Independence of waves
 

Owen Duffy wrote:
For avoidance of doubt, power is not a quantity to be superposed, though
presumably if it can be deconstructed to voltage or current or electric
field strength or magnetic field strength (though that may require
additional information), then those components may be superposed.

The single bit of additional information required is the phase
angle between the voltages (or currents or fields). Optical
physicists deduce the relative phase angle by the ratio of
intensity (power density) in the bright rings vs the dark rings.
We hams can deduce the relative phase angle by the ratio of
forward power (density) to reflected power (density). Our task
as hams looking at a one-dimensional transmission line is much
easier than the task of optical physicists looking at visible
light in three-dimensional space. Our transmitted CW signals are
coherent and collinear in a transmission line, something that
optical physicists can only dream of.

The resultant fields at a point though seem to not necessarily contain
sufficient information to infer the existence of a wave, just one wave,
or any specific number of waves, so the superposed resultant at a single
point is by itself of somewhat limited use. This one way process where
the resultant doesn't characterise the sources other than at the point
seems to support the existence of the source waves independently of each
other, and that there is no merging of the waves.

That is the case in a majority of examples. But in the case
of two coherent collinear waves superposed in a one-dimensional
transmission line where the resultant is the same at every point,
we can safely assert that those two waves have ceased to have an
existence independent of each other. The idea of two waves
canceling all up and down the transmission line yet continuing
their separate existences until their combined zero energy level
is dissipated (or not) is a pipe dream. If ExB = 0, the energy
in those canceled waves went the other direction a long time ago
and those waves have ceased to exist in their original direction
of travel, i.e. they have interacted and canceled.

When two waves combine to a zero energy level, the pre-existing
energy in those two waves is "redistributed in the direction
of constructive interference". In a one-dimensional transmission
line, there are only two possible directions. If waves superpose
to zero energy in one direction, their energy components are
"redistributed" in the only other direction possible. If the energy
ceases to flow in the reverse direction, then it must flow in the
forward direction. That's why Pforward = Psource + Preflected.
Anything else would violate the conservation of energy principle.
--
73, Cecil http://www.w5dxp.com

Independence of waves
 

Owen Duffy wrote:

Fine Roy, the maths is easy, but you don't discuss the eligible
quantities.

As I learned the superposition theoram applying to circuit analysis, it
was voltages or currents that could be superposed.

Presumably, for EM fields in space, the electric field strength and
magnetic field strength from multiple source can be superposed to obtain
resultant fields, as well as voltages or currents in any circuit elements
excited by those waves.

For avoidance of doubt, power is not a quantity to be superposed, though
presumably if it can be deconstructed to voltage or current or electric
field strength or magnetic field strength (though that may require
additional information), then those components may be superposed.

The resultant fields at a point though seem to not necessarily contain
sufficient information to infer the existence of a wave, just one wave,
or any specific number of waves, so the superposed resultant at a single
point is by itself of somewhat limited use. This one way process where
the resultant doesn't characterise the sources other than at the point
seems to support the existence of the source waves independently of each
other, and that there is no merging of the waves.

Is anything above contentious or just plain wrong?

No, I agree entirely, except for

This one way process where
the resultant doesn't characterise the sources other than at the point
seems to support the existence of the source waves independently of
each other, and that there is no merging of the waves.

which I don't understand. We lose information when we add or otherwise
operate on two numbers to get one. (Which I think is what you might be
saying.) Given a number which is the sum of two others, we can't tell
from that sum alone what the two original numbers were. The same is
naturally true of superposed or added, if you prefer, waves or fields.
Power has the same problem (among others) -- given even an instantaneous
power, we can't tell without some other information (such as the complex
impedance) what the constituent voltage and current were.

Roy Lewallen, W7EL

Independence of waves
 

"Cecil Moore" wrote in message
.. .
Roy Lewallen wrote:
Superposition means the following: If f(x) is the result of excitation x
and f(y) is the result of excitation y, then the result of excitation (x
+ y) is f(x) + f(y). . .

Now the big question is: Is superposition always reversible?
If not, it implies interaction between f(x) and f(y).
--
73, Cecil http://www.w5dxp.com

as long as everything is linear, yes.

Independence of waves
 

Dave wrote:
yes, superposition is meant to work directly on voltage, current, electric
fields, and magnetic fields. it can be extended by adding appropriate extra
phase terms to power or intensity as cecil prefers to use.

That seems to be common knowledge except for some
(narrow-minded?) posters here. Powers do not superpose
but there is a method of adding power (densities)
that has been acceptable to physicists for at least
a century and may date back to Young, Fresnel, and
Huygens.
--
73, Cecil http://www.w5dxp.com

Independence of waves
 

Roy Lewallen wrote:
Given a number which is the sum of two others, we can't tell
from that sum alone what the two original numbers were.

