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What is the speed of a small
longitudinal disturbance along the same spring?
A cable under tension is a spring with very large k.
to behave like a rope or cable.
What is "compliance per unit length [1 / nt]"?
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 for flexion).
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 irrelevant in another. I would expect the longitudinal
wave in a slinky to depend on T rather than on s.
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?
Each of them, however, should behave like
a cable, or a rope, whose T and mu are identical.
The textbooks I know totally ignore contribution
of stiffens to the speed of transverse waves.
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 .....
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.
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.
In particular, it's bad luck to speak of "the" wave
equation. There's lots of wave equations.
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.
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.
... 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.
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.
...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".
Biologists can't even agree on the definition of "plant"
and "animal".
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)?
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.