__Feb.2 /2019__

__Part IX__

__A Human Equation of State at
a Thermodynamic Limit __

__A Scientific Requirement for Spirit?____ __

__Summary__

A need in neuroscience, for a scientific
theory to explain observations of brain temperature patterns and variations, points
to a thermodynamic Equation of State. The classic one is the Ideal Gas Law [pv = constant]. This, however, has a requirement for an infinity for its
variable n, the number of equal energy states, and so it is is
always used in physics as a close approximation
only.

However, if applied to humans
under the classical philosophic ‘body and soul’ theory of human nature, then the equation could
fit exactly, because spirit in rational philosophy is intrinsically
unquantified and can assume any value required.

This is rather startling, as is
the fact that the predictions of such a Human State Equation would be readily verified or falsified by observation. Thus, a canonical scientific
theory, which is predicated
on a spiritual essence for human nature, exists.

Experimental verification of the
theory with human neurological and cognitive data is discussed.

__Contents__

** •Introduction to a Thermodynamic Problem**

** •Thermodynamic Equations of State**

** **

**
•Proposed Unique Solution: An Ideal/ Isothermal Human Equation of State
at a Thermodynamic Limit**

** **

** • Properties of The
Proposed Human Thermodynamic State **

**•Experimental Verification of the Proposal**

** • Relevant
Further Questions**

** •
Caveat**

**
• Conclusions**

** • References**

**Appendix A:
Outline of Compressible Fluid Flow**

** **

__Introduction to a
Thermodynamic Problem__

In the course of a study of five cosmological fields, namely:
ordinary compressed matter of the visible universe, the dark matter, dark
energy, quantum and electromagnetic fields, and gravity [1, 2 3], a review was made of their respective thermodynamic equations of state. The simplest
of these state equations is the hyperbolic equation **pv**** = constant** for the matter of the To repeatvisible
cosmos. This equation is the familiar Ideal Gas Law which has such wide application
in compressible fluid
physics, gas dynamics and
cosmology [4].

.

In a derivation, this same Ideal Gas Law, pv = const., emerges sequentially from the adiabatic gas equation series,
pv^{k}
= constant , at the theoretical
limit where the adiabatic index k
becomes equal to + 1 [ i.e. pv^{k}^{=1} = pv^{1} = pv = const.] and the
equation becomes simply pv =constant. However, since this also requires that ** n**,
the associated number of degrees of
freedom, or number of ‘microstates’, must become

It seems, however, odd that such a
fundamental scientific relationship as the ideal gas law must, strictly
speaking, exist
only theoretically, and so the matter was studied in some depth. To repeat, the derivation of the ideal law as
the limit of the adiabatic series, where k, the adiabatic index, becomes unity,
is ;

pv^{k} = pv^{k}^{=1} = pv^{1}
= pv = const.

But, at __this limit, when k = +1, n becomes infinity__, since n = **2/(k
− 1) = 2/(1− 1) = 2/0 =
∞.**

Since infinity is not possible in
physical systems, it then seems clear that the exact ideal gas law can not actually
exist in the physical cosmos.

Toward a possible solution, we first
look at compressible flow and thermodynamic equations of state.

__Thermodynamic Equations of
State __

__ Thermodynamic
Variables __

Compressible systems have the thermodynamic
variables, specific density
ρ, pressure p, specific volume v = 1/ ρ and temperature
T (4).

The **relationships** among these thermodynamic state variables are given in an ** Equation of State**.

__The Ideal Gas Equation of State__

Principal among the equations of
state is __the__** Ideal Gas Law**,

where **k **is the ratio
of the two specific heats , k = c_{p}/c_{v}
and is called the ‘adiabatic index,’ and
**n** is the number of ‘microstates’ and
is related to k as n = 2/[k − 1].

__The Isothermal Equation__

When the constant in the gas law, pv = const, is set to equal RT, with T as the temperature and R as the universal
gas constant, then we get the __isothermal
equation of state:__

**pv**** = RT= const.**

This equation also requires that **n** must equal infinity.

In physic,s this isothermal equation is applied to expansion
or compression of gases which are immersed in an environmental heat bath to
maintain the required constant temperature. Moreover, these processes are
required to take place slowly enough to maintain this temperature constancy by
heat diffusion into or out of the working gas.

