Wave Equation Derivation

The wave equation is the fundamental mathematical result that explains how electromagnetic fields propagate through space and matter. It emerges directly from Maxwell's equations and governs the behavior of electric and magnetic fields as traveling waves. Understanding this derivation is essential for every GATE aspirant and forms the theoretical backbone of all electromagnetic wave analysis.

Core Concept Explanation

Maxwell's four equations completely describe how electric and magnetic fields are generated and how they interact with each other and with matter. When these equations are combined algebraically, they yield second-order partial differential equations in E and H. These are known as the wave equations and they confirm that electromagnetic fields must propagate as waves through any medium.

The derivation begins with Faraday's law: curl E = -dB/dt. Taking the curl of both sides introduces the term curl(curl E). Using the standard vector identity, curl(curl E) equals grad(div E) minus the Laplacian of E. In a source-free region with no free charges, div E equals zero, so the first term vanishes. The curl H term is replaced using Ampere's law, giving the full wave equation in E. The same procedure applied to Ampere's law yields the wave equation in H.

For a lossless, source-free medium with permittivity epsilon and permeability mu, the wave equations are: Laplacian E = mu*epsilon * d2E/dt2 and Laplacian H = mu*epsilon * d2H/dt2. The quantity mu*epsilon determines how fast the wave propagates. In free space this becomes mu0*epsilon0, and the wave velocity equals the speed of light. This is one of the most celebrated results in all of physics.

Mathematical Expression

In phasor form for time-harmonic fields with angular frequency omega, the time derivative d/dt is replaced by j*omega. The wave equation then becomes the Helmholtz equation: (Laplacian + k^2) E = 0, where the wave number k = omega * sqrt(mu*epsilon). For a lossy medium with conductivity sigma, the propagation constant becomes complex: gamma^2 = j*omega*mu*(sigma + j*omega*epsilon). The real part of gamma is the attenuation constant alpha and the imaginary part is the phase constant beta.

For a lossless medium sigma equals zero, so alpha equals zero and gamma equals j*beta with beta = omega*sqrt(mu*epsilon). A plane wave solution for a wave traveling in the positive z direction is E(z,t) = E0 * cos(omega*t - beta*z). This confirms sinusoidal spatial and temporal variation, which is the defining character of a propagating wave.

Practical Understanding

The wave equation tells engineers that any disturbance in the electromagnetic field does not act instantaneously at a distance. Instead it propagates at a finite velocity determined by the medium. In free space this velocity is approximately 3 x 10^8 m/s. In a dielectric medium with relative permittivity epsilon_r, the velocity reduces to c divided by sqrt(epsilon_r). This is why optical fibers made of glass slow light down and confine it to the fiber core.

In lossy media, the wave equation predicts that the amplitude of E and H decay exponentially with distance. This is the mathematical origin of the skin effect in conductors, where high-frequency currents concentrate near the surface. For GATE problems, recognizing whether the medium is lossless or lossy from given parameters is the first step in selecting the correct wave equation form.

Given:
Medium with epsilon_r = 4, mu_r = 1 (lossless dielectric)
frequency f = 300 MHz, sigma = 0

Why this formula applies:
Lossless medium means alpha = 0, gamma = j*beta
beta = omega * sqrt(mu * epsilon)

Formula:
beta = (2*pi*f) * sqrt(mu0 * mu_r * epsilon0 * epsilon_r)

Substitution:
beta = (2*pi * 300e6) * sqrt(4 * pi * 1e-7 * 8.854e-12 * 4)
beta = (1884.96e6) * sqrt(4.443e-18)

Calculation:
beta = (1884.96e6) * (6.666e-9 * 2)
     = (1884.96e6) * (1/(3e8/2))
     = 2 * (2*pi * 300e6) / (3e8)
     = 4*pi = 12.57 rad/m

Final Answer:
beta = 12.57 rad/m
Phase velocity = omega/beta = (2*pi*300e6)/12.57 = 1.5 x 10^8 m/s = c/2

Exam Tip: In GATE, if the medium is lossless (sigma = 0), the wave equation simplifies to Laplacian E = mu*epsilon * d2E/dt2 and gamma is purely imaginary. If sigma is given and is not zero, use gamma^2 = j*omega*mu*(sigma + j*omega*epsilon) and compute alpha and beta separately. Confusing these two cases is the most common GATE error in wave propagation problems.

• The wave equation is derived by taking curl of Faraday law and substituting Ampere law, eliminating H to get an equation purely in E.
• The vector identity curl(curl E) = grad(div E) - laplacian E is the critical algebraic step. In source-free regions div E equals zero.
• For lossless media, the propagation constant gamma equals j*beta and the wave travels without attenuation. For lossy media, gamma has both real part alpha (attenuation) and imaginary part beta (phase).
• The phasor-domain version of the wave equation is the Helmholtz equation: (laplacian + k^2) E = 0.
• Free space wave velocity c = 1/sqrt(mu0*epsilon0) = 3 x 10^8 m/s emerges naturally from this derivation.

Quick Revision

• Wave equation: laplacian E = mu*epsilon * d2E/dt2 for lossless source-free medium.
• Helmholtz form: (laplacian + k^2) E = 0, where k = omega*sqrt(mu*epsilon).
• Lossy medium: gamma^2 = j*omega*mu*(sigma + j*omega*epsilon), giving complex gamma = alpha + j*beta.
• Lossless medium: sigma = 0, alpha = 0, gamma = j*beta.
• Free space: c = 1/sqrt(mu0*epsilon0) = 3 x 10^8 m/s.
• Exam trap: Do not set div E = 0 in a region with free charges. The source-free condition must be confirmed before applying the simplified wave equation.
• GATE shortcut: If mu_r = 1 and epsilon_r is given, phase velocity = c/sqrt(epsilon_r). Wavelength in medium = free-space wavelength / sqrt(epsilon_r).