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Hyperbolic waves

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'Any recognizable signal that is transferred from one part of the medium to ... Shockwave lithotripsy. Shockwave development. Surfing. Hydraulic jump. River surfing ... – PowerPoint PPT presentation

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Title: Hyperbolic waves


1
Hyperbolic waves
  • Shock and awe

2
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.

3
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.

4
Shock waves in gas
5
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6
Detonation waves
7
Detonation Waves

8
Shockwave lithotripsy
9
Shockwave development
10
Surfing
11
Hydraulic jump
12
River surfing
13
Tidal bores (really a soliton)
14
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.

15
Model conservation equation
Flux in
Flux out
dx
16
Waves when flux is function of density
  • Traffic
  • River floods
  • Glaciers
  • Sedimentation
  • Chromatography (chemical separations)
  • Granular flows

17
The simplest wave
18
Simple wave
19
Space-time
density
20
What if speed depends on density?
21
Space-time
Note all features propagate at constant v
22
Some cool math
23
Provides easy method for solution
24
Function cannot be multi-valued.Conservation
matters!
25
Evolution single valued
Energy is dissipated!
26
Flow through the Lincoln tunnel
(cars/hour)
(cars/mile)
27
Sub-critical congestion
28
Sub-critical congestion
29
Super-critical
30
Super-critical
31
What prevents single valued-ness?
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