In the proposed classical human ‘body
and soul’ case, the behavior of the temperature during some human activities is
likely to be unusual. As with all warm-blooded animals the human general body temperature
is quasi- constant and may supply the equivalent of a general constant
temperature heat bath. However, the restriction to an extremely
slow reaction process may not
apply, and would be a matter for observation and analysis. The effects on the
stability of wavc types and their amplitudes could also
be unusual in humans.

In general in both the ideal state
and the derived isothermal state the waves are stable only if they are low
amplitude acoustic types; finite waves are compressive ,
unstable and grow to shocks.

__The Linear Wave “Tangent” Equation of State__

The Linear Wave Equation of State is: **p = ** **Av + B
, w**here A is a negative constant. ** **It graphs as a straight line with negative
slope on the pv diagram.
When it is set tangent to the hyperbolic ideal gas law, it is known as the Tsien/ Tangent Gas [4] or ‘The Tangent Gas’. In the non- tangent case, it is called the Chaplygin gas (4).

A principal feature of this state
equation is that its waves can be either compression or rarefaction types and they are uniquely stable at all amplitudes,
large or small. Shocks are not possible. Since rarefaction waves are producers
of negative entropy, this state is of obvious special interest to human neurology
and cognitive science
for possible intellectual
applications.

This state occurs with **k = −1 ;and n
= −1 . Again, as with the infinity requirement for the ideal gas state,
we have physically
exotic or unrealizable values**.

**Fig. 1.****
Thermodynamic Equations of State: The Ideal Gas Law and the Linear, Tsien/ Tangent Equation **

[Note:
A more detailed outline of compressible flow matters is given in **Appendix A:
Outline of Compressible Fluid Flow****]**

__Proposed Unique____ Solution:An
Ideal Isothermal Human Equation of State at a Thermodynamic Limit __

** The Problem:** This Proposed Ideal Human Equation of State, [ pv = const
= RT] is only possible with n = infinity

To meet this requirement for an infinity, so that the ideal equation of state can
actually exist, we note that
philosophically, a ** spiritual dynamism**
has the property of being i

Now, over millennia, in classical
and pre-modern times, we
humans have been, on the ** rational**
evidence, considered to be

__A Solution: __

It is surely rational if we now propose, as a
starter, that humans possess an ideal thermodynamic equation of state **pv**** = const****.**

As we have seen, this equation
requires an infinity for the state variable **n**, but if we further propose that humans are ‘body and soul
animals’, then, since the proposed human spirit would be intrinsically non-quantified,
it would be able to meet the infinity requirement for n, and so would qualify humans as unique embodiments of this thermodynamic
equation of state.

This simple but demanding equation
of state would then describe aspects of an inner state or inner core to our human activities, physical and intellectual. Remarkably, it ** also appears to be a true scientific proposal by being open to experimental testing
for its acceptance or rejection**. It may or may not be real, depending on
the results of experiment data of human neurology and physiology.

In summary, a** valid thermodynamic human equation of state of
format**, pv = constant = RT can now emerge,

The logic and rational evidence of
the classical and pre-modern philosophical position on the nature of man as a
spiritual/rational animal has always
been overwhelming. The rise of our scientific and technological way of life, however, has
routinely excluded spirit from
discussion because of its requirement
for scientific proof, which requires experimental proof as well as logical, physical theory.

The proposed human equation of
state pv = RT predicts thermodynamic and dynamical effects
which, if real, will be observable experimentally. This particular, human ‘body and
soul’ hypothesis is thus a scientific
one.

__[ __Note**: The Tangent, **

__Properties of The
Proposed Human Thermodynamic State __

** The Proposal: **Neurology observes animal brain temperatures, variations
and patterns, and

The most widely used Equation of
State is probably the
Ideal Gas Law: **pv**** = const. = RT. **

This equation is always used as a
close approximation only, since the exact conformity would require an infinite numerical value for one state
variable, namely for **n** the number
of equal energy states or ‘microstates’.

Our proposal is that the Ideal Law
would apply ** exactly** to humans if their
nature is taken as ‘body and soul’, that
is to say , as in the classical
definition. This proposal is also a scientific one, since it is open to
experimental testing.

