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WAVES, VERY LONG



I hope I am not the only one who learned from John's
contributions to a recent thread on waves. So for the
benefits of those who might be interested I edited the
thread (making arbitrary cuts, etc.) in order to print
it. Here it is, in the form of a dialogue (plus David's
message). Ludwik Kowalski


*******************************************
Ludwik
******
The speed of a small transverse disturbance along a
coiled spring is v=sqr(T/mu), where T is the constant
tension and mu is the linear mass density. What is
the speed of a small longitudinal disturbance along
the same spring? In other words, what is the relation
between the speeds of P waves and S waves in the
one-dimensional case?

John
****
Reeeeally? I would have thought that was the speed for
waves on a *string* not a spring. That is, that formula
neglects stiffness. ...

What is the speed of a small
longitudinal disturbance along the same spring?

What's wrong with
1 / sqrt(rho s)
where
rho = density per unit length [kg / m] and
s = compliance per unit length [1 / nt]

Ludwik
******
In my version the spring (slinky) is always stretched
to behave like a rope or cable. A cable under tension is
a spring with very large k.

What is "compliance per unit length [1 / nt]"?

John
****
A cable under tension is a spring with very large k.

Maybe. I'm not sure what you mean
or€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m*€m(slinky)
is always stretched
to behave like a rope or cable.

In context, I assume that means you intend to treat
stiffness as negligible compared to tension. I doubt
that is correct for slinkies; that is, I suspect that
for any tension that does not do irreversible damage
to the slinky, stiffness is non-negligible.

What is "compliance per unit length [1 / nt]"?

According to my dictionary, compliance is strain per
unit stress. It has dimensions of [N/m] and for springs
it is the reciprocal of the spring constant. ...

Ludwik
******
To predict the transverse speed we must know the
tension T (in addition to linear density rho, which
I call mu); the magnitude of the spring constant k
has nothing to do with it. But, as you "proved", to
predict the longitudinal speed we simply replace T
by s, the spring stiffness (which appears in Hookes
law).

On the purely intuitive basis I would not expect this.
After all in both cases, longitudinal and transverse,
the restoring force is directly proportional to the
corresponding displacement. If k is not relevant in
one case then I would expect s to be irrelevant in
another. I would expect the longitudinal wave in a
slinky to depend on T rather than on s.

It turns out that s can be measured as easily as k.
Right? Thus your mathematically derived formula
can be subjected to an acceptable experimental
validation. A good student project or lab. Does
anybody do this?

By the way, my first classroom waves demo is
based on a long stretched spring. I firmly attach
one end to a hook, increase the length of the spring
by stretching it (about 30%) and wiggle the other
end up and down. A standing wave (over a distance
of about 10 m) can easily be produced. The speed
was never measured but I assume that the standard
formula, v=sqr(T/mu), is applicable. Would you agree?

John:
****
Be careful, there are three pieces of physics here, and
you mentioned only two; we have not only
a) transverse waves driven by tension, and
b) longitudinal waves driven by compression/extension,
but also
c) transverse waves driven by stiffness

To predict the transverse speed we must know the
tension T (in addition to linear density rho, which I
call mu);

Yes, if you insist on ignoring flexional stiffness.

the magnitude of the spring constant k has nothing
to do with it.

OK; evidently k is the compressive spring constant
and you are still ignoring flexional stiffness.

But, as you "proved", to predict the longitudinal speed
we simply replace T by s, the spring stiffness (which
appears in Hookes law for flexion).

Ooops, now you mention flexion and stiffness, which is
inconsistent with the foregoing statements and
assumptions. My (s) is absolutely not the flexional
stiffness, for at least two reasons:
*) My (s) is a *compliance* per unit length, so it
measures how unresisting a spring is, not how
resisting it is, and
*) My (s) has to do with pure compression, not flexion.

After all in both cases, longitudinal and transverse,
the restoring force is directly proportional to the
corresponding displacement.

I agree.

If k is not relevant in one case then I would expect s
to irrelevant in another. I would expect the longitudinal
wave in a slinky to depend on T rather than on s.

I disagree. Here's why:

The tension from the left termination and the right
termination apply forces to every element of the spring,
but as long as we consider only longitudinal motions
these two forces are equal and opposite at all places
and all times.

Applying tension to a spring might change its length,
but this does not (to first order) change its spring
constant. (This is direct consequence of the linearity
of Hooke's law.) It is the compressive spring constant
that drives the compressive longitudinal waves.

By the way, my first classroom waves demo is based on
a long stretched spring. I firmly attach one end to
a hook, increase the length of the spring by stretching
it (about 30%) and wiggle the other end up and down.
A standing wave (over a distance of about 10 m) can
easily be produced. The speed was never measured but
I assume that the standard formula, v=sqr(T/mu), is
applicable. Would you agree?

As I said in previous messages, it depends on what
medium you're using. Any slinky I've ever owned would
be destroyed if extended to 10 m length. Depending on
the medium, I wouldn't be surprised to find that the
flexional stiffness makes a nontrivial contribution to
the speed of the transverse waves.

Each of them, however, should behave like
a cable, or a rope, whose T and mu are identical.

Not unless you can arrange to:
*) Ignore the compressional spring constant, and
*) Ignore flexional stiffness.

Ludwik
******
I should have said the "s, the spring compliance".
The word "proved" was quoted to emphasize what
you wrote yesterday about differences between
proving things in math and in science. Your derivation
is general enough to apply to both longitudinal and
transverse speeds. Bringing torsional disturbances
would not be appropriate for a one dimensional
approximation.

I am puzzled by the fact that the derived formula,
v=sqr(k/mu), for the speed [stiff transverse wave]
does not agree with v=sqr(T/mu), which students
often test in physics labs (via standing waves). Why
should a spring disturbance propagate at a different
speed than a string disturbance when T and mu are
exactly the same? Next time I do the class demo with
the long spring I will try to see which of the two
formula is in agreement with reality.

Ludwik
******

What a coincidence, John's message flushed on my
screen only several seconds after my reply. Thanks
for additional clarifications. ....

I just discovered a misconception in my ways of
thinking about transverse waves. Am I the only one
who was not aware of the third kind of waves in a
stretched spring? The textbooks I know totally
ignore contribution of stiffenss to the speed of
transverse waves. ...

John
****
The textbooks I know totally ignore contribution
of stiffens to the speed of transverse waves.

Possibly that's because the physics of
a) transverse waves driven by tension, and
b) longitudinal waves driven by compression/extension,
is basically the same physics, and is easy and elegant,
while the physics of
c) transverse waves driven by stiffness
is different physics, requires a more sophisticated analysis,
and leads to more complicated results.

Here's the deal: for case (a) or (b), we get a wave equation
of the form
(d/dt) (dy/dt) - (d/dx) (dy/dx) c^2 = 0
which admits running-wave solutions of the form
f(x,t) = F(x+-ct)
for any F(), and which has the simplest possible dispersion
relation
omega = +- c k

IN CONTRAST... the physics of flexure is more complicated.
See _The Feynman Lectures on Physics_ volume II section
38-4 for an introduction. ...

For flexy waves we have an equation of the form
(d/dt)^2 y - const * (d/dx)^4 y = 0
and yes, that's a fourth-order spatial derivative. This is
a linear differential equation (linear in y) so we retain
the superposition principle (which means this is much
simpler than, say, fluid dynamics, where the equations
are nonlinear). But it is highly anharmonic, i.e. highly
dispersive. Because of the dispersion, there cannot be
running-wave solutions that retain their shape. The
dispersion relation is omega = const * k^2 ......

John
****

Yesterday Ludwik said that transverse waves driven
by tension were relatively familiar, while those
driven by other restoring forces were relatively
unfamiliar.

I suspect that another way of classifying waves gets
more to the point: some wave equations are (more or
less) nondispersive, while others are markedly
dispersive.

Examples of (more or less) nondispersive waves
include:
visible light in air or vacuum
audible sound in air or water
transverse waves driven by tension in a taut string
longitudinal compression/extension waves in a spring

In contrast, examples of markedly dispersive waves
include:
visible light in media such as flint glass (e.g. prisms)
radio waves in the ionosphere
low-energy solutions of the Schrödinger equation
surface waves at the air/water interface
transverse waves driven by stiffness in a spring

Many high-school physics books blissfully ignore dispersion.
They can get away with this by considering only
nondispersive media and/or considering only monochromatic
waves if the medium is dispersive.