Seems as though those two numbers interacted and
then lost their separate identities, huh?
--
73, Cecil http://www.w5dxp.com

Independence of waves
 

Dave wrote:
Now the big question is: Is superposition always reversible?
If not, it implies interaction between f(x) and f(y).

as long as everything is linear, yes.

This is really interesting. Given the following:

b1 = s11(a1) + s12(a2) = 0

Let P1 = |s11(a1)|^2 = 1 joule/sec

Let P2 = |s12(a2)|^2 = 1 joule/sec

Therefore, Ptot = |b1|^2 = 0 joules/sec

Ptot = P1 + P2 + 2*SQRT(P1*P2)cos(180)

Ptot = 1 + 1 - 2 = 0 joules/sec = |b1|^2

Can one reverse the superposition whose result is
zero to recover the original two component waves?
If not, isn't that proof that the two original
component waves interacted?
--
73, Cecil http://www.w5dxp.com

Independence of waves
 

"Cecil Moore" wrote in message
...
Dave wrote:
Now the big question is: Is superposition always reversible?
If not, it implies interaction between f(x) and f(y).

as long as everything is linear, yes.

This is really interesting. Given the following:

b1 = s11(a1) + s12(a2) = 0

Let P1 = |s11(a1)|^2 = 1 joule/sec

Let P2 = |s12(a2)|^2 = 1 joule/sec

Therefore, Ptot = |b1|^2 = 0 joules/sec

Ptot = P1 + P2 + 2*SQRT(P1*P2)cos(180)

Ptot = 1 + 1 - 2 = 0 joules/sec = |b1|^2

Can one reverse the superposition whose result is
zero to recover the original two component waves?
If not, isn't that proof that the two original
component waves interacted?
--
73, Cecil http://www.w5dxp.com

no, because you have done a non-linear operation on them by converting to
powers. obviously at the start 'a1' and 'a2' are separate.

Independence of waves
 

"Dave" wrote in message news:7RoWh.109$Zm.79@trndny03...

"Cecil Moore" wrote in message
...
Dave wrote:
Now the big question is: Is superposition always reversible?
If not, it implies interaction between f(x) and f(y).

as long as everything is linear, yes.

This is really interesting. Given the following:

b1 = s11(a1) + s12(a2) = 0

Let P1 = |s11(a1)|^2 = 1 joule/sec

Let P2 = |s12(a2)|^2 = 1 joule/sec

Therefore, Ptot = |b1|^2 = 0 joules/sec

Ptot = P1 + P2 + 2*SQRT(P1*P2)cos(180)

Ptot = 1 + 1 - 2 = 0 joules/sec = |b1|^2

Can one reverse the superposition whose result is
zero to recover the original two component waves?
If not, isn't that proof that the two original
component waves interacted?
--
73, Cecil http://www.w5dxp.com

no, because you have done a non-linear operation on them by converting to
powers. obviously at the start 'a1' and 'a2' are separate.


i should expand a bit more. all your equations above have done is shown
that at the point where you are doing your analysis s11(a1) and s12(a2),
which add up to 0... also produce a net 0 power at that point. this is as
expected for destructive interference AT THAT POINT. as such your s
parameter analysis is insufficient to separate the individual components
after you combine them into a power. however, at the begining they are
obviously separate waves since you have represented them with separate input
values, and given a linear transfer function for your point on the wire, or
in space, they can always be kept separate. it is only your act of
calculating the power at that point that combines them.

Independence of waves
 

K7ITM wrote:
On Apr 20, 10:10 pm, Roy Lewallen wrote:
Correction:

Roy Lewallen wrote:

Superposition means the following: If f(x) is the result of excitation x
and f(y) is the result of excitation y, then the result of excitation (x
+ y) is f(x + y). . .
That should read:

Superposition means the following: If f(x) is the result of excitation x
and f(y) is the result of excitation y, then the result of excitation
(x + y) is f(x) + f(y). . .
^^^^^^^^^^^
I apologize for the error. Thanks very much to David Ryeburn for
spotting it.

Roy Lewallen, W7EL

I guess that's the definition of linearity. I'm not sure I've heard
it called superposition before, but rather that the superposition
theorem is a direct result of the linearity of a system. I trust
that's a small definitional issue that doesn't really change what
you're saying.

Cheers,
Tom



Tom,

For most purposes the terms superposition and linearity are
interchangeable. However, for the purists there is a difference.

A system that is deemed linear requires that it has the properties of
both superposition and scalability. These properties are essentially the
same for simple systems, but they are not necessarily the same when
considering complex values. I found some clear examples in a book, "The
Science of Radio", by Paul Nahin.

