CONTENTS 2
Contents
1 Supernova basics and observational classification 3
2 Introduction to astrophysical fluids 19
3 Hydrodynamic Equations 31
4 Surface waves 43
5 Perturbations at a two-fluid interface 54
6 Density fluctuations and Jeans instability 69
7 Shock waves and supernova envelope expansion 79
8 Shocks jump conditions and envelope expansion 87
9 Particle acceleration in SNRs 98
10 Massive star evolution 110
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 3
1 Supernova basics and observational classification
1.1 Main Course Topics:
1. Supernovae: classification and basic properties
2. Astrophysical Fluids
3. Hydrodynamic instabilities
4. Shocks and particles acceleration
5. Supernovae models and observations
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 4
1.2 Literature
• Prialnik, D., An Introduction to The Theory of Stelalr Structure and
Evolution, 2000 simple introduction into the subject
• Landau & Lifshitz, Fluid Mechanics, 1987 cover most fluid topics
• some other books and web sources indicated in the slides
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 5
1.3 What is a supernova ?
Stars undergoing a tremendous explosions, during which their lumi-
nosity becomes comparable to that of an entire galaxy called supernovae.
Figure 1: Left: Multiwavelength X-ray image of the remnant of Kepler’s Supernova, SN
1604. (Chandra X-ray Observatory); Right: Nova Persei (exploded 1901) from US Naval
Observatory in Flagstaff
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 6
Historically, nova was the name for an apparently new star, e.g. brighten
suddenly by many orders of magnitude. Nova characteristically increases
in brightness some ten magnitude. The rise is very rapid.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 7
1.4 Supernova vs nova
Figure 2: From Zeilik and Gregory, Introductory Astronomy and Astrophysics, 1998
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 8
1.5 SN 1987A
Figure 3: NASA Image of 1987A supernova: The above two photographs are of the
same part of the sky. The photo on the left was taken in 1987 during the supernova
explosion of SN 1987A, while the right hand photo was taken beforehand. Supernovae
are one of the most energetic explosions in nature, making them like a 1028 Megaton
bomb (i.e., a few octillion nuclear warheads).
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 9
1.6 Historical observations of supernovae
Supernovae have been witnessed in our galaxy but only in the histori-
cal past. Now we observe supernova remnants.
Notable supernovae in the Milky Way (our galaxy):
• 1006 - the brightest ever recorded (1/4 Moon!)
• 1054 -the best known (Crab nebula nowadays), first recorded Japanese,
but also by Koreans and then Chinese
• 1572 -Tycho’s supernovae (Tycho Brahe)
• 1604 -a student of Tycho, Kepler, also observed a supernova
• 1680 - Cassiopeia A (allegedly seen by Flamsteed, the first Royal
Astronomer for England) is a gaseous remnant that could have been
a under-luminous supernovae (the last discovered supernovae in
our galaxy).
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 10
1.7 First image of a supernova?
SN 1054, the best known a Crab
nebula, (M1);
It was probably also recorded by
Anasazi Indian artists (in present-
day Arizona and New Mexico), as
findings in Navaho Canyon and
White Mesa
Milton (1978) points that on the
morning of July 5, 1054 the cres-
cent moon came remarkably close to
the supernova, as seen (only) from
Western North America.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 11
1.8 Observations of supernovae: How frequent ?
Figure 4: Image of SN 1987A
There were no supernova
observed in our Galaxy since
1680, although supernovae oc-
cur once per hundred years in
a single spiral galaxy like the
Milky Way.
1987A from the Large Mag-
ellanic Cloud is probably the
best studied supernova of all
times (see Figure 4).