__A ____Human____ ____Inner____ ____State__** : **The next relevant question is :What is this “State” to which the Ideal Equation pv = c would apply? Is it, for example, a compressible medium of
some special sort? A medium of what? A compressible energy field? A wave field

Let us look at the ‘__body and soul
interaction field or state’ choice .__ We have mentioned above that the
interaction of a wave having velocity **c**
with a flow having an impulse velocity **u,
produces ** **a specific energy ( i**.e. an
energy per unit mass) **E _{sp}**
expressible as:

**E _{sp}**

where the term **2cu representts an energy of** **interaction between wave speed c
and the pulse velocity u**

If now we postulate that the human
soul, or spiritual human , induces the wave
motion **c **and the body supplies the pulse action **u, ** then the 2cu interaction energies
summed over the entire body would form a
new energy field or state, one that is a true human unity of spirit and matter, free of any Cartesian Dualism . The energy of the interaction
field, E_{Ψ, might} then be expressed
symbolically as:

E_{Ψ = }**∑
(2cu)**

This energy field
would form an
inner statc, central to our human activities, physical,
intellectual and spiritual.

If this 2cu energy is conceived of as a 2cu human field, analogous to an atom field , a molecular field, a quantum
field, and etc. then we could conceive
of it as an “interaction energy field’
or human compressible state to which the Ideal Law, pv
= constsnt = RT, applies. This law would then describes the thermodynamic observables, i.e. temperature,
entropy, pressure, specific volume and
density, and would support the dynamic
observables of wave action, oscillations, resonances and the like.

We emphasize that this inner field
is neither a
physical field nor a ‘soul field.’. Instead,
it is a new __interaction__ entity and field with its own observable
thermodynamic and dynamic variables.

We might further envisage the living state as involving a constant generation of interaction energy,
2cu, as each atom, molecule, tissue or
fluid pulsates, vibrates or flows. In
non-human living things, these interactions would be purely physical waves and pulses interacting to produce a physical living being. With
humans, the 2cV field relates to a new and distinct ‘soul and body’entity
[5].

The Human Ideal thermodynamic Equation of State, pv = c, which requires an infinity for its ‘number of microstates’
or equipartition
parameter **n**, could therefore describe exactly , i.e. without
approximation , the thermodynamic and dynamic neurological and
cognitive properties of a ‘body and soul’ human inner energy state or inner field.

__ ____Experimental Verification
of the Proposal __

A
proposition is only scientific if, in addition to its rational, scientific logic
or theory, there is also at least one experimental test open for validity or falsification.

** Generally:
**

Furthermore, it has been reported [ 6,7] that some
general theoretical scientific theory seems required to properly account for the
neurological temperature field observed in
the brain and for the observed temperature correlations with other brain properties.

Therefore,
our present proposal that there is__ a human thermodynamic
equation of state which embodies such thermodynamic properties, is a valid
candidate to meet the expressed needs of
the science for a suitable formal theory. __

** Principal Quantitative Observables: ** Brain temperatures
and neural waves and oscillations would appear to be among the main observables.

Since
the proposed equation of state has the i__sothermal__ form pv = RT = constant, then the stability of brain tempersture
and the magnitude of thermal fluctuations in the human brain could __quantitatively__
test the
predicted isothermal behaviour under different
physical and mental activitie. The correlations
established would sharpen comparative data.

Neural
waves and oscillations could now be studied for patterns of wave types and
quantitative properties permitted by the proposed exact equation of state, such
as frequency, amplitude, wave speed, etc.

** Specifically; Some** specific areas suitable for quantitative testing might include the following:

(a): A requirement to explain the existence and qualities of **states
of inner calm** or serenity (8), which
would appear to mesh with **isentropic **states
of the state equation, pv = const. = RT.

(b): **Orderly states** for logical and intellectual activities. These might be accompanied by a stable rarefaction wave state which would mean an emergence of the
decreased entropy or **increased order, **which
is required.

(e): **Wave types
**which would match various observed neural waves or oscillations. The isothermal
state’s waves[ pv
= RT = constant] should be stable for low amplitude acoustic type waves, but gives
unstable compressions for large amplitude waves; the
linear wave state [ p = Av +B] could match any desired stable wave type,
compression or rarefaction.[See **Appendix A: Outline of Compressible Fluid Flow** Section 2.10 Wave Types and Stability].

[ We suggest that comparisons of humans with non-human primates will not provide a
sustainable objection, since, for the other primates, a purely
physical adiabatic equation of state [
e.g. pv^{k}>^{1}
= const.) with k greater then 1 and n less than infinity, may well exist to match
their neurological and cognitive manifestations. The proposed
differences for humans ought
to be readily observable and verifiable].

[Note,
that the inescapable inexactitude in observational data is not an obstacle to determining whether of
not the proposed human equation of sate
is intrinsically ‘exact’. All that is required is a unique degree of exactitude
in the human case].

__Relevant Further Questions__

** In General,** further relevant matters which arise, are: :

(a): “**The hard** **problem of human consciousness**” in cognitive science.

