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Gravity Waves

Gravity Waves

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Gravity Waves

Gravity Waves

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Kayak Surfing on ocean gravity waves Oregon Coast

Waves:sea & ocean waves

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Sound Waves

Sound Waves:

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Sound Waves

Linear Waves

Sound Waves

compression

rarefaction

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H

L

H

L L

Phase & GroupVelocity

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Doppler Effect

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Shock Waves

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Shocks1. Shocks are sudden transitions in flow properties

such as density, velocity and pressure;

2. In shocks the kinetic energy of the flow is convertedinto heat, (pressure);

3. Shocks are inevitable if sound waves propagate overlong distances;

4. Shocks always occur when a flow hits an obstacle supersonically

5. In shocks, the flow speed along the shock normalchanges from supersonic to subsonic

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Wave BreakingHigh-pressure/densityregions move faster

( 1)/2

0

0

0

21

1

s

s

cu

c

Shock must form

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Chelyabinsk Meteorite (Feb. 2013):Sonic Boom 

Examples of Astrophysical shocks

Cometary bow-shocks

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Earth’s bow shock

Heliosphere

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Supernova Remnant Cassiopeia A

Supernova blast waves

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Tycho’s Remnant (SN 1572AD)

Tsar Bomba Nuclear Explosion

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Tsar Bomba Nuclear Explosion

Tsar Bomba Nuclear Explosion

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Hiroshima, the Shockwave

Radio galaxyCygnus A

Radio picture

X-ray picture

Hot spotsare shocks!

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‘Knots’ in jet of Galaxy M87 are shocks!

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Summary :Shock Physics

Across an infinitely thin steady shock you have, in the shock frame where the shock is at rest, the following Rankine-Hugoniot Jump conditions:

Mass-flux conservation

Momentum-flux conservation

Energy-flux conservation

1 1 2 2n nV V

2 2

1 1 1 2 2 2n nV P V P

1 2t tV V

2 21 21 11 22 2

1 2( 1) ( 1)n n

P PV V

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Summary: Rankine-Hugoniot relations

(for normal shock)

22

21

22

1

1 1

1 2 1

2 1

1

s

s

sP

P

1

1

shock speed

sound speedss

V

c Fundamental parameter:

Mach Number

R-H Jump Conditionsrelate the up- and downstreamquantities at the shock:

From normal shock to oblique shocks:

All relations remain the same if one makes the replacement:

1 n1 1 1

n n1 1 1

cos ,

/ cosS s S

V V V

V c

is the angle between upstream velocity and normal on shock surface

Tangential velocity along shock surface is unchanged

t1 1 1 t2 2 2sin sinV V V V

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Example from Jet/Rocket engines

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Supernova Remnants

Tycho SNR(1572)

SN1006 SNR(1006)

Cas A(1680 ?) 

Kepler(1604)  

Cas A 

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Cas A SNR flythrough

Theory of Supernova Blast Waves

Supernovae:

Type Ia Subsonic deflagration wave turning into a supersonic detonation wave in outerlayers.

Mechanism: explosive carbon burning ina mass-accreting white dwarf

Type Ib-Ic& Core collapse of massive star

Type II

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Supernova II Explosion: SN1054

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Supernova Ia Explosion

Blast waves

Main properties:

1. Strong shock propagating through the Interstellar Medium,or through the wind of the progenitor star;

2. Different expansion stages:

- Free expansion stage (t < 1000 yr) R t

- Sedov-Taylor stage (1000 yr < t < 10,000 yr) R t 2/5

- Pressure-driven snowplow (10,000 yr < t < 250,000 yr) R t 3/10

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Tsar Bomba Nuclear Explosion

Sedov-TaylorExpansion Law

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Blast waves

Main properties:

1. Strong shock propagating through the Interstellar Medium,or through the wind of the progenitor star;

2. Different expansion stages:

- Free expansion stage (t < 1000 yr) R t

- Sedov-Taylor stage (1000 yr < t < 10,000 yr) R t 2/5

- Pressure-driven snowplow (10,000 yr < t < 250,000 yr) R t 3/10

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Radio mapCassiopeia A(VLA)

Remnant of Tycho’s supernova of 1572 AD

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An old supernova remnant (age ~ 10,000 years)

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Free-expansion phaseEnergy budget:

2533

grav 5

99% into neutrino's10 erg

1% into mechanical energyc

c

GME

R

-1/21/2

ejmech mechexp 51

ej

23000 km/s

10 erg 10

ME EV

M M

Expansion speed:

Sedov-Taylor stage

‐Expansion starts to decelerate due to swept-up mass

- Interior of the bubble is reheated due to reverse shock

- Hot bubble is preceded in ISM by strong blast wave

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1/ 2 1/ 2

snr03 3

ej d d

2 1 1

1 / 1 /s

EV V

M R R R R

3/ 2Rt

2/ 5t

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Shock relationsfor strong

(high-Mach number)shocks:

22

21

2 22 1 1 12

221 1

1

22 1 1

1 1

1 2 1

as 2 1 2

1 +1

2

1

s

s

ssss

V VP c PP

P V

2 22 1 ism

2 2

1 1s sP P V

SNRi i

3S

4

3

1 1E

P eR

Pressure behind strongshock (blast wave)

Pressure in hot SNR interior

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At contact discontinuity:equal pressure on bothsides!

2 SNRism

3s

4

3

21

1 s

EV

R

This procedure is allowed because of high sound speedsin hot interior and in shell of hot, shocked ISM:No large pressure differences are possible!

At contact discontinuity:equal pressure on bothsides!

2 SNRism

3s

4

3

21

1 s

EV

R

1/ 2

3/ 2snr2

ism

8

3 1s

s s

dR EV R

dt

Relation betweenvelocity and radiusgives expansion law!

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1/ 2

3/ 2 snrs s 2

ism

8

3 1

ER dR dt

Step 1: write the relationas difference equation

1/ 2

3/ 2 snrs s 2

ism

1/ 2

5/2 snrs 2

ism

8

3 1

2 8

5 3 1

ER dR dt

Ed R dt

Step 2: write as totaldifferentials and………

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1/ 2

3/ 2 snrs s 2

ism

1/ 2

5/2 snrs 2

ism

1/5

2 /5snrs

ism

1/52 /5

2

8

3 1

2 8

5 3 1

( ) ,

5 81.96

2 3 1

ER dR dt

Ed R dt

ER t C t

C

……integrate to find theSedov‐Taylor solution

Sedov & Taylor