[i] Any Waves
-------------
The Google search tool returns ~6/49 references,
in the rec.radio.amateur.antenna newsgroup,
for "Kurokawa", the author of the paper
"Power Waves and Scattering Matrix",
IEEE Transactions on Microwave Theory and Techniques,
vol. MTT-13, March 1965, pp 194-202.

This search result indicates a subject of some interest.

Let us take Kurokawa's conclusion for granted:

"...the power waves are the result
of just one of an infinite number
of possible linear transformations
of voltage and current...".

Following this,
it seems that,
at least mathematically,
we can choose
any four complex constants,

p, q, r, s

and define

a, b

by the linear transformation

a = p.V + q.I
b = r.V + s.I

But as far as we consider both of

V, I

to be waves,

since they are solutions of the
-notably the same-
"One-Dimensional Wave Equation":

X'' - (g^2).X = 0

where X stands for either V or I,
it is trivial to show that each of the just defined

a, b

verify exactly the same equation.

Therefore,
it seems legitimate to call

a, b

"Any Waves", too.

Thus, as we see, we can define,
from a unique couple of (V, I) Voltage and Current waves
"an infinite number" couples of (a, b) "Any Waves",
since, the four mentioned constants can take obviously
an infinite number of values.

[II] Mathematical Restrictions
-----------------------------
If we intend to use the Any Waves mathematically,
e.g. perhaps to facilitate a manipulation of formulas
which requires in some step
the inverse expression of V, I from the a, b,
then we have to impose
a restriction for the existence
of the inverse transformation,
that is

(1): p.s =/= q.r

After that
the

V, I

can be result from the

a, b

as

V = A.a + B.b
I = C.a + D.b

where the new four complex constants

A, B, C, D

are given in terms of the old constants

p, q, r, s

by the relations

A = [ s/(p.s - q.r)]
B = [-q/(p.s - q.r)]
C = [-r/(p.s - q.r)]
D = [ p/(p.s - q.r)]

since the denominator is non-zero.

The condition (1)
although restricts somehow the otherwise totally independent cases,
"The Any Waves" remain of "an infinite number".

[III] Dimension Balance Restrictions
------------------------------------
Now if we would like to certify physically,
the validity of the introduced transformations,
then we have to establish
the "(Dimensional) Unit Balance" of these equations.

To do this,
let us note by

{y}

the Unit(s) of any Physical Quantity y.

Then, from the above equations,
it is almost obvious that
the following relations must be valid:

(2.1): {q} = {Ohms}.{p}
(2.2): {s} = {Ohms}.{r}
(2.3): {a} = {Volt}.{p}
(2.4): {b} = {Volt}.{r}

The added conditions (2),
although reduce the number of
the independently varied Physical Quantities (Units) to two,
they do not affect in essence "The Any Waves"
which are still of "an infinite number".

[IV] Restrictions referenced to Physical Principles
---------------------------------------------------
Next let us impose some specific
physical considerations or principles
on "The Any Waves",
e.g. that of the conservation of energy.

As an example of this
we can consider
as special cases of "The Any Waves"
those mentioned in the referenced paper,
that is the
"Power Waves" and "Traveling Waves".

Indeed;
these waves fulfill the linear transformations,
with all constants specifically expressed
in terms of some impedances,
measured as follows

{p} = {r} = 1/Sqrt(Ohms} = {Ohms}^(-1/2),
{q} = {s} = Sqrt{Ohms} = {Ohms}^(+1/2)

and from them,
the "Physical Units" of both the "Waves" are the same:

(3): {a} = {b} = Sqrt{Watt}.

Once again the additional restriction (3)
although forces the a and b to be of the same physical entity,
one which somehow is connected to the power,
still keeps "The Any Waves" to "an infinite number".

[V] Measurement Restrictions
----------------------------
Further we can assume, for the sake of completeness,
that maybe there is a -necessarily finite- number
of appropriate materials and/or devices,
other than the well known reflectometer,
capable to set apart the Two Any Wave Components
on the basis of some Physical Properties.

If this is the case then it is reasonable to try to build
a linear transformation
in terms of a couple of wave quantities appropriate to
express the mentioned physical properties.

But neither this case can reduce the possibly of
"The Any Waves" "of an infinite number".

[VI] Physical Hypotheses Restrictions
-------------------------------------
Finally, we can imagine that
we ingeniously predict some extraordinary Any Waves,
with the proof of their existence
to become a subject of a life-long heavy research activity.

For instance;
if for some reason we define the constants
by using some "extreme" expressions of the Impedance,
such as the following

{p} = (Ohms}^k
{q} = {Ohms}^n

with k, n equal to any other non-zero rational number,
e.g.
.... -100, -1, -1/3, 1, 1/3, 100 ...

then "The Any Waves" have now
the extraordinary Physical Meaning
suggested by their corresponding physical units expressions

{a} = {Ohms^(k+(1/2)}.{W^(1/2)}
{b} = {Ohms^(n+(1/2)}.{W^(1/2)}

Fine;
but "The Any Waves" stay "of an infinite number".

[VII] Practical Applications
----------------------------
Let us drop from the clouds...

In the practical application of the Transmission Line,
"The Subtle Detail" which discriminates
the two instances of "The Any Waves",
"The Power Waves" and "The Traveling Waves",
with two different physical meanings for the same physical phenomenon
and the concrete result of "The Third Term" disappearance
in the "Beloved" Complex Characteristic Impedance case,
increases the confusion instead of comprehension.

Accordingly and as far as we are not in place
to reduce the number of "The Any Waves",
regardless of their more or less well established physical meaning,
in something less than infinity,
any further attempt to discuss about "what actually happens"
in the totality of the related practical applications
becomes redundant and worthless.

It seems that we have to content ourselves
with what it is already known...

Sincerely,

pez
SV7BAX
TheDAG