You can make a good classroom demonstration of
nondispersive propagation by stringing a strong but flexible
rope across the room under plenty of tension. (I've got some
nice Dacron kernmantle yacht braid that works great.) Then
smack it near one end with a bat or a hammer. This will form
a nice pulse which can be seen to retain its shape as it runs
down the rope, in accordance with the running wave solution

f(x,t) = F(x-ct)

The directly-contrasting demonstration is more difficult
and less pleasing, because it tries to demonstrate the
NONexistence of such a solution -- and it's always hard to
prove a negative. Get a long metal or fiberglass rod and
clamp it at one end so it sticks across the classroom.
It needs to be stiff enough to keep itself off the floor, but
floppy enough to allow flexional waves. If you smack it
with a bat, you will *not* see a shape-preserving wave
packet run to the other end. The wavespeed for the
high-frequency components is so much faster than the
wavespeed for the low-frequency components that the
wave packet tears itself apart.
--------
A better contrast would be to measure the dispersion
relation for the two media directly, by measuring the
wavelength versus frequency.

Of course this would require students to actually *do*
stuff (measuring lengths, counting and timing cycles,
drawing graphs) rather than just sitting there and
watching a qualitative demonstration.

Ludwik
******

It is good that stiffness waves are ignored in
introductory courses. The tension and compression
waves are already quite difficult for students to
grasp. These questions below are for my own
education only.

Thinking about one-dimensional waves we say that
any function of x and t in which these two independent
variables appear in the form of a (x-c*t) argument
is a mathematical description of a traveling wave.
Thus any function f(x-c*t) is a description of a
possible wave. It satisfies the differential wave
equation. And vice versa, any function which satisfies
the equation must have the f(x-c*t) form, where c is
a positive or negative constant called speed. If we plot
that function versus x, for different moments of time,
we see "a moving shape". The shape is preserved, if it
is an equilateral triangle at t=0 then it must also be
the equilateral triangle (same base length) at t>0,
far away.

I am assuming the dissipation of energy is negligible,
so that the "far away" really means about ten characteristic
distances (such as initial shape's length), or more. In my
model only energy dissipation can be responsible for
changes of shape.

Here my dilemma. The stiffness wave satisfies the
differential wave equation, as demonstrated by John.
Therefore the initial shape must be preserved, as the
wave travels to the left or to the right in a long stiff
road. I am assuming that the rod is uniform (mu and s
do not change along the x axis). This conflicts (in my
mind) with what John later said about dispersion.
Dispersion implies that the shape is not preserved
(each harmonic travels with different speed) and that
changes in shape are not caused by energy dissipation. ...

John
****

Thinking about one-dimensional waves we say that
any function of x and t in which these two independent
variables appear in the form of a (x-c*t) argument is
a mathematical description of a traveling wave. Thus
any function f(x-c*t) is a description of a possible
wave. It satisfies the differential wave equation.

The foregoing paragraph seems to assume a totally
nondispersive situation. That way of looking at things
will lead to trouble as soon as there is dispersion.

In particular, it's bad luck to speak of "the" wave
equation. There's lots of wave equations.

And vice versa, any function which satisfies .....

Again: Shape-preserving travelling waves are
associated with nondispersive propagation.

Here my dilemma. The stiffness wave satisfies the
differential wave equation, as demonstrated by John.

Nope. If you re-read my derivation you'll see it explicitly
applies to ***longitudinal*** waves driven by
compression. It isn't even close to right for transverse
waves driven by stiffness.

Therefore the initial shape must be preserved, as the
wave travels to the left or to the right in a long stiff
road.

Not true for stiffness waves.

This conflicts (in my mind) with what John later said
about dispersion. Dispersion implies that the shape is
not preserved (each harmonic travels with different
speed) and that changes in shape are not caused by
energy dissipation.

Yes, that's what dispersion implies. ....

Ludwik
******
In particular, it's bad luck to speak of "the" wave
equation. There's lots of wave equations.

Probably because many quite different physical phenomena
are called waves. Can somebody bring an acceptable general
definition of a wave for classical physics? My impression
was (up to now) that all waves (by definition ?) must satisfy
the familiar second order derivative equation. At least that
is what I teach: wave is a moving disturbance (shape for
transverse waves). Waves of probability do not belong to
classical physics.

It provides a lesson on the unimportance of definitions.
Students often demand a definition of this or that, and they
get really ticked off if they don't get one. But I suspect most
real-world physicists get along just fine without having a
precise definition of "wave". Biologists can't even agree on
the definition of "plant" and "animal".

Can somebody bring an acceptable general definition
of a wave for classical physics?

The usual pat answer is:
A wave is a disturbance that propagates leaving the
medium (if any) behind.

My impression was (up to now) that all waves
(by definition ?) must satisfy the familiar second
order derivative equation.

