11: Wave Generation and Absorption Zones

[philipnickel]

Description

Implement relaxation zones for generating waves at the inlet and absorbing waves at the outlet. This prevents reflections and allows simulation of propagating waves.

Background

The relaxation method adds a forcing term to the evolution equations:

∂q/∂t = N(q) + (1 - γ(x)) · τ⁻¹ · (q_a - q)

where:

• q is η or φ̃
• γ(x) = 0 in relaxation zone, 1 in physical domain
• q_a is target solution (input wave for generation, zero for absorption)
• τ = Δt (relaxation time scale)

Tasks

• Implement blending function γ(x)
  • Smooth transition (cosine or polynomial ramp)
  • Configurable zone lengths
• Implement wave generation zone
  • q_a from stream function wave solution
  • Phase-matched traveling wave
• Implement absorption zone
  • q_a = 0 (still water target)
• Modify RHS to include relaxation terms
• Test wave generation (clean input wave)
• Test absorption (minimal reflection)

Acceptance Criteria

• Generated wave matches target amplitude within 5%
• Generated wave matches target phase speed within 2%
• Reflection coefficient < 5% at absorption zone
• No spurious disturbances from zone boundaries

Configuration Parameters

Generation zone: x ∈ [0, L_gen], typically 1-2 wavelengths
Absorption zone: x ∈ [L - L_abs, L], typically 1-2 wavelengths
Blending: γ(x) = 0.5·(1 - cos(π·x/L_zone)) or similar