Introduction to Astrophysical Gas Dynamics

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Introduction to Astrophysical Gas Dynamics
Part 3

• Bram Achterberg
• a.achterberg_at_astro.uu.nl

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Waves
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Astrophysical manifestations of waves
Spiral Density Waves
Solar Oscillations
Large-Scale Structure
Accretion DiskWaves
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Simple (linear) waves

• Properties
• Small perturbations of velocity, density and
  pressure
• Periodic behavior (sines and cosines) in
  space and time
• No effect of boundary conditions

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Perturbations
Perturbations are small
Perturbations are periodic the plane wave
assumption!
with displacement vector
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Perturbation analysis simple mechanical example

• Small-amplitude
• motion
• Valid in the vicinity
• of an equilibrium
• position

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Perturbation analysisfundamental equations
Equilibrium position
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Perturbation analysismotion near x0
Taylor expansion
Near x0
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Perturbation analysismotion near x0
Equation of motion near x0
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Perturbation analysismotion near x0
Solutions
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Waves, wavelength and the wave vector
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Who measures what?
Two fundamental types of observer Observer
fixed to coordinate system measures the Eulerian
perturbation Observer moving with the
flow Measured the Lagrangian perturbation
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Lagrangian labels
Lagrangian Labels are carried along by the flow
Conventional choice position fluid-element at
some reference time
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Re-interpretation of time-derivatives
At a fixed position
Comoving with the flow
Lagrangian and Eulerian perturbations
Lagrangian
Eulerian
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What does it all mean?

• If you observe from a fixed position in space,
  and
• a small-amplitude wave passes, you measure
• A pressure perturbation ?P
• A density perturbation ??
• A velocity perturbation ?V
• If you observe while moving along with the
  unperturbed
• flow, and a small-amplitude wave passes, you
  measure
• A pressure perturbation ?P
• A density perturbation ??
• A velocity perturbation ?V

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Relation between the two perturbations
Effect of the position shift of the co-moving
observer
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Important Commutation Relationsfor taking
derivatives
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Important Commutation Relationsfor taking
derivatives
Small perturbations in position fluid-element
Associated variations
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Important Commutation Relationsfor taking
derivatives
Small perturbations in Position fluid-element
General relation between Lagrangian and
Eulerian Changes
Associated variations
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Important Commutation Relationsfor taking
derivatives
Small perturbations in Position fluid-element
General relation between Lagrangian and
Eulerian Changes
Associated variations
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Application velocity perturbation due to
small-amplitude wave
Commutation Rules
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Application velocity perturbation due to
small-amplitude wave
Commutation Rules
Definition comoving derivative
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Eulerian and Lagrangian velocity perturbations
General relation between the two kinds of
perturbations
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Density perturbations and the deformation tensor
A ?X , B ?Y, C ?Z
The vectors A, B and C are carried along, and
deformed by the flow!
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Definition Deformation Tensor Dij
Small position change
Associated change in position differences
Same animal In tensor notation
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Small displacement (e.g. wave motion)
Component notation
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Small displacement (e.g. wave motion)
Component notation
Matrix notation
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Physical Interpretation of the components of
the Deformation Tensor
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Deformation of an infinitesimal volume-element
After deformation
Before deformation
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Density variation from mass conservation volume
change law
Starting volume orthogonal cube
Deformation due to displacement
New, non-orthogonal volume
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Mathematical IntermezzoDeterminants, vector
products and the totally antisymmetric
Levi-Cevita symbol
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Properties of Levi-Cevita synbol
Also useful the vector cross-product in
?-language
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Volume change in terms of deformation
Alternative notation cross-product
New (deformed) volume
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Volume change in terms of deformation
Alternative notation cross-product
New (deformed) volume
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Volume change in terms of deformation
Alternative notation cross-product
New (deformed) volume
Summation convention for repeated indices i, j
and k!
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Deformation tensor determinant
Definition of the Deformation Tensor
Definition of the determinant of the Deformation
Tensor
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Deformation tensor determinant
Definition of the Deformation Tensor
Definition of the determinant of the Deformation
Tensor
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Deformation tensor determinant
Definition of the Deformation Tensor
Definition of the determinant of the Deformation
Tensor
If displacement is small
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Density perturbation
Relation between old and new (deformed) volume
Deformation Tensor determinant
Definition divergence
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Density perturbation
New volume
Mass conservation
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Density perturbation
New volume
Mass conservation
New density
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Lagrangian and Eulerian density change
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Lagrangian and Eulerian density change
Lagrangian and Eulerian pressure change
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Summary changes due to a small displacement
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Linear sound waves in a homogeneous, stationary
gas