Nope. There's lots of different waves and wave equations...
-- some have significant nonlinearity, some don't;
-- some have significant dispersion, some don't;
-- some have significant damping, some don't.

Example: The most familiar waves of all, waves on the
surface of a pond, are damped, nonlinear, and dispersive.
You wouldn't want to define them out of existence.

David
****
Can somebody bring an acceptable general definition
of a wave for classical physics?

The usual pat answer is:
A wave is a disturbance that propagates leaving the
medium (if any) behind.

My (maybe idiosyncratic) definition of a wave is an
oscillatory behavior of a *field* such that the
oscillations occur in the value of the field (or its
components) about a mean value in *both* space *and*
time. Thus at each fixed point in space the field oscillates
in time about its mean value, and at each fixed instant of
time the field oscillates in space about its mean value.

Such a definition would include propagating and standing
waves, damped and positive gain waves, and solitons, etc.
as actual waves. But it would consider purely evanescent
"waves" (i.e. oscillations characterized by purely imaginary
wave numbers but real frequencies) not as waves since
they do not oscillate in space (they, rather, only decay/grow
in space) at a given instant of time. In order to count as
a wave in my book the oscillation would need to have a
nonzero real part of the frequency *and* a nonzero real
part of the wave number/vector, or be some sort of
superposition of multiple components made of such things.

If the oscillatory function is only a function of time (such
as the position of the mass center for a harmonic oscillator
or the AC voltage on the terminals of an electrical outlet)
but is not a field which is a function of space as well, then
it is not a wave to my way of thinking. If we don't have a
field defined over (at least one dimension of) space we
don't have a wave--just an AC time dependence of some
function.

This definition can make the status of a propagating shock
front somewhat problematic as such a situation need not
involve oscillations in time or space at all, but rather
might only include a propagating localized change in the
mean value of the field. My definition would, I guess, not
count such a shock as a wave.

OTOH, a propagating localized pulse can be thought of as
a legitimate superposition of spatially oscillatory waves
of a field about a fixed mean value. Such a pulse *does*
oscillate in space and time, albeit with possibly few
total excursions in its value. But in such a case the
field's value has at least one oscillation of both a rise
and a fall in space. So I would count a pulse as a wave.

Sometimes one hears that a wave needs a medium to
propagate in. The way I see it the 'medium' is really the
field itself (defined over space and time), and the
wiggles/oscillations in the value of the field are the
wave. The wave's "wave function" is just the field
considered as a function (or a multiplet set of functions
if the field has multiple components) of space and time.

John
****

My (maybe idiosyncratic) definition of a wave is an
oscillatory behavior of a *field* such that the oscillations
occur in the value of the field (or its components) about a
mean value in *both* space *and* time. Thus at each fixed
point in space the field oscillates in time about its mean
value, and at each fixed instant of time the field oscillates
in space about its mean value.

That definition has many strengths and a few weaknesses.

1) A favorable example is a big block of metal subjected to
alternate heating and cooling on one face. The internal
temperature will be a function of space and time, and it will
oscillate as a function of time, but it will never oscillate
in space. So the proposed definition successfully excludes
this non-wavelike situation.

2a) On the other hand, a legitimate wave need not oscillate.
Consider a long string under tension, with absorbers at each
end. Suppose I pluck it. The result is a little wave packet
with an everywhere-upward displacement that runs along.
Most people would consider it a wave.

2b) You can consider example (2a) as a superposition of
oscillatory waves, but there are solitons (solitary waves)
which *cannot* be considered superpositions. A soliton
in a trough of water is stable if it has a rising edge; the
corresponding falling-edge waveform is unstable.

... it would consider purely evanescent "waves" (i.e.
oscillations characterized by purely imaginary wave
numbers but real frequencies) not as waves since
they do not oscillate in space (they, rather, only
decay/grow in space) at a given instant of time.

They don't "only" decay... A typical evanescent wave (such
as in a waveguide beyond cutoff, or when light is totally
internally reflected from the side of an aquarium) oscillates
in space and time for many cycles while it is decaying.

So the proposed definition _includes_ evanescent waves,
which suits me fine.

In order to count as a wave in my book the oscillation
would need to have a nonzero real part of the frequency
*and* a nonzero real part of the wave number/vector, or
be some sort of superposition of multiple components
made of such things.

Are you assuming linearity? There are lots of nonlinear
waves. Without linearity it's tricky to talk about
superposition.

