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Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit

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Central concepts: Phase velocity: velocity with which surfaces of constant phase move Group velocity: velocity with which slow modulations of the wave amplitude move

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Definition phase S

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Definition phase-velocity

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Definition phase S Definition phase-velocity

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This should vanish for constructive interference!

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Wave-packet, Fourier Integral

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Phase factor x effective amplitude

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Wave-packet, Fourier Integral Phase factor x effective amplitude Constructive interference in integral when

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1.Incompressible, constant density fluid (like water!) 2.Constant gravitational acceleration in z- direction; 3.Fluid at rest without waves

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SAME as for SOUND WAVES!

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1.At bottom ( z=0) we must have a z = 0:

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2. At waters surface we must have P = P atm :

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2. At waters surface we must have P = P atm :

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Shallow lake: Deep lake:

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shallow lake deep lake

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Situation in rest frame ship: quasi-stationary

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wave frequency: wave vector: Ship moves in x -direction with velocity U 1: Wave frequency should vanish in ships rest frame: Doppler:

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wave frequency: wave vector: Ship moves in x -direction with velocity U 2: Wave phase should be stationary for different wavelengths in ships rest frame:

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Ship moves in x -direction with velocity U

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Wave phase in ships frame: Wavenumber:

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Ship moves in x -direction with velocity U Stationary phase condition for

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Situation in rest frame ship: quasi-stationary

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Shocks occur whenever a flow hits an obstacle at a speed larger than the sound speed

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1. Shocks are sudden transitions in flow properties such as density, velocity and pressure; 2.In shocks the kinetic energy of the flow is converted into heat, (pressure); 3.Shocks are inevitable if sound waves propagate over long distances; 4.Shocks always occur when a flow hits an obstacle supersonically 5.In shocks, the flow speed along the shock normal changes from supersonic to subsonic

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Time between two `collisions `Shock speed = growth velocity of the stack.

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Go to frame where the `shock is stationary: Incoming marbles: Marbles in stack: 12

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Flux = density x velocity Incoming flux: Outgoing flux: 1 2

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Conclusions: 1. The density increases across the shock 2. The flux of incoming marbles equals the flux of outgoing marbles in the shock rest frame:

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Generic conservation law:

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Change of the amount of Q in layer of width 2 e: flux in - flux out

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Infinitely thin layer: What goes in must come out : F in = F out

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Infinitely thin layer: What goes in must come out : F in = F out Formal proof: use a limiting process for 0

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Starting point: 1D ideal fluid equations in conservative form; x is the coordinate along shock normal, velocity V along x -axis! Mass conservation Momentum conservation Energy conservation

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Mass flux Momentum flux Energy flux Three equations for three unknowns: post-shock state (2) is uniquely determined by pre-shock state (1)! Three conservation laws means three fluxes for flux in = flux out!

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1D case: Shocks can only exist if M s >1 ! Weak shocks: M s =1+ with << 1; Strong shocks: M s >> 1.

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Sound waves:

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Approximate jump conditions: put P 1 = 0!

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