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

Waves

Simple (linear) waves Properties: Small perturbations of velocity, density and pressure 1. Small perturbations of velocity, density and pressure 2. Periodic behavior (“sines and cosines”) in space and time 3. No effect of boundary conditions

1.Wave amplitude is small: unperturbed position small displacement

2.Sinusoidal behavior in space and time: plane wave representation Complex amplitude vector Phase factor

Waves, wavelength and the wave vector

3.Wave amplitude is small in the following sense: -|a| is much smaller than the wavelength λ; -|a| is much smaller than gradient scale of the flow; -Density and pressure variations remain small:

Perturbation analysis: -Expand fundamental equations in displacement ξ(x,t); Neglect all terms of order ξ 2 and higher! -Express density and pressure variations in terms of ξ(x,t); Neglect all terms of order ξ 2 and higher! -Find equation of motion for ξ(x,t) where only terms linear in ξ(x,t ) appear; - Substitute plane wave assumption.

Perturbation analysis: simple mechanical example 1.Small-amplitude motion; 2.Valid in the vicinity of an equilibrium position;

Perturbation analysis: fundamental equations Equilibrium position:

Perturbation analysis: motion near x=0 Taylor expansion: Near equilibrium position x=0 :

Perturbation analysis: motion near x=0 Equation of motion near x=0 :

Perturbation analysis: motion near x=0 Solutions:

Two fundamental types of observer in fluid mechanics: Observer fixed to coordinate system measures the Eulerian perturbation: Observer moving with the flow Measured the Lagrangian perturbation

Lagrangian Labels are carried along by the flow

Conventional choice: position x 0 of a fluid-element at some fixed reference time t 0 As always:

At a fixed positionComoving with the flow

At a fixed positionComoving with the flow Lagrangian and Eulerian perturbations: Lagrangian: Eulerian:

Stay at old position! Follow the fluid to new position!

Effect of position shift ξ Small change induced by ξ in Q at fixed position

Formal calculation:

Commutation Rules

Commutation Rules Definition of the comoving derivative:

General relation between the two kinds of perturbations:

Special simple case: stationary unperturbed fluid that has V = 0 :

Another special case: unperturbed fluid has uniform velocity V ≠ 0 :

Mass conservation:

One dimension: Three dimensions:

Adiabatic flow: From general relation between Lagrangian and Eulerian perturbations:

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

Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave:

Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave: This is KINEMATICS, not DYNAMICS!

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.

To find the equation of motion governing small perturbations you have to perturb the equation of motion!

Unperturbed gas is uniform and at rest: Apply a small displacement

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 !

Use commutation rules again:

I have used:

What do we know at this point:

Linearized equation of motion = wave equation Definition sound speed: C s 2 =  P / 

Linearized equation of motion = wave equation Definition sound speed: C s 2 =  P /  Put in plane wave assumption:

An algebraic relation between vectors! Plane wave assumption converts a differential equation into an algebraic equation!

In matrix notation for k in x-y plane:

Algebraic wave equation: three coupled linear equations

Algebraic wave equation: three coupled linear equations Solution condition: vanishing determinant, an equation for ω given k

Dispersion relation

H L H L L

Sound waves in a uniform medium: Wave equation: Plane-wave solution:

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