Wave Equation

d'Alembert in 1747 solved the vibrating string problem, laying the foundation of PDE theory. A century later Kirchhoff extended the solution to R^3. Two centuries later the Huygens principle explained why a sharp sound in R^3 is heard once and then vanishes, with no 'sound echo'.

• Seismology: S and P waves solve the wave equation in heterogeneous media
• Electromagnetism: Maxwell equations reduce to the wave equation in vacuum
• Gravitational waves: linearized Einstein equations are wave equations for h_mu_nu

Предварительные знания

• Previous lesson

Wave Equation and d'Alembert Formula

d'Alembert in 1747 solved the wave equation u_tt = c^2 u_xx via substitution xi=x+ct, eta=x-ct: u(x,t) = f(x-ct) + g(x+ct). Every solution is a superposition of traveling waves. Propagation speed c. In R^3: u_tt = c^2*Delta u, Kirchhoff formula. Dispersion relation: omega = c|k|.

Wave Equation in R^3 and Huygens Principle

Kirchhoff formula: u(x,t) = d_t(t M_{ct}[phi]) + t M_{ct}[psi], where M_r is spherical mean. Huygens principle (strict): in R^3 the support of the solution propagates exactly on the cone; in R^2 it fills the cone. Decay rate: O(1/t) in 3D, O(1/sqrt(t)) in 2D. Poisson formula in 2D: via descent from 3D.

Key Ideas

• d'Alembert: u = f(x-ct) + g(x+ct) in 1D
• E(t) = (1/2)*integral(u_t^2 + c^2 u_x^2) dx = const
• Dispersion: omega = c|k| -- non-dispersive equation
• Kirchhoff formula in R^3: via spherical mean on |y-x|=ct
• Huygens principle (strict): support only on cone in R^{2n+1}

Further Directions

These ideas open paths to deeper mathematics.

• de-29-einstein-equations — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of mathematics?