If the oscillatory function is only a function of time
(such as the position of the mass center for a harmonic
oscillator or the AC voltage on the terminals of an
electrical outlet) but is not a field which is a function
of space as well, then it is not a wave to my way of
thinking.

Agreed.

This definition can make the status of a propagating
shock front somewhat problematic as such a situation
need not involve oscillations in time or space at all,
but rather might only include a propagating localized
change in the mean value of the field. My definition
would, I guess, not count such a shock as a wave.

I'd want to see a stronger argument before declaring
shockwaves to be not waves.

OTOH, a propagating localized pulse can be thought of
as a legitimate superposition of spatially oscillatory
waves of a field about a fixed mean value. Such a pulse
*does* oscillate in space and time, albeit with possibly
few total excursions in its value. But in such a case the
field's value has at least one oscillation of both a rise
and a fall in space. So I would count a pulse as a wave.

Actually it is easy to construct a pulse that has a rise
but no fall.

I think we don't yet have a simple bug-free definition of
"wave" -- but we're getting close. I'm surprised how hard
it is to find one.

Ludwik
******
John wrote:

...It provides a lesson on the unimportance of definitions.
Students often demand a definition of this or that, and
they get really ticked off if theydon't get one. But I
suspect most real-world physicists get along just fine
without having a precise definition of "wave".

If definitions are unimportant how can we use
mathematics in physics? We must define concepts before
analyzing them mathematically.


Biologists can't even agree on the definition of "plant"
and "animal".

Can this be an illustration for saying that biology is not
as scientific as physics? Probably not.

A poet would say "I see waves in a field of wheat". Heads
of plants oscillate is space and time. A physicist would
say that a set of unlinked oscillators is not a wave. I am
assuming plants do not touch each other. Each plant is
driven only by the gusts of wind.

John
****

The condition for the validity of your derivation [for
compression waves] was Hooke's law, the restoring force
must be proportional to displacement. Is this not
true for an ideal transverse wave (small amplitude)?

The derivation depends not only on Hooke's law, but also
on other bits of physics. In the compressional case the
key notion is that a force applied at some point causes
a uniform extension of everything to the left, and a
uniform compression of everything to the right. The
resulting displacement field is graphed in the bottom
curve in

http://www.monmouth.com/~jsd/physics/greenfun.gif

That is, the displacement ramps up linearly and then ramps
down linearly. The displacement for compressional waves
is oriented along the axis of the spring; we graph it in a
perpendicular direction just because that's how graphs
are made.

Note that the same curve is an equally good description
of a string under tension (with zero stiffness). In this
case, the displacement is perpendicular to the string, so
the graph is practically a picture of the physical situation.

If you differentiate this function twice, you get a delta
function at the point where the force is applied. This
delta function represents the force per unit length.
We know the mass per unit length, so in conjunction
with this force per unit length we can immediately
write down the wave equation that describes this case:
(d/dt)^2 y - (d/dx)^2 y = 0
which is the elementary wave equation. It is linear and
nondispersive.

What prevents me from applying the same to a transverse
displacement and from using k in Hooke's law (rather than s)?
I am assuming that there is no tension in a long string of
tiny masses connected by tiny relaxed springs.

The foregoing analysis applies to compression waves and
tension-driven waves on a string, but...

It _cannot_ be reused for flexy waves on stiff springs.
There is a simple reason for this: a force at a point on
a flexy rod does _not_ cause a ramp-like displacement
field. What you actually get is shown in the top curve
in the URL cited above.

Note that the flexy rod is clamped at each end so as to
ensure y=0 and dy/dx=0 at each end. The right end is
free to slide in a tube, so there won't be any tension.

In the upper curve, let's look at the regions where no
force is applied. We see that the first, second, and
third derivatives are nonzero. This is clear from the
graph, because there is a slope (first derivative), a
curvature (second derivative), and the curvature is
changing (third derivative) -- even in places where
no force is applied! You need to differentiate that
function at least _four_ times in order to get
anything related to the local force per unit length.
(In fact the fourth derivative is the right thing, but
you can't tell that just by looking.) So we get a
different wave equation in this case:
(d/dt)^2 y - (d/dx)^4 y = 0
which is highly dispersive. (Note that it's still
linear -- linear in y.)

Here's another dead giveaway that we're dealing
with a fourth-order system: For the flexy waves
we needed to specify _four_ boundary conditions
(y=0 and dy/dx=0 at both ends). Contrast this with
the compressional waves (a second-order system)
for which we needed to specify only _two_ boundary
conditions (y=0 at both ends).