,

(b ): The origin and
nature of the universally experienced
intellectual **appetite or innate
drive towards understanding** in Science and Technology research (9).

(c): The
nature and source of the **“****Eureka****”** **Moments of “Joyous” Insight and Discovery** which
occur sporadically during intellectual
and scientific investigations (9).

(d): Human Contemplative and Meditative States (6).

(e) The human sense of
“self”, individuality, unity. and identity.

__Thermodynamic Tests:__

The observables
here would seem to be principally body temperature and, especially, brain
temperature.

Entropy
change values would be readily available via temperature changes since the two
are proportionally related.

This
ideal, human equation
of state should variously be
thermodynamically entropic, isentropic or
negentropic according to the particular human
activity involved.

Thermodynamically, entropy/temperature
signals should be related to neural
signals and to harmonic neural
oscillations.

Distinctive entropy changes, in particular any
presence of a negentropy being associated with
intellectual activities, would be of
central importance.

Any temperature (entropy)
effects accompanying the “Eureka
Moments” of Insight that occur during
research. .

Do the
data fit so as to verify or suggest an __exac____t
__ Hunan Equation of State? [Note,
that the inescapable inexactness in observational data is not an obstacle to determining whether of not the
proposed human equation of sate is
intrinsically ‘exact’. All that is required is a unique degree of exactitude in
the human case].

__Dynamic Tests____:__

Here we
have the state equations’ wave speed values, and especially their unique wave
properties. Many varied wave tests are possible, including a wide range of oscillations,
resonances and pulses.

In neurological and
cognitive science a massive amount of observational
data is already available, and there is an expressed need for
theoretical thermodynamic and oscillatory insight.

Finite, stable Wave Amplitudes

Absence
of Shock Waves

Wave
types: In the pv = consant state, low
amplitude acoustic compressions would be the rule. Rarefaction waves are not possible.

In the
pv = RT or isothermal state, stable waves of both
compression or rarefaction
may be possible, depending on the precise temperature control.
The wave action in the exact human case nay be different than that
in a purely physical, approximate case.

Wave
types: In the linear State, both stable compressions and
stable rarefactions of finite amplitude are
present. The latter
could give rise to negentropy. Shocks
are ruled out.

__A Caveat__

Because of the highly unusual nature of the subject matter
of the present report, it seems prudent to offer some cautions.

First, the views presented above are preliminary,
conditional on a particular logical path chosen, and therefore tentative. That
said, it is felt that they solidly support further investigation by
neurologists, thermodynamicists, and physiologists, among others.

In addition to the pursuit of the Relevant Further Questions put
forward above, there will arise, if the experimental data are supportive of the
thesis, the interesting matter of the details of the unity of soul and body. I
have mentioned the flow/wave pulse energy
example (c + u )^{2 } = c^{2}
+ u^{2} + **2cu ** where the interaction energy term 2cu is given
as an instance of a complete union on the energy level of a spiritual wave c
and a physical flow pulse u, or vice versa. The investigation and presentation of other such interaction effects will require care.

Hopefully, this whole endeavour
will bring about a replacement of the present rather disengaged situation, with
a cooperative endeavour to expand the factual knowledge
on human nature.

* *

.[See **Appendix A: Outline of
Compressible Fluid Flow** __Section
2.10 Wave Types and Stability__].

__Conclusions__

The Ideal Gas Law, pv
= const., occurs sequentially at the theoretical limit of the adiabatic
equation series, pv^{k} = const., where the adiabatic index k becomes equal to
+ 1. and the equation becomes pv
= const.. However, since this also requires that ** n**,
the associated number of degrees of freedom , or number of ‘microstates’, must
become an infinity, then the ideal gas
law is not physically possible, and evidently must exist only theoretically or only as a very
useful approximation.

However, on the classical and
pre-modern consensus that humans are a real composite of matter and __spiri__t,
body and __soul__, this infinity problem could conceivably vanish. This is because a** valid exact thermodynamic human equation of
state of
format**, pv = constant, could then emerge,

The logic and evidence of the
classical and pre-modern philosophical position on the nature of man as a
spiritual/rational animal
has always been overwhelming. The rise of our scientific and
technological way of life, however, has routinely excluded spirit because of a
requirement for scientific proof, which means experimental verification as well
as logical theory. The proposed human equation of state theory predicts thermodynamic,
dynamic and entropy effects
which should be observable experimentally. The present hypothesis
is therefore a scientific one.