Before 1987 around 30 su-
pernovae were observed yearly,
now it is around a few hundred
per year
For the list of the latest discovered supernovae see: this webpage.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 12
1.9 The latest supernova in the Galaxy...
Cassiopeia A Image: X-ray: NASA/CX-
C/SAO; Optical: NASA/STScI; Infrared:
NASA/JPL-Caltech
The last nearby supernova
explosion occurred in 1680,
It was thought to be just
a normal star at the time,
but it caused a discrepancy
in the observer’s star cata-
logue which historians finally
resolved 300 years later, af-
ter the supernova remnant
(Cassiopeia A) was discov-
ered and its age estimated.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 13
1.10 Observational classification of supernovae
Supernovae explosions are divided into two types in accordance with
spectral properties (observation of some spectral lines in absorption):
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 15
1.12 Lightcurves of supernovae
Schematic light curves for super-
novae of Types Ia, Ib, II-L, II-P (from
http://www.talkorigins.org/faqs/supernova/)
V-band Magnitudes of Type Ia
(from the Perlmutter group (Su-
pernova Cosmology Project)) Note
that absolute magnitude are almost
identical
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 16
1.13 Supernova location within Galaxies
• Type Ia supernovae appear in all sort of galaxies: spiral, elliptical,
and irregular. They tend to avoid the arms of spiral galaxies (regions
of star formation), and hence explode from old (long-lived) stars.
The only type to explode in elliptical galaxies (no star formation)
• Type II supernovae have never appeared in elliptical galaxies, occa-
sionally in irregular galaxies, but mostly in the arms of spiral galax-
ies, e.g. stars leading to Type II supernovae are short-lived =⇒
massive stars
Note: Tycho Brahe and Kepler both witnessed Type Ia supernovae.
1987A was Type II-P.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 17
1.14 Supernova progenitors
Type Ib and Ic only seem to explode in the arms of spiral galaxies (like
type II but unlike Ia). Also are likely to be associated with massive stars
probably Wolf-Rayet stars.
Type Ib and Ic are also strong radio sources unlike Type Ia (presumably
collision of a shock with the media =⇒ strong mass loss)
• Type II and Type Ib/c are likely to represent the explosions of mas-
sive stars
• Type Ia are likely to be a part of double star systems (to increase
white dwarf above Chandrasekhar limit)
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 18
1.15 Example: supernova progenitor
Figure 6: The Hubble Space Telescope’s Advanced Camera for Surveys acquired
these images of a small field inside M51. The left image, taken in January, shows the
field prior to the eruption of Supernova 2005cs. The right image, taken on July 11th,
shows the magnitude-14 supernova. 2005cs supernovae progenitor is a red supergiant
of 7-10 solar masses
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 19
2 Introduction to astrophysical fluids
2.1 Description of N-particle system
Generally a system of N particles can be easily described by the sys-
tem of 6 × N Hamiltonian equations. However, as soon as N ∼ 10
24
À 1
(Avogadro number), the solution of the system becomes impossible to find
and alternative (simplified) methods of description should be applied.
There are a number of simplifications possible:
• Test particles description: Exact solution for M ¿ N particles while
the rest of the particles are treated as an external slow varying me-
dia.
• Fluid (HD) description of plasma: plasma is assumed to be a con-
tinues media at L À l, where L is a scale of processes to consider,
and l is the mean free path of a particle in a plasma.
• Classical kinetics is the study of the relationship between motion
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 20
and the forces affecting motion introducing statistical tools for de-
scription.
Other methods (or combination of the above) are possible.
2.1.1 Fluid Description
Fluid dynamics studies the motion of fluids (liquids and gases) on
macroscopic scale, a fluid is regarded as a continuous medium.
The mathematical description of the state of a moving fluid is via macro-
scopic parameters fluid density ρ, fluid velocity
−→
v , and for instance pres-
sure p.
All macroscopic parameters are functions of position
−→
r and time t .
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 21
2.2 Conservation of matter
Figure 7: Fluid volume V
and surface element d
−→
s .
Let us consider a fluid with mass density ρ
in volume V (Figure 7). The mass of fluid is
simply
M =
Z
V
ρdV
where integration is over the volume V.
The mass of the fluid flowing in unit time
through an surface element d
−→
s is
ρ
−→
v d
−→
s
(
>0 outflow
<0 inflow
Given that there are no sources or sinks of fluid inside the volume V
(i.e. no chemical or nuclear reactions to produce/absorb fluid), the total
mass of fluid flowing out of the volume V in unit time is
I
∂V
ρ
−→
v d
−→
s
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 22
where ∂V denotes integration over the surface bounding volume V .
