Hyperbolic waves

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Hyperbolic waves

• Shock and awe

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What is a wave?

• Any recognizable signal that is transferred from
  one part of the medium to another with a
  recognizable velocity of propagation. The
  signal may distort, change its magnitude, and
  change its velocity provided it is still
  recognizable Whitam, 1974.

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What are hyperbolic wave?

• Classification of PDE related to hyperbolas.
• Acoustics, gas dynamics, elasticity, EM
• We have studied linear hyperbolic waves
• Non-linear phenomena shocks and wave breaking.

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Shock waves in gas
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Detonation waves
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Detonation Waves

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Shockwave lithotripsy
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Shockwave development
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Surfing
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Hydraulic jump
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River surfing
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Tidal bores (really a soliton)
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Lots of nonlinearity

• In each case, the physics is different.
• Wave speed depends upon value itself.
• Must account for details of each process, but
  many qualitative features are the same.
• Today, we will take a simplest case kinematic
  waves.
• Kinematic waves are determined by conservation
  not acceleration.

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Model conservation equation
Flux in
Flux out
dx
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Waves when flux is function of density

• Traffic
• River floods
• Glaciers
• Sedimentation
• Chromatography (chemical separations)
• Granular flows

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The simplest wave
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Simple wave
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Space-time
density
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What if speed depends on density?
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Space-time
Note all features propagate at constant v
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Some cool math
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Provides easy method for solution
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Function cannot be multi-valued.Conservation
matters!
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Evolution single valued
Energy is dissipated!
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Flow through the Lincoln tunnel
(cars/hour)
(cars/mile)
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Sub-critical congestion
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Sub-critical congestion
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Super-critical
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Super-critical
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What prevents single valued-ness?