One example, y(t)=Re{x(t)} describes a system which obeys superposition,
but not scaling. Hint: try a scaling factor of "j". That system is not
linear.

Another example is y(t)=[1/x(t)]*[dx/dt]^2. That system obeys scaling,
but not superposition. Again, it is not linear.

The bottom line is that superposition is necessary, but not sufficient
to ensure linearity.

You are correct that the definitional issue is not relevant to the
current RRAA discussion.

73,
Gene
W4SZ

Independence of waves
 

Owen Duffy wrote:
There has been much discussion about wave cancellation, anihalation,
interaction etc. The discussion was initially about waves confined to a
transmission line (but would apply also to a waveguide in a sense) and
then progressed to radiation in free space.

Let me initially explore the case of radiation in free space. I am
talking about radio waves and the radiation far field.

If we have two widely separated antennas radiating coherent radio waves
don't they each radiate waves that travel independently through space. (I
have specified wide separation so as to make the effect of one antenna on
the other insignificant.

If we were to place a receiving antenna at a point in space to couple
energy from the waves, the amount of energy available from the antenna is
the superposition of the response of the antenna to the wave from each
source. This is quite different to saying that the electric field (or the
magnetic field) at that point is the superposition of the field resulting
from each antenna as is demonstrated by considering the response of
another recieving antenna with different directivity (relative to the two
sources) to the first receiving antenna.

A practical example of this is that an omni directional receiving antenna
may be located at a point where a direct wave and a reflected wave result
in very low received power at the antenna, whereas a directional antenna
that favours one or other of the waves will result in higher received
power. This indicates that both waves are independent and available to
the receiving antenna, the waves do not cancel in space, but rather the
superposition occurs in the antenna.

Though we frequently visualise nodes and antinodes in space, or talk of
nulls in space (eg have you ever noticed that when you stop a car at
traffic lights, you are smack in the middle of a null), whereas it seems
to me that the realisation of a null involves the response of the
receiving antenna.

This explanation IMHO is more consistent with the way antennas behave
than the concept that waves superpose in space, it allows waves to
radiate outwards from a source, passing through each other without
affecting each other. Whilst we routinely look at plots of the
directivity of an antenna, and assume that the plotted directivity is
merely a function of polar angle, we overlook that the plotted pattern
assumes an isotropic probe at a distance very large compared to the
dimensions of the antenna (array). Tracing the position of a pattern
minimum in towards the array may well yield a curved path rather than a
straight line, and a curved path is inconsistent with waves anihalating
each other or redistributing energy near the antenna and radiating
outwards in true radial direction from some virtual antenna centre.

So, it seems to me that coherent waves from separated sources travel
independently, and the response of the probe used to observe the waves is
the superposition of the probe's response to each wave. (A further
complication is that the probe (a receiving antenna) will "re-radiate"
energy based on its (net) response to the incoming waves.)

Now, considering transmission lines, do the same principles apply?

A significant difference with uniform TEM transmission lines is that
waves are constrained to travel in only two different directions.

Considering the steady state:

If at some point two or more coherent waves travelling a one direction,
those waves will undergo the same phase change and attenuation with
distance as each other and they must continue in the same direction
(relative to the line), and the combined response in some circuit element
on which they are incident where superposition is valid (eg a circuit
node) will always be as if the two waves had been superposed... but the
response is not due to wave superposition but superposition of the
responses of the circuit element to the waves. It is however convenient,
if not strictly correct to think of the waves as having superposed.

That convenience extends to ignoring independent coherent waves that
would net to a zero response. For example, if we were to consider a
single stub matching scheme, though one there might consider that
multiple reflected waves arrive at the source, if they net to zero
response, then it is convenient to regard that in the steady state there
are no reflected waves, the source response is as if there were no
reflected waves. An alternative view of that configuration is that
superposition in the circuit node that joins the stub, the line to the
load and the line to the source results in conditions at that end of the
source line that do not require a reflected wave to satisfy boundary
conditions at that point, and there really is no reflected wave.

Steady state analysis is sufficiently accurate and appropriate to
analysis of many scenarios, and the convenience extends to simplified
mathematics. It seems that the loose superposition of waves is part of
that convenience, but it is important to remember the underlying
principles and to consciously assess the validity of model
approximations.

Comments?

Owen



If a probe is confined to the null
"space" consisting of the plane between
the antennas, it seems clear it cannot
distinguish between radiation from the
two antennas.

But no probe so constrained can detect
radiation from either antenna alone (or
from any other radiating source).

Seems somewhat tautological then that
the probe cannot be designed to
distinguish between the two antennas.

If the assumption so constrains the
probe to fatal infirmities under any
circumstances, then the observance of
failure in a particular circumstance is
void of information.

No?

Chuck

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