The decrease per unit time in the mass of fluid in the volume V can be
written as a time derivative
−
∂
∂t
Z
V
ρdV
Provided that the mass in volume V is conserved, one can equate the two
expressions, so we can write
∂
∂t
Z
V
ρdV = −
I
∂V
ρ
−→
v d
−→
s (2.1)
The surface integral can be transformed to volume integral using Green’s
formula
I
∂V
ρ
−→
v d
−→
s =
Z
V
∇(ρ
−→
v ) dV
Therefore Equation (2.1) can be re-written:
Z
V
µ
∂ρ
∂t
+ ∇(ρ
−→
v )
¶
dV = 0
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 23
note that ∇
−→
v ≡ div
−→
v .
Since the equation above is true for any volume, the integration must
vanish:
∂ρ
∂t
+ ∇(ρ
−→
v ) = 0 (2.2)
This is the equation of continuity.
Expanding second term in the continuity equation, one finds
∂ρ
∂t
+ ρ∇
−→
v +
−→
v ∇ρ = 0
The vector ρ
−→
v =
−→
j is called mass flux density.
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 24
2.3 Equation of motion
Figure 8: Fluid volume and
pressure.
Let us again consider some volume (a ele-
ment of fluid) in a fluid (Figure 8). The total force
acting on this volume is equal to the integral of
the pressure
−
I
∂V
pd
−→
s
where p(
−→
r (t), t) is the pressure. The integral is
taken over the surface ∂V bounding the volume
V .
Transforming this integral into a volume integral, one finds
−
I
∂V
pd
−→
s = −
Z
V
∇pdV
where ∇p ≡ grad p is the pressure gradient.
In other words, we can say that a force −∇p acts on unit volume of
the fluid. We can now write the equation of motion of a volume element in
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 25
the fluid, recalling second Newton law, m
−→
a =
−→
F , we can write for the fluid
element:
ρ
d
−→
v
dt
= −∇p (2.3)
where d/dt denotes the rate of change of velocity of a given fluid volume
and not the rate of change of the fluid velocity at a fixed point in space.
2.3.1 Lagrangian and Eulerian descriptions
If we ‘follow the fluid’ (as above in Equation (2.3)), an arbitrary function
f changes as we track a particular fluid element, called Lagrangian fluid
element. Such derivative is called the convective derivative or Lagrangian
derivative.
Description at given (fixed) spatial positions (as we used in the conti-
nuity equation (2.2) ) is called Eulerian description.
Let us relate these two descriptions. Consider function f (t,
−→
r (t)) of a
fixed Lagrangian fluid element. The derivative of f with respect to time t
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 26
can be written
d f
dt
=
∂f
∂t
+
∂f
∂
−→
r
d
−→
r
dt
since
d
−→
r
dt
=
−→
v , we can write
d f
dt
=
∂f
∂t
+
−→
v
∂f
∂
−→
r
or in operator form as relation between Lagrangian and Eulerian
derivatives
d
dt
=
∂
∂t
+
−→
v ·
∂
∂
−→
r
(2.4)
The convective derivative (2.4) gives the Lagrangian rate of change in
time in terms of Eulerian measurements.
Using equation (2.4), the equation of motion (2.3) can be written:
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 27
∂
−→
v
∂t
+ (
−→
v · ∇)
−→
v = −
1
ρ
∇p (2.5)
This equation of fluid motion was first obtained by Euler in 1755 and
is called Euler equation.
If the fluid is in the gravitational field additional force ρ
−→
g , where
−→
g is
acceleration due to gravity, acts on any unit volume. Equation (2.5) can
be re-written:
∂
−→
v
∂t
+ (
−→
v ∇)
−→
v = −
1
ρ
∇p +
−→
g
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 28
2.4 Incompressibility
Consider a closed surface ∂V in a fluid. The net volume rate of the
fluid that is leaving volume V is given by the surface integral over ∂V
I
∂V
−→
v d
−→
s
Using Gauss’s integral theorem, one finds
I
∂V
−→
v d
−→
s =
Z
V
∇
−→
v dV
For incompressible fluid the volume rate should be zero. Since this must
be true for all possible volumes in the fluid, we can write
∇ ·
−→
v = 0 (2.6)
This is so-called incompressibility condition.
