Prof dr A Achterberg Astronomical Dept IMAPP Radboud
Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit Gas Dynamics, Lecture 6 (Waves: applications) see: www. astro. ru. nl/~achterb/
Linear sound waves in a homogeneous, stationary gas Main assumptions: 1. Unperturbed gas is uniform: no gradients in density, pressure or temperature; 2. Unperturbed gas is stationary: without the presence of waves the velocity vanishes; 3. The velocity, density and pressure perturbations associated with the waves are small
Immediate consequence: perturbations are “simple”: Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave: This is KINEMATICS, not DYNAMICS!
The equation of motion for ξ(x, t):
Plane-wave solution for sound waves
The wave equation and its solution: now finally the use of exponential functions in plane waves pays off:
Plane-wave solution for sound waves An algebraic relation between vectors! Plane wave assumption converts a differential equation into an algebraic equation!
Plane-wave solution for sound waves In matrix notation for k in x-y plane:
Wave dispersion relation for : Algebraic wave equation: three coupled linear equations
Wave dispersion relation for : Algebraic wave equation: three coupled linear equations Solution condition: vanishing determinant, an equation for ω given k
Wave dispersion relation for : Dispersion relation
Wave polarization: direction of amplitude vector a:
H H L L L
Another useful example: relations between perturbations and the wave amplitude
Summary: Sound waves in a uniform medium: Wave equation: Plane-wave solution:
A quick summary of plane-wave method: If you have a wave equation for , then the plane wave assumption corresponds to “simple substitution”:
Generalization to moving medium: Medium with uniform velocity V: Comoving time derivative In unperturbed flow
Use comoving derivative
Wave equation Use comoving derivative
Wave equation Use comoving derivative Doppler-shifted frequency Plane wave assumption
Phase- and group velocity Central concepts: Phase velocity: velocity with which surfaces of constant phase move 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
Group velocity: the case of a “narrow” wave packet
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
Summary and example: sound waves in a moving fluid
Summary and example: sound waves
Summary and example: sound waves
Application: Kelvin Ship Waves
Waves in a lake of constant depth
Fundamental equations: 1. Incompressible, constant density fluid (like water!) 2. Constant gravitational acceleration in z-direction; 3. Fluid at rest without waves
Unperturbed state without waves:
Small perturbations:
Equation of motion small perturbations: SAME as for SOUND WAVES!
Solve for pressure perturbation first!
Solution for pressure perturbation:
Solve equation of motion:
Solve equation of motion:
There are boundary conditions: #1 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 :
Dispersion relation from boundary conditions:
Dispersion relation from boundary conditions:
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
- Slides: 63