Wave Equation

Definition and Basic Form

Standard Form

One-dimensional wave equation: second-order linear PDE. Expresses wave-like phenomena. Standard form:

∂²u/∂t² = c² ∂²u/∂x²

Where: u(x,t) is the wave function; c > 0 is wave speed; x spatial variable; t time variable.

Multidimensional Generalization

In ℝⁿ, wave equation generalizes to:

∂²u/∂t² = c² ∇²u

∇² is Laplacian operator: sum of second partial derivatives in spatial dimensions.

Linearity and Homogeneity

Wave equation is linear, homogeneous: superposition principle applies. Solutions can be added to form new solutions.

Physical Interpretation

Wave Propagation

Models propagation of mechanical, electromagnetic, acoustic waves. Disturbances travel at finite speed c.

Examples of Physical Systems

Strings under tension, sound waves in air, electromagnetic waves in vacuum, seismic waves in Earth’s crust.

Energy Conservation

Energy density and flux conserved in absence of damping. Wave equation derivation often from energy principles.

Derivation of the Wave Equation

From Newton’s Second Law: Vibrating String

Consider infinitesimal string element. Tension T, linear density ρ. Transverse displacement u(x,t).

Balance of forces yields differential equation:

ρ ∂²u/∂t² = T ∂²u/∂x²

Divide by ρ → standard form with c² = T/ρ.

From Conservation Laws: Electromagnetic Waves

Maxwell’s equations imply wave equation for electric and magnetic fields in vacuum with c speed of light.

From Continuum Mechanics

Elastic media obey wave equation for small disturbances; relates stress and strain via Hooke’s law.

Classification of PDEs: Hyperbolic Nature

Types of Second-Order PDEs

General PDE form: A ∂²u/∂x² + 2B ∂²u/∂x∂t + C ∂²u/∂t² + ... Classification by discriminant D = B² - AC.

Hyperbolic Classification

Wave equation: B=0, A = -c², C = 1, D = 0 - (-c²)(1) = c² > 0 → hyperbolic PDE.

Implications of Hyperbolicity

Finite speed of propagation, well-posed initial value problem, existence of characteristic curves.

Initial and Boundary Conditions

Initial Conditions

Specify displacement and velocity at t=0:

u(x,0) = f(x), ∂u/∂t (x,0) = g(x)

Boundary Conditions

Common types: Dirichlet (fixed), Neumann (free), Robin (mixed), periodic. Essential for unique solutions.

Well-Posedness

Existence, uniqueness, continuous dependence on data ensured by appropriate initial-boundary conditions.

Methods of Solution

Analytical Techniques

d'Alembert formula, separation of variables, integral transforms, Fourier methods.

Qualitative Analysis

Energy methods, maximum principles, domain of dependence, and influence.

Numerical Approaches

Finite difference, finite element, spectral methods for approximating solutions.

d'Alembert's Solution

One-Dimensional Infinite String

Solution for initial value problem on ℝ without boundaries:

u(x,t) = ½[f(x - ct) + f(x + ct)] + (1/2c) ∫_{x - ct}^{x + ct} g(s) ds

Interpretation

Superposition of two traveling waves moving left and right at speed c.

Limitations

Applies only in one dimension, infinite domain, no boundary conditions.

Separation of Variables

Assumption and Procedure

Assume solution u(x,t) = X(x)T(t). Substitute into PDE to obtain ODEs:

∂²u/∂t² = c² ∂²u/∂x² → T''(t)/[c²T(t)] = X''(x)/X(x) = -λ

Eigenvalue Problem

Spatial part: Sturm-Liouville problem with eigenvalues λ and eigenfunctions X(x).

Superposition Principle

General solution is sum over eigenmodes weighted by initial data coefficients.

Fourier Series and Eigenfunction Expansion

Fourier Series Representation

Periodic boundary condition solutions expressed as:

u(x,t) = Σ [A_n cos(c n π t / L) + B_n sin(c n π t / L)] sin(n π x / L)

Coefficients Determination

Coefficients A_n, B_n determined from initial conditions via orthogonality relations.

Convergence and Smoothness

Regularity of initial data affects convergence speed and smoothness of solution.

Wave Equation in Higher Dimensions

Formulation in ℝ² and ℝ³

Equation: ∂²u/∂t² = c² (∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) or ∂²u/∂t² = c² ∇²u.

Radial Symmetry

For radial solutions u(r,t), reduce PDE to:

∂²u/∂t² = c² (∂²u/∂r² + (n-1)/r ∂u/∂r)

Huygens’ Principle

In odd spatial dimensions ≥3, signals propagate sharply on wavefront; in even dimensions, tail effects occur.

Applications in Physics and Engineering

Acoustics

Sound waves modeled by wave equation with pressure or velocity potential as unknown.

Electromagnetics

Electric and magnetic fields satisfy wave equation in vacuum and non-conductive media.

Structural Vibrations

Strings, beams, membranes: vibrational modes governed by wave equation variants.

Numerical Methods

Finite Difference Methods (FDM)

Discretize space and time derivatives. Explicit, implicit schemes. Stability conditions (e.g., CFL condition).

Finite Element Methods (FEM)

Spatial discretization by piecewise polynomial basis. Flexible geometry handling.

Spectral Methods

High accuracy for smooth solutions; global basis functions (Fourier, Chebyshev).

Stability and Convergence

Time step and mesh size critical. CFL condition: c Δt / Δx ≤ 1 for explicit schemes.

Example: 1D explicit finite difference schemeu_i^{n+1} = 2u_i^n - u_i^{n-1} + (c Δt / Δx)² (u_{i+1}^n - 2u_i^n + u_{i-1}^n) 

References

• Evans, L.C. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010, pp. 237-245.
• Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 2. Wiley, 1962, pp. 123-160.
• John, F. Partial Differential Equations. Springer-Verlag, 1982, pp. 112-130.
• Strauss, W.A. Partial Differential Equations: An Introduction. Wiley, 2007, pp. 210-230.
• Rauch, J. Partial Differential Equations. Graduate Texts in Mathematics, Vol. 128, Springer, 1991, pp. 75-95.