• Main assumptions
• Gas is uniform no gradients in density,
  pressure or
• temperature
• Gas is stationary without the presence of waves
• the gas velocity vanishes
• The velocity, density and pressure perturbations
• associated with the waves are small

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Uniform background no differencebetween
Lagrangian and Eulerian perturbations!!!!
Vanishes if Q is uniform in absence of waves
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Aim
To derive a linear equation of motion for the
displacement vector ??(x,t) by linearizing
the equation of motion for the gas.
Method
Take the Lagrangian variation of the equation of
motion.
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Perturbing the Equation of Motion
To find the equation of motion governing
small perturbations you have to perturb the
equation of motion!
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Unperturbed gas is uniform and at rest
Apply a small displacement
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Unperturbed gas is uniform and at rest
Apply a small displacement
Because the unperturbed state is so simple, the
linear perturbations in density, pressure and
velocity are also simple!
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Effect of linear perturbations on the equation
of motion
Use commutation rules
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Effect of linear perturbations on the equation
of motion
Use commutation rules
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Effect of linear perturbations on the equation
of motion
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Effect of linear perturbations on the equation
of motion
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Effect of linear perturbations on the equation
of motion
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What have we learned about smallperturbations in
a ideal uniform gas?
Small displacement of fluid elements
Equation of motion for the displacement vector
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The wave equation and its solution
Linearized equation of motion wave equation
Definition sound speed Cs2 ? P / ?
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The wave equation and its solution
Linearized equation of motion wave equation
Definition sound speed Cs2 ? P / ?
Trial solution plane wave
Handy because
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Example relations between perturbations and the
wave amplitude
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Example relations between perturbations and the
wave amplitude
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H
H
L
L
L
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Plane-wave solution
Wave equation
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Plane-wave solution
Wave equation
Algebraic relation
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Plane-wave solution
Wave equation
Algebraic relation
Matrix form (assume kz 0)
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Wave dispersion relation for ?
Algebraic wave equation three linear equations
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Wave dispersion relation for ?
Algebraic wave equation three linear equations
Solution condition vanishing determinant
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Wave dispersion relation for ?
Dispersion relation
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Wave polarization direction of a
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Alternative derivation of the wave properties
Wave equation
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Alternative derivation of the wave properties
Wave equation
Divergence of a gradient is the Laplace operator!
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Alternative derivation of the wave properties
Wave equation
Pressure perturbation
Sound waves are pressure waves!
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Alternative derivation of the wave properties
Wave equation
Rotation of a gradient vanishes!
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Alternative derivation of the wave properties
Wave equation
Dispersion relation for compressive perturbations
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Alternative derivation of the wave properties
Wave equation
Dispersion relation for divergence-free
perturbations
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Summary the road to sound waves
Assume a small displacement Calculate linear
response of fluid Find linear
equation-of-motion for the perturbations
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Plane wave assumption convert a partial
differential equation into algebraic equations
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Plane wave assumption convert a partial
differential equation into algebraic equations
Condition for non-trivial solutions
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Sound waves in a moving medium

• Assumptions
• Uniform pressure and density in unperturbed flow
• Constant flow velocity in unperturbed flow

Main effect of flow appearance of the comoving

time-derivative in equations
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Use comoving derivative
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Wave equation
Use comoving derivative
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Wave equation
Plane wave assumption
Use comoving derivative
Doppler-shifted frequency
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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
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Phase velocity
Definition phase S
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Phase velocity
Definition phase S
Definition phase-velocity
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Phase velocity
Definition phase S
Definition phase-velocity
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Group Velocity
Wave-packet, Fourier Integral
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Group Velocity
Wave-packet, Fourier Integral
Phase factor x effective amplitude
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Group Velocity
Wave-packet, Fourier Integral
Phase factor x effective amplitude
Constructive interference in integral
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Summary and example sound waves
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Summary and example sound waves
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Summary and example sound waves
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Summary the path to linear (small-amplitude)
waves

• Calculate the density- and pressure
  perturbations
• given a small displacement
• Find the equation of motion for the displacement
• by perturbing the equation of motion
• Substitute a plane-wave for the perturbation
• 4. Solve the resulting algebraic set of equations.