In
essence, our proposal is for the existence in humans of an ideal thermodynamic
equation of state which would necessitate a dynamic capacity for numerical
infinity. This equation, [pv
= const.] and a
second associated exotic, linear wave
equation [p = −Av + B] , would govern such things as
compression/rarefaction waves in human tissues, and would describe temperature changes
and their associated entropy changes.

But,
the existence of such an
ideal equation of state in humans would also require an accompanying
__human spirit or soul__
because of the equation’s thermodynamic
requirement for a physical infinity which could only be supplied by the intrinsically non-quantified dynamism of
spirit or soul. Human mental and intellectual activity would necessarily
generate unique thermodynamic entropy signals which would then be observable through associated characteristic brain/body temperature variations. This
proposal is thus open to experimental examination and judgment, and so this argument for the
reality of human spirit is canonically scientific.

Such
thermodynamically described entropy/temperature signals should then relate to
the data of neural signals and harmonic neural oscillations.

To the classical and pre-modern
philosophical position on a soul/body human nature there may now be added a
scientific theory i.e. one which is
experimentally verifiable. The classical
philosophical and theological position still holds on the weight of its rational argument, while the post modern, scientifically
conditioned position of thought may now
join the discussion on its own formal,
scientific terms.

-----------------------------

The above Conclusions
represent the completion of the
First Step in this R&D project,
namely the Proposal of a Human Equation
of State designed to meet an expressed need
in neurological science for a theory of brain temperature behaviour.

The Second Step is for its examination by neurologists for
correspondence, or otherwise, of
existing data with the proposed state
equation. If this also is
supportive, then an examination of Relevant Further Questions by neurologists and cognitive
scientists could be made. If the fit of human data to predictions is close or unique, then the
validation of the **e xact** human
equation of state would be established. In this case, the ‘body and soul ‘
theory of human nature already established from classical philosophy and
theology would become scientifically supported as well.

---------------------------------

__References__

1. Power, Bernard A,, www.energycompressibility.info:
Part VIII. *Equations of State for Cosmic Fields*.** 2017.**

2. …………………………………:Part
VI: ** How
Did Life Originate? The Role of Cell Waves and their Negative
Entropy.** 2016.

3. ………………………………… Part VII B *What is Human Life? A Classical Viewpoint with
a Verifiable Scientific Hypothesis. Nov. 2016.*

4. Shapiro, A. S.
*The Dynamics and Thermodynamics of
Compressible Fluid Flow**.** * 2 vols. John Wiley and Sons,

5. Catechism of the Catholic
Church: Section **365***:*** **Publications
Service, Canadian
Conference of Catholic Bishops,

5a. Moreland, J.P., The Soul. Moody Publishers,

6. Beauregard, Mario and Vincent
Paquette. “Neural correlation of a
mystical experience in Carmelite Nuns” Neuroscience Letters, 405, 186-190
(2006).

7. Carhart-Harris,
Robin L., R. Leech, P.J. Hellyer, M. Shanahan, A. Feilding,
E. Tagliazucchi, D.R. Chivalo, D. Nutt “ The entropic brain: A Theory of
conscious states informed by neuroimaging
research data with psychedelic drugs”. Frontiers in Human Neuroscience.

8.
Note: In a current Op Ed in the New York Times ( Int’nl
Ed. June 5, 2018, pp 10-11) Prof.
Stephen Asma
wrote, on the irrationality of religion to him personally, but of its
value as the unique soother of mangled human emotions. While both points are
arguable, the second is evocative,

It brings to mind a marked characteristic of religion: For example, **Inner Peace** has been the hallmark of the Christian religion since
its beginnings. *“……………the Peace of
the Lord which surpasses
all understanding”. *

This peace experience was so
universal that a standard liturgical greeting among Christians
from those earliest days down to the present day was, and is : “**May the Peace of the Lord be always with
you”, **with the reply:
“*And* * with
your spirit”*.

*[‘Pax Domini sit semper vobiscum’*; and ‘*Et
cum spiritu tuo’*.]

This peace that Paul emphasizes
was not seen as a soother of emotions in time of need, such as that which Prof Asma
cites as a religious accompaniment. It was rather a spiritual ** state**
that was a
simple fact accompanying acceptance of the new religious
faith. Paul himself, an intellectual,
could not ‘understand’
it. So it was clearly a state other than an ordinary mental notion, or any ordinary
emotion; rather it was deeply immaterial, human and spiritual.

