The Wave Equation

The Wave Equation

1. Initial Value Problem

   • δy/δt at t=0
     • p(x)
   • y(x,t) at t=0
     • h(x)
   • y(x,t) = ½h(x-ct) + ½h(x+ct) +1/2c∫p(s)ds integral from (x-ct) to (x+ct)

2. Harmonic Solutions

   • y(x,t) = A(x)cos(ωt)
     • A(x) = A₁cos(ωx/c) + A₂sin(ωx/c)

3. D'Alembert's Solution

   • y(x,t) = f(x-ct) + g(x+ct)

4. δ²y/δt² = c² δ²y/δx²

   • c² = T/ρ
   • 2nd order, linear

5. Group and Phase Velocity

   • Cph = ω(k)/k
   • [graph wave]
     • y(x,t) = A₀ exp[iθ]; θ = ωt-kx
   • [graph wave packets]

6. Complex Solution

   • y(x,t) = A₀ exp[i(ωt-kx)]
   • ω² = c²k²
     • τ = 2π/ω
     • λ = 2π/k
     • λ = cτ

7. Two K Solutions

   • Produces wave packets
   • y(x,t) = a cos(ω₁t-k₁x) + a cos(ω₂t-k₂x) = {2a cos[t(ω₂-ω₁)/2 - x(k₂-k₁)/2]} *cos[t(ω₂+ω₁)/2 - x(k₂+k₁)/2]
   • [graph sweet]
     • Rapidly varying signal
       • Frequency: (ω₂+ω₁)/2
       • Wavenumber: (k₂+k₁)/2
       • Point of constant phase have speed: δx/δt = (ω₂+ω₁)/(k₂+k₁) ≈ ω₁/k₁ = Cph (if ω₂≈ω₁ and k₂≈k₁)
     • Amplitude envelope
       • Speed: (ω₂-ω₁)/(k₂-k₁) ≈ δω/δk|k₁ = Cg

8. Fourier Solutions

9. Wave Reflection