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Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit. Gas Dynamics, Lecture 6 (Waves & shocks) see: www.astro.ru.nl/~achterb/. Phase- and group velocity. Central concepts: Phase velocity: velocity with which surfaces of constant phase move
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Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, RadboudUniversiteit Gas Dynamics, Lecture 6(Waves & shocks)see: www.astro.ru.nl/~achterb/
Phase- and group velocity Central concepts: Phase velocity: velocity with which surfaces of constant phasemove Group velocity: velocity with which slow modulations of the wave amplitude move
Phase velocity Definition phase S
Phase velocity Definition phase S Definition phase-velocity
Phase velocity Definition phase S Definition phase-velocity
Group velocity: the case of a “narrow” wave packet (cntd) This should vanish for constructive interference!
Group Velocity Wave-packet, Fourier Integral
Group Velocity Wave-packet, Fourier Integral Phase factor x effective amplitude
Group Velocity Wave-packet, Fourier Integral Phase factor x effective amplitude Constructive interference in integral when
Fundamental equations: Incompressible, constant density fluid (like water!) Constant gravitational acceleration in z-direction; Fluid at rest without waves
Equation of motion small perturbations: SAME as for SOUND WAVES!
There are boundary conditions: #1 At bottom (z=0)we must have az = 0:
There are boundary conditions: #2 2. At water’s surface we must have P = Patm:
There are boundary conditions: #2 2. At water’s surface we must have P = Patm:
Limits of SHALLOW and DEEP lake Shallow lake: Deep lake:
Universal form using dimensionless variables for frequency and wavenumber: deep lake shallow lake
Finally: ship waves Situation in rest frame ship: quasi-stationary
Case of a deep lake wave frequency: wave vector: Ship moves in x-direction with velocity U 1: Wave frequency should vanish in ship’s rest frame: Doppler:
Case of a deep lake (2) wave frequency: wave vector: Ship moves in x-direction with velocity U 2: Wave phase should be stationary for different wavelengths in ship’s rest frame:
Case of a deep lake (3) Ship moves in x-direction with velocity U
Case of a deep lake (4) Ship moves in x-direction with velocity U Wave phase in ship’s frame: Wavenumber:
Case of a deep lake (5) Ship moves in x-direction with velocity U Stationary phase condition for
Kelvin Ship Waves Situation in rest frame ship: quasi-stationary
Shocks: non-linear fluid structures Shocks occur whenever a flow hits an obstacle at a speed larger than the sound speed
Shock properties • Shocks are sudden transitions in flow properties • such as density, velocity and pressure; • In shocks the kinetic energy of the flow is converted • into heat, (pressure); • Shocks are inevitable if sound waves propagate over • long distances; • Shocks always occur when a flow hits an obstacle • supersonically • In shocks, the flow speed along the shock normal • changes from supersonic to subsonic
Time between two `collisions’ `Shock speed’ = growth velocity of the stack.
1 2 Go to frame where the `shock’ is stationary: Incoming marbles: Marbles in stack:
2 1 Flux = density x velocity Incoming flux: Outgoing flux: