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

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

Definition phase S

Definition phase-velocity

Definition phase S Definition phase-velocity

This should vanish for constructive interference!

Wave-packet, Fourier Integral

Phase factor x effective amplitude

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

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

SAME as for SOUND WAVES!

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

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

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

Shallow lake: Deep lake:

shallow lake deep lake

Situation in rest frame ship: quasi-stationary

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

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:

Ship moves in x -direction with velocity U

Wave phase in ships frame: Wavenumber:

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

Situation in rest frame ship: quasi-stationary

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

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

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

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

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

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:

Generic conservation law:

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

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

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

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

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!

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

Sound waves:

Approximate jump conditions: put P 1 = 0!