____9. Lonergan, S.J., Bernard. *Insight: A Study of Human Understanding**.* Philosophical Library Inc.,

---------------

*‘To
attain any assured knowledge about the soul is one of the most difficult things*

* in the
world’……Aristotle*

*‘Man,
the Rational Animal’…….** ………… Aristotle*

*“What is real is that which can be
intellectually grasped and reasonably affirmed.”…Bernard Lonergan,
S.J.*

*“
Insight** is
Joyous” ….**. Richard
Feynman*

__Human Ideal Equation of State__

pv^{k}^{=}^{1} = pv^{1} = **pv**** = C = RT**

c_{p} / c_{v}
= k = 1

**n = **__2/(k-1)__** = 2/0 = ∞**

^{ }

**Copyright,
Bernard A. Power,**

** February 2019**

** **

__2.0 Outline of
Compressible Fluid Flow__

The following is
a listing of some relevant
basic compressible flow
principles. For more
complete treatment see texts on
compressible fluid flow or gas dynamics.

__2.1 Steady____ State Energy Equation __

c^{2} = c_{o}^{2} − V^{2}/n
(1)

where c is the local wave speed, c_{o}
is the static [i.e at V = 0] or maximum wave speed, V
is the relative flow speed, n is the number of ways the energy of the flow
system is divided (i. e. the number of degrees
of freedom) of the system and [ n = 2/(k−
1), where k = c_{p}/c_{v} _{ }is the’ adiabatic constant’ or ratio of specific heats.]

Here, ‘relative’
means referred to any (arbitrarily) chosen physical flow boundary. The equation is for unit
mass, that is, it pertains to ‘specific energy’ flow.

The case where V = c = c* is called the critical state. The
ratio (V/c) is the Mach number M of the flow.
The ratio (V/c_{o}) is also a quantum state variable. The maximum
flow velocity V_{max} (when c = 0) is the escape speed
to a vacuu:

V_{max}_{ }= √_{ }n c_{o}.

_{ }

__2.2 Unsteady State Energy Equation __

c^{2} = c_{o}^{2}
– V^{2}/n – 2/n (dφ/dt)
(2a)_{}

_{ }

where φ is a velocity potential, and dφ/dx
= u is a perturbation of the relative
velocity. Therefore, in three
dimensions, substituting V for u, we have dφ/dt
= V (dx/dt) = Vc,
and

c^{2} = c_{o}^{2}
– V^{2 }/n
− 2 (cV)/n
(2b)

__2.3 Lagrangian____ Energy Function L __

L = (Kinetic energy) – (Potential energy)
= ( c^{2}+ V^{2}/n) – c_{o}^{2}
, and so, from (2b)

L = − (2cV)/n
(3)

**2.**__4 Equation____ of State for Compressible, Ordinary
Matter Systems__** **

The
equations of state link the thermodynamic quantities of pressure
p, specific volume ( volume per unit mass v = 1/ρ] and temperature T. The
basic equation of state **for ordinary
gases** is the equilateral hyperbola of the Ideal Gas Law:

**pv**** = RT ;
p/ρ = RT = constant** (4)

Equation 4 is
seen to be isothermal
(T= constant) . For **adiabatic
changes** it becomes **pv _{}^{k}**

Here, each point
on the curve presents the values for a particular pressure and volume pair and shows how the two relate to each other
when one or the other is changed. In
this hyperbolic equation, the product of the two -- i.e. **the pv****-
energy** -- has a constant value as
set out by the equation of state.

Hyperbolic Equation of State (Ideal Gas Law)

Equations of State can be formed
for gases, liquids or solids. Here, we shall be concerned mainly with those for
the highly compressible states i.e. for gases.

__2.5 Waves and Flow__

__2.5.1 The Classical Wave Equation __

∆^{2}ψ = 1/c^{2 }[∂^{2}ψ/∂t^{2}]^{ }_{ }^{ }(5)

where ∆^{2} = ∂^{2}../∂x^{2
} + ∂^{2}..^{.}/∂y^{2}
+ ∂^{2}../∂z^{2};
**ψ** is the wave amplitude, that is, it is the
amplitude of a thermodynamic state variable such as the pressure p or the
density ρ. The local
wave speed is c.

The general
solution of (5) is

**ψ** = φ_{1} (x – ct) + φ_{2
}( x+ ct)
(6)

Equation 6 is a
linear, approximate equation for the case of low-amplitude waves
in which all small terms (squares, products of differentials, etc.) have been
dropped.