Note: Deriving the equation of motion, we have ignored the processes
of energy and momentum dissipation due to internal friction (viscosity)
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 29
and heat exchange between different parts of fluid (thermal conductivity).
Such incompressible fluids are called ideal.
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 30
2.5 Fluid equations in Lagrangian description
Using Equation (2.4), we can write our first hydrodynamic equation
(continuity equation) with Lagrangian derivative:
∂ρ
∂t
+ ρ∇
−→
v +
−→
v ∇ρ =
Eulerian
z }| {
µ
∂
∂t
+ (
−→
v ∇)
¶
ρ +ρ∇
−→
v =
Lagrangian
z}|{
dρ
dt
+ρ∇
−→
v = 0 (2.7)
Similarly, we can write equation of motion including gravity
ρ
Lagrangian
z}|{
d
−→
v
dt
= ρ
Eulerian
z }| {
µ
∂
∂t
+ (
−→
v ∇)
¶
−→
v = −∇p + ρ
−→
g
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3 HYDRODYNAMIC EQUATIONS 31
3 Hydrodynamic Equations
3.1 Fluid vorticity
Newtonian gravity is a conservative force, so the gravitational accel-
eration may be expressed as the gradient of a scalar-valued gravitational
potential ψ(
−→
r ):
−→
g = −∇ψ
Assuming that ρ = constant, the hydrodynamic equation of motion can be
written
∂
−→
v
∂t
+ (
−→
v · ∇)
−→
v = −∇
µ
p
ρ
+ ψ
¶
To simplify the equation above, let us consider
−→
v × (∇ ×
−→
v ) term. We
can re-write using the vector triple product property
1
−→
v × (∇ ×
−→
v ) = ∇(
↓
−→
v ·
−→
v ) −
↓
−→
v (∇ ·
−→
v )
1
For any three vectors
−→
A × (
−→
B ×
−→
C) =
−→
B(
−→
A ·
−→
C) −
−→
C(
−→
A ·
−→
B)
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3 HYDRODYNAMIC EQUATIONS 32
where notation
↓
−→
v explicitly states that
−→
v should be differentiated. The first
RHS term is ‘half derivative’, where only the first
−→
v (and not the second)
should be differentiated. Indeed, since
∇
−→
v
2
= ∇(
−→
v ·
−→
v ) = ∇(
↓
−→
v ·
−→
v ) + ∇(
−→
v ·
↓
−→
v ) = 2∇(
↓
−→
v ·
−→
v ),
we can write
∇(
−→
v ·
↓
−→
v ) =
1
2
∇
−→
v
2
The term
↓
−→
v (∇ ·
−→
v ) can be re-arranged, so that the differential operator ∇
acts to the right
↓
−→
v (∇ ·
−→
v ) = (∇ ·
−→
v )
↓
−→
v = (
−→
v · ∇)
−→
v
Combining these two expression, we can write
−→
v × (∇ ×
−→
v ) =
1
2
∇(
−→
v
2
) − (
−→
v · ∇)
−→
v
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3 HYDRODYNAMIC EQUATIONS 33
and re-arranging
(
−→
v · ∇)
−→
v =
1
2
∇(
−→
v
2
) −
−→
v × (∇ ×
−→
v ) =
1
2
∇(
−→
v
2
) + (∇ ×
−→
v ) ×
−→
v
Using this mathematical expression, the equation of motion becomes:
∂
−→
v
∂t
+ (∇ ×
−→
v ) ×
−→
v = −∇
µ
p
ρ
+ ψ+
1
2
−→
v
2
¶
where
−→
ω ≡ ∇ ×
−→
v is called fluid vorticity. To derive equation for fluid vor-
ticity, let us apply ∇× to both parts of the equation of motion, so we can
derive
∂
−→
ω
∂t
+ ∇ × (
−→
ω ×
−→
v ) = 0
Consider the second term. Since
∇×(
−→
ω ×
−→
v ) = ∇×(
↓
−→
ω ×
−→
v )+∇×(
−→
ω ×
↓
−→
v ) =
↓
−→
ω (
−→
v ·∇)−
−→
v (∇·
↓
−→
ω )+
−→
ω (∇·
↓
−→
v )−
↓
−→
v (
−→
ω ·∇)
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3 HYDRODYNAMIC EQUATIONS 34
which can be simplified using ∇ ·
−→
v = 0 for incompressible fluid and ∇(∇ ×
