Simple Wave¶
A simple wave is a smooth, one-characteristic-family region of a hyperbolic flow in which state changes while the other relevant invariant stays fixed.
Core Idea¶
A simple wave is a smooth region of hyperbolic flow in which states vary through one characteristic family while the other relevant invariant stays fixed. In one-dimensional shallow water, a 1-rarefaction has fixed u+2√(gh) and fan rays x/t=u−√(gh). These are model-specific formulas, not universal wave equations.[^ref-75e3b656339e]
Scope of Application¶
Clawpack's h_L=4, h_R=1, zero-velocity dam break has a 1-rarefaction component and a separate 2-shock. The smooth fan, not the whole solution, is a simple wave. In steady supersonic gas turning around a convex corner, a Prandtl–Meyer expansion fan is another case: a flow-angle/Mach invariant relation constrains changing states while Mach rises and pressure falls.[ref-75e3b656339e][ref-b566955a1833][^ref-e6693c0be7ff]
Clarity¶
The left shallow-water state h=4,u=0 fixes u+2√(gh)=4√g; different fan depths determine corresponding velocities, and each local state lies on its characteristic ray. At the gas corner, ν(M₁)=ν(M₀)+a relates downstream Mach to turn angle a. The shared structure is one-family variation, not identical state variables or formulas.[ref-75e3b656339e][ref-b566955a1833]
Manages Complexity¶
The fixed invariant reduces a multi-variable flow region to one state curve with characteristic geometry. This fails as a description of a shock, a uniform state with no active wave, or a composite region involving both families. Compressing characteristics may require a discontinuous shock and entropy condition rather than a continued smooth-fan construction.[^ref-75e3b656339e]
Abstract Reasoning¶
Identify the model's characteristic families, choose a reference state, test whether one invariant remains fixed as state varies continuously, then locate states along the appropriate characteristics. If both invariants vary or a jump intervenes, the proposed region is not one simple wave. Do not transfer the shallow-water invariant formula to gas dynamics.[ref-75e3b656339e][ref-e6693c0be7ff]
Knowledge Transfer¶
Shallow-water rarefaction and gas-corner expansion share a way of recognizing restricted characteristic structure. Their equations, physical meanings and boundary conditions do not transfer. The simple-wave name is therefore domain-specific to hyperbolic-flow analysis. The live Differential Equation entry is the governing-system prerequisite, not a genus of which this solution regime is a subtype.[ref-4dc83326cc1d][ref-75e3b656339e]
[^ref-75e3b656339e]: Clawpack Riemann book, shallow-water chapter, integral curves and dam-break correct solution.
[^ref-b566955a1833]: NASA Glenn, “Centered Expansion Fan”, convex-corner expansion.
[^ref-e6693c0be7ff]: K. Niemeyer, Gas Dynamics Notes, Method of Characteristics, θ±ν(M) invariants.
[^ref-4dc83326cc1d]: P. Germain, NACA Technical Note 3299, centered simple-wave corner discussion.
Relationships to Other Abstractions¶
Current abstraction Simple Wave Domain-specific
Parents (1) — more general patterns this builds on
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Simple Wave presupposes Differential equation Domain-specific
A simple-wave solution presupposes a differential-equation system.
Hierarchy paths (2) — routes to 2 parentless roots
- Simple Wave → Differential equation → Derivative → Function (Mapping)
- Simple Wave → Differential equation → Derivative → Convergence
Neighborhood in Abstraction Space¶
Simple Wave sits in a sparse region of the domain-specific corpus (81st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Galloping Instability — 0.83
- Control-Theoretic Orbit — 0.83
- Discrete rate simulation — 0.82
- Brunt–Väisälä Frequency — 0.82
- Kinematic Wave — 0.82
Computed from structural-signature embeddings · 2026-10-08