The natural
graphical representation of steady state compressible flows and their waves is
on the (x,t), or **space-time diagram. **

The classical
wave equation corresponds to isentropic conditions. It represents a stable,
low-amplitude wave disturbance, such as an acoustic–type wave.

__Unsteady State ( Accelerating)
Compressible Flow __

In compressible
flow theory, **forces**, when present, introduce a
curvature of the characteristic lines for velocity on the space-time or xt-diagram. Space- time curvature thus indicates velocity
acceleration and the presence
of force .

**(a) Straight characteristics and path
lines show steady flow and absence of force.**

**(b) Curved characteristics and path lines
show acceleration and presence of force**

In the case of
compressible flow and 3-space (x,y,z)
, a curved path line dx/dt
= v (path) may be ‘transformed away’ to a straight line Lagrange representation
[ dh/dt = 0].

[Note: In General Relativity the analogous
distortion of its 4-space (x,y,z,t)
to obtain a force- free representation is a **tensor distortion. **

However, it should be noted that
general relativity is a continuous field theory, and, as such, excludes
discontinuities or singularities such as shocks. Therefore, it appears to be
fundamentally incompatible with quantum physics.

On the other hand, compressible
flow as shown below in Section 3.1 on Visible Matter predicts shock
discontinuities as the physical mechanism for the emergence of the elementary
particles of matter by shock compression of an energy flow. Thus, compressible flow is compatible with
quantum theory.

__2.5.2 The
Exact Wave Equation __

Ñ^{2 }ψ^{ } =^{ }1/c^{2}
∂^{2}ψ/∂t^{2} [ 1 + Ñψ ]^{(k
+ 1)
}(7)

where k, the
adiabatic exponent, is k = c_{p}/c_{v}
= ( n + 2) /n; and ( k + 1 ) = 2( n + 1)/n =
2(c_{o}/c*)^{2}.
Here, pressure is a function of density only. This wave is isentropic, **non-linear, unstable, and grows to a non-isentropic discontinuity
called a shock wave.**

__2.5.3
Shock Waves __

All finite
amplitude, compressive waves are non-linear and grow in amplitude with time to
form shock waves. These shocks are
discontinuities in flow, across which the flow variables p, ρ, V, T and c
change abruptly. (Note p = pressure, ρ = momentum).

_{ }

__2.5.3.1
Normal Shocks __

V_{1} > V_{2} (8)_{ }

p_{1}, ρ_{1},
T_{1} < p_{2}, ρ_{2, }T_{2} (9) _{ }

_{ }

_{ }

__Entropy Change Across Shock: __

∆S = S_{1} – S_{2} =
− ln(ρ_{02}/ρ_{01})
(10)

__Maximum Condensation Ratio:__

ρ_{1}/ρ_{2} = [n+1]^{1/2}
= V_{max}/c*
(11)

__2.5.3.2 Oblique Shocks __

If the
discontinuity is inclined at angle to the direction of the oncoming or upstream
flow, the shock is called **oblique**.

Oblique Shock

V_{1N}
> V_{2N}

p_{1}ρ_{1}T_{1} <
p_{2}ρ_{2}

Since the flow V
is purely relative to the oblique shock front, the shock may be transformed to
a normal one by rotation of the coordinates, and the equations for the normal
shock may then be used instead.

__2.5.3.3
Strong____ and Weak Oblique Shock Options: The Shock
Polar __

For each inlet
Mach number M_{1} ( = V_{N1}/c), and
turning angle of the flow θ, there are **two physical options**:

1) the strong shock ( intersection S) with strong compression
ratio and large flow velocity reduction
(p_{2 }>> p_{1}; V_{2}
<< V_{1}, or

2) the weak shock (intersection W, with small pressure rise and
small velocity reduction.

Which of the two
options occurs depends on the boundary conditions: low back i.e. low downstream
pressure favours the weak shock occurrence; high downstream
pressure favours the strong shock.

When the turning
angle θ of the oncoming flow is zero, the strong shock becomes the normal
or maximum strong shock, and the weak shock becomes an infinitesimal,
low-amplitude, acoustic wave.

__2.6 Types of Compressible Flow: __

a) Steady, subcritical flow ( e.g. subsonic,
V< c), governed by elliptic, non-linear, partial differential equations.

b) Steady,
supercritical flow ( e.g. supersonic, V_{1} > c) governed by hyperbolic,
nonlinear, partial differential equations.

c) Unsteady flow (either subcritical or supercritical). These are wave equations
governed by hyperbolic, non-linear, partial differential equations. They are
often simplified to linear approximations, for example to the classical wave
equation (5); if of finite amplitude they grow to shocks..