...) = 0. Re-arranging, we have
∇ × (
−→
ω ×
−→
v ) = (
−→
v · ∇)
−→
ω − (
−→
ω · ∇)
−→
v
Finally one finds the equation for vorticity
∂
−→
ω
∂t
+ (
−→
−→
v · ∇)
−→
ω = (
−→
ω · ∇)
−→
v (3.1)
In Lagrangian description, the vorticity equations has a more simple
form
d
−→
ω
dt
= (
−→
ω · ∇)
−→
v
In 2D fluids (Figure 9), the term (
−→
ω · ∇)
−→
v vanishes and we have
d
−→
ω
dt
=
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
ω = 0
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3 HYDRODYNAMIC EQUATIONS 35
Figure 9: Left: US Satellite image of Katrina hurrcane.Right: Jupiter’s Great Red Spot
as seen by a Voyager spacecraft from NASA/JPL-Caltech.
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3 HYDRODYNAMIC EQUATIONS 36
Hence, in the absence of friction, vorticity is conserved along trajectories.
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3 HYDRODYNAMIC EQUATIONS 37
3.2 Energy equation
The first law of thermodynamics states that
dQ = dU + pdV
where
dQ is the heat (energy) that has been added to the system
dU is the change in the internal energy
pdV is the work done by the system
We want to adapt the first law of thermodynamics to a continuous fluid
in which there may be motions.
Let us first introduce q and ε – the heat added and the change of
internal energy per unit of mass:
dQ = δmdq, dU = δmdε
The volume per mass unit is 1/ρ, so that the last term in the first law
of thermodynamics can be written pδmd(1/ρ). Putting these three terms
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3 HYDRODYNAMIC EQUATIONS 38
together one finds:
dq = dε + pd
µ
1
ρ
¶
and per unit of time
dq
dt
=
dε
dt
+ p
d
dt
µ
1
ρ
¶
Using the conservation of matter (our first hydrodynamic equation in La-
grangian form, see Equation 2.7), one finds
p
d
dt
µ
1
ρ
¶
= −
p
ρ
2
dρ
dt
=
p
ρ
∇ ·
−→
v
we can derive the energy conservation equation
ρ
dε
dt
= ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = −p∇ ·
−→
v + ρ
dq
dt
(3.2)
where the term ρdq/dt is the rate of heat gain/loss per unit volume, which
could be either external to fluid (e.g. energy loss due to radiation) or
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3 HYDRODYNAMIC EQUATIONS 39
internal (e.g. due to thermal conduction within the fluid), and the term
p∇ ·
−→
v is the work done per unit volume.
3.2.1 Heat conduction
Let us consider a small volume of fluid. The heat flux
−→
F inside a
continuous fluid can be assumed to be proportional to the temperature
gradient ∇T
−→
F = −k∇T
where T(
−→
r , t ) is fluid temperature, k is the thermal conductivity coefficient
of fluid. The heat loss rate from a volume of fluid is equal to the heat flux
integrated over the bounding surface ∂V of the volume V
I
∂V
−→
F d
−→
s =
Z
V
∇ ·
−→
F dV
Using −ρdq/dt = ∇ ·
−→
F , one finds in Euler’s coordinates:
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3 HYDRODYNAMIC EQUATIONS 40
ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = ∇(k∇T) − p∇ ·
−→
v (3.3)
This is the equation for energy conservation including thermal con-
duction.
