \) indicates a virtual image (same side as object). This equation forms the basis of geometric optics and optical instrument design -- microscopes, telescopes, cameras, and the human eye all obey this relation.

12. Rayleigh Criterion (Resolution Limit)

Diffraction limits resolution. A circular aperture of diameter D produces an Airy diffraction pattern (the 2D analog of the sinc-squared pattern). The angular radius of the first dark ring is obtained from the zeros of the Bessel function \( J_1 \):

The Rayleigh criterion states that two point sources are just resolvable when the central maximum of one falls on the first minimum of the other -- that is, when their angular separation equals \( \theta_{\min} \).

This is a fundamental limit: no matter how perfect the optics, diffraction sets the ultimate resolution. For a telescope with D = 10 cm observing at \( \lambda \) = 550 nm:\( \theta_{\min} \approx 6.7\times 10^{-6} \) rad ≈ 1.4 arcseconds. Larger apertures and shorter wavelengths give better resolution -- this drives the construction of large telescopes and the use of electron microscopy (shorter de Broglie wavelength) for higher resolution imaging.