The solutions to
the above hyperbolic equations are called characteristic solutions. If linear, they correspond to the eigenfunctions and eigenvalues of
the linear solutions to the various wave equations of quantum mechanics ( Sect. 3.7), or, equally, to the diagonal solutions of the
matrix equation of Heisenberg’s formulation of quantum mechanics.

__2.7 Wave Speeds __

c = [ c_{o}^{2}
– V^{2}/n]^{1/2 } (steady flow) (12)

c = c_{o}^{2} – V^{2}/n
– [(2/n)cV ]^{1/2 }(unsteady flow) (13)

c^{2} = (dp/dρ)_{s}, [ c = (dp/dρ)_{s}^{1/2}]_{
} where s is an isentropic state.

Since V is
relative, it may be arbitrarily set to zero to give a stationary or “local”
coordinate system moving with the flow; this automatically puts c = c_{o}
and transforms the variable wave speed to any other relatively moving
coordinate system.

The shock speed U
is always supercritical (U > c) with respect to the upstream or oncoming
flow V_{1}.

__2.8 Wave Speed Ratio c/c _{o} and The Isentropic Thermodynamic Ratios__

c/c_{o} = [**1
-1/n(V/c _{o})^{2}]^{1/2}**

__All the basic thermodynamic parameters of a compressible
isentropic flow are therefore specified by the wave speed ratio c/c _{o}.
. __

__2.9 Relativity Effects in Compressible Flow__** : **In compressible flows all velocities [V or u] are relative only,
and, moreover, the wave speed c is a variable which is dependent on V and n; it decreases for larger
velocities V. and it reaches its maximum
value c

Interestingly, Equation
(14)

c/c_{o} = [**1
-1/n(V/c _{o})^{2}]^{1/2}**

shows
that the correction factor for the
effect of flow speed on wave speed
c on the right hand side of the equation
** has the same form as the Lorentz
Transformation of special relativity**. If n = 1 the two correction
factors become formally identical.

The differences from
special relativity are
that the wave speed **c** is now a
variable and a function of the flow velocity **V**, and that there is
the energy partition constant **n**. Since the wave speeds are low ( c = 334 m/s for air at m.s.l.), the ‘Lorentz’ corrections for physical compressible
systems such as gases are relatively
large. Also, the flow speeds can exceed the wave speed (
supersonic flow), whereas in special relativity theory, the
wave speed c is a constant ( 3 x 10^{8}
m/s) which can never be exceeded.

Photon shocks are thus impossible in special relativity, whereas
in compressible flow they furnish a quantum physical theory for the origins of
matter itself via the
formation of the elementary particles of matter at compression shock discontinuities.

__2.10
Wave____ Type and Stability __

Compression waves
are the rule in the baryonic physical world ( i..e.
in Quadrant I on the pressure-volume diagram) Here, the usual equation of state
is hyperbolic and its density waves are always compressive and all compression waves of finite amplitude
grow towards shocks. Here, only **acoustic** compression waves ( i.e. infinitely low-amplitude compressions) are stable.

Rarefaction waves
require an elliptical equation of state. Again these rarefactions only stable
for acoustic infinitely low amplitudes. Finite amplitude grow
to rarefaction shocks.

Stable waves of
both rarefaction and/or compression and of finite amplitude are possible with
the linear equation of state [ p = Av +B].

**II**

+p -v

+v

__Equations of State for Various Wave Types__

**The criterion for wave behaviour** is the curvature (d^{2}p/dv^{2} ) of the various isentropic equations of state:

1. If d^{2}p/dv^{2} is > 0 (i.e. the hyperbolic curve) compression waves form and steepen to form
compression shocks, while rarefaction waves flatten and die out. Only compression waves of infinitely small
amplitude (“acoustic” or “sound” waves” ) are stable. Compression shocks form from
finite amplitude waves..

2. If
d^{2}p/dv^{2} is < 0 , ( i.e. the elliptical curve) rarefaction waves form and steepen to form rarefaction shocks, while compression waves flatten and die out.
Only rarefaction waves of infinitely low amplitude are stable.

3. If d^{2}p/dv^{2} = 0, (i.e. the l**inea**r state ) waves of either
compression or rarefaction are stable, all amplitudes are supported and shocks
are ruled out.

** **

**Copyright
**Bernard** A. Power**

**February
2019**