For incompressible fluid, ∇ ·
−→
v = 0 and for fluid with thermal conduc-
tivity k = 0, one can simplify the equation (3.3), so we have in Lagrandian
and Eulerian descriptions
dε
dt
=
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = 0
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3 HYDRODYNAMIC EQUATIONS 41
3.3 Hydrodynamic equations
To sum up hydrodynamic equations, we can write all equations to-
gether (Eulerian coordinates)
mass:
µ
∂
∂t
+ (
−→
v · ∇)
¶
ρ = −ρ∇
−→
v
momentum: ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
v = −∇p + µ4
−→
v
energy: ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = ∇(k∇T) − p∇ ·
−→
v
where µ4
−→
v is the viscosity term (we did not derive) and µ is the
dynamic viscosity.
The equations are known as the Navier-Stokes Equations. These
equations describe how the velocity, pressure, temperature, and density
of a moving fluid are related. The equations were derived independently
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3 HYDRODYNAMIC EQUATIONS 42
by G.G. Stokes, in England, and M. Navier, in France, in the early 1800’s.
The equations are nonlinear and very difficult to solve.
In case of ideal fluid, e.g. ν = 0, k = 0, and ∇ ·
−→
v = 0, the equations
are simplified to:
mass:
µ
∂
∂t
+ (
−→
v · ∇)
¶
ρ = 0
momentum: ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
v = −∇p
energy:
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = 0
In both cases, these equations are incomplete to describe the fluid, the
equation of state of the fluid needs to be added to complete the system.
In case of ideal gas, the pressure has simple expression p = (γ − 1)ρε.
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4 SURFACE WAVES 43
4 Surface waves
4.1 Surface gravity waves
Figure 10: Fluid surface and
coordinates used.
Let us consider the free fluid surface in
a gravitational field
−→
g (e.g. ‘stellar surface’).
−→
g is antiparallel to
−→
z and is perpendicular
to the fluid surface in Figure 10.
Hydrodynamic equations can be written
(assuming ideal fluid)
µ
∂
∂t
+ (
−→
v · ∇)
¶
ρ = 0
ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
v = −∇p + ρ
−→
g
In addition, if there is no vorticity inside of the fluid [note that the vorticity
equation (3.1) says that velocities induced remain irrotational], we can
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4 SURFACE WAVES 44
assume that velocity can be presented as a gradient of a scalar function
−→
v = −∇ϕ,
where ϕ(
−→
r , t ) is the velocity potential.
Combining second hydrodynamic equation and the relation (
−→
v ·∇)
−→
v =
1
2
∇(
−→
v
2
)+(∇×
−→
v )×
−→
v , where we note that when
−→
v = −∇ϕ, ⇒ ∇×
−→
v = 0,
hence one finds
−
∂∇ϕ
∂t
+ ∇
µ
1
2
−→
v
2
¶
= −∇
µ
p
ρ
+ ψ
¶
,
where ψ is gravitational field potential. This equation can be re-written in
the form of an integral
∇
F(t) - some constant in space
z }| {
µ
−
∂ϕ
∂t
+
1
2
−→
v
2
+
p
ρ
+ ψ
¶
= 0,
where F(t) is a constant in space, but can be a function of time. For the
geometry (Figure 10), the gravitational potential is ψ = gz.
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4 SURFACE WAVES 45
Assume further:
1. Let a constant pressure p
0
act on the surface of fluid
2. We will consider small perturbations, so that
−→
v
2
≈ 0
3. The constant p
0
acting on the surface can be eliminated by re-
defining the velocity potential ϕ
Adding to ϕ a quantity independent on coordinates p
0
t/ρ, e.g. ϕ → ϕ +
p
0
t/ρ, we simplify equation for F(t)
F(t) =
µ
−
∂ϕ
∂t
+ gz
¶
= 0.
