Questions: Hyperbolic PDE Theory (Wave Propagation and Characteristics)

Question 1

The domain of dependence for the wave equation u_tt = c²Δu at a point (x₀, t₀) is:

• A. The intersection of the backward characteristic cone with the initial surface
• B. The entire initial surface t = 0
• C. A single point on the initial surface
• D. The forward characteristic cone

Answer: (A)

The solution u(x₀, t₀) depends only on initial data within the ball |x - x₀| ≤ ct₀ on the initial surface. This is the intersection of the backward light cone with t = 0. Data outside this region has no influence—information cannot travel faster than speed c.

Question 2

Huygens' principle holds in all spatial dimensions.

• T. True
• F. False

Answer: False

The strong form of Huygens' principle—that a sharp initial pulse produces a sharp signal—holds only in odd dimensions ≥ 3. In even dimensions and in 1D, signals leave 'tails': the wave equation in 2D produces a residual signal after the main wavefront passes (this is why a splash in a pond creates spreading ripples rather than a single sharp ring).

Question 3

What does energy conservation for the wave equation imply about solution regularity?

Model answer: Solutions are exactly as regular as the initial data — they neither smooth out nor develop new singularities

Since E(t) = ½∫[u_t² + c²|∇u|²]dx is conserved, the H¹ norm of the solution is preserved in time. This means initial data in H^k produces a solution in H^k for all time — no smoothing (unlike parabolic) and no loss of regularity.

Question 4

A symmetric hyperbolic system is a first-order system of the form A₀u_t + Σ Aⱼu_{xⱼ} = f where:

• A. A₀ is symmetric positive definite and all Aⱼ are symmetric
• B. All matrices are diagonal
• C. The system is scalar
• D. A₀ = I (identity)

Answer: (A)

Symmetric hyperbolicity is a structural condition guaranteeing well-posedness via energy estimates. The symmetry of Aⱼ enables energy methods: multiplying by u^T A₀ and integrating gives a conserved energy. Maxwell's equations, linearized Euler equations, and Einstein's equations can all be written as symmetric hyperbolic systems.