Figure 11: Fluid surface perturbation with amplitude ξ(x, z, t)
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4 SURFACE WAVES 46
Without loss of generality, we can set F(t) = 0, since F(t) could always
be absorbed into velocity potential. Hence at a perturbed fluid surface
z = ξ, we can write
µ
−
∂ϕ
∂t
+ gz
¶
¯
¯
¯
¯
z=ξ
= 0 =⇒ ξ =
1
g
∂ϕ
∂t
¯
¯
¯
¯
z=ξ
where ξ is the amplitude of a small perturbation (see Figure 11).
Since the perturbation is small, we can assume that the vertical com-
ponent of the velocity v
z
of the points at the surface is simply the time
derivative of ξ, so we have
v
z
=
dξ
dt
'
∂ξ
∂t
Note that dξ/dt ' ∂ξ/∂t due to smallness of ξ. Using the definition of the
velocity potential, z-component of fluid velocity is v
z
= −
∂ϕ
∂z
, therefore one
finds at z = ξ
−
∂ϕ
∂z
¯
¯
¯
¯
z=ξ
=
∂ξ
∂t
=
1
g
∂
2
ϕ
∂t
2
¯
¯
¯
¯
z=ξ
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4 SURFACE WAVES 47
Finally we have the following equation at the surface z = ξ:
∂ϕ
∂z
¯
¯
¯
¯
z=ξ
+
1
g
∂
2
ϕ
∂t
2
¯
¯
¯
¯
z=ξ
= 0 (4.1)
In addition due to incompressibility of fluid ∇ ·
−→
v = 0, we have the second
equation for velocity potential ϕ:
4ϕ = 0, (4.2)
where 4 is the Laplace operator or Laplacian, 4 ≡ ∇
2
, e. g. the diver-
gence of the gradient of velocity potential.
The system of equations (4.1, 4.2) describe small perturbations at fluid
surface.
Let us now find the solution of the equations (4.1, 4.2). Consider waves
on the surface of a fluid whose area is unlimited and the wavelength is
small in comparison with the depth of the liquid. So we seek for a solution
of the form
ϕ(z, x, t) = f (z)cos(kx − ωt),
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4 SURFACE WAVES 48
where ω is the angular frequency and k is the wavenumber of a wave
prorogating in x direction.
From Equation (4.2), one finds
d
2
f
dz
2
− k
2
f = 0
The solution of this equation, which vanishes at z → −∞ is ∼ exp(kz).
Therefore, we can write
ϕ(z, x, t) = A exp(kz)cos(kx − ωt)
where A is a constant. Note that the wave amplitude decreases exponen-
tially with depth, hence the waves are surface waves.
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4 SURFACE WAVES 49
4.2 Dispersion relation, group and phase velocities
Using Equation (4.1) and substituting the solution ϕ(z, x, t), we can
find
ω
2
= k g (4.3)
Equation (4.3) is the dispersion relation between the wavenumber k and
the wave frequency ω for surface gravity waves. As we see from the
dispersion relation the presence of g plays the key role.
The phase and group velocities of the surface gravity waves
phase velocity : v
phase
≡
ω
k
=
r
g
k
=
s
gλ
2π
group velocity : v
grou p
≡
dω
dk
=
1
2
r
g
k
=
1
2
s
gλ
2π
where λ = 2π/k is the wavelength.
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4 SURFACE WAVES 50
The dispersion relation helps us to understand some important prop-
erties of waves. In the above, we assumed deep water while deriving the
wave dispersion relation. When the water becomes shallow the wave-
length should decrease. At the same time, the wave energy flux v
grou p
A
2
should remain constant along the wave path. So when v
grou p
decreases,
the wave amplitude A must increase (Figure 12).
Surface gravity waves are common for stellar surfaces. For example
the solar g-mode or gravity waves are density waves confined to the in-
terior of the Sun below the convection zone. f-mode or surface gravity
waves are also gravity waves occurring at or near the photosphere.
The surface gravity waves are neither longitudinal nor transverse. In-
deed, using our solution for ϕ(z, x, t), we can calculate two non-vanishing
Pulsars and Supernova I