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Re: Speed of Mechanical Wave (long)



Regarding David Abineri's question:

I understand that the speed of a sound wave in air is inversely
proportional to the density (possibly the square root of the density?)
of the air, all other factors being the same.

Can anyone supply a metaphor or way of seeing that this is true that
might help high school students to understand this relationship. I
started to explain it and found that my metaphors were showing the
reverse so I clearly have a misunderstanding of the phenomenon.

Any help appreciated. Thanks, David Abineri

I'm not sure of how much it will help, but consider the following
crude hand waving argument.

Consider the behavior of any wavelike disturbance where the wave-like
oscillations are energy-conserving non-dissipative phenomena.
Typically such behavior can be considered as a result of the
application of Newton's 2nd law (in an appropriately generalized
form) to the response of some medium to a disturbance. The medium
typically has 2 most important characteristics--elastic stiffness
and inertia.

First, when it is not disturbed it is in a state of some sort of
stable equilibrium. This means that to disturb the medium in any way
away from that equilibrium it requires work to be done on the medium
such that for sufficiently small disturbances the medium responds to
the disturbance by producing some sort of generalized restoring force
that tends to drive the medium back toward the undisturbed state, and
this restoring 'force' is proportional to the amount of the
disturbance. It is a generalized condition of elasticity or a
generalized Hooke's Law if you will. For the wave-like phenomena
considered this restoring 'force' is assumed to be the dominant
response of the medium which dominates over other possible responses
such as dissipative behavior, nonlinear response, instabilities, etc.
This elasticity is quantified in terms of some sort of tension,
stiffness or bulk(, shear, or Young's,, etc.) modulus of rigidity
against deformations.

Now usually the amount of the restoring 'force' depends on not only
on the overall magnitude of the disturbance but on its spatial
concentration--the more locally concentrated a disturbance is in
space the stronger the restoring force. We typically find that the
potential energy associated with a local region of distortion is
(to leading order) proportional to the square of the gradient of the
medium's distortion. There are a couple of reasons for this. To
understand them let's imagine expanding the potential energy density
as a generic Taylor expansion in distortion and its spatial
derivatives. Since it doesn't do any work to uniformly translate the
whole medium by any arbitrary amount we see that the potential energy
density can't depend explicitly on the bare magnitude of the
displacement itself of the medium from its equilibrium value (the
medium typically is subject to a translational invariance condition
for the equilibrium value). This means that the potential energy
density must depend only on the spatial derivatives of the distortion
field rather than on the distortion field itself. If the medium is
spatially uniform there is no potential energy regardless of the
background value of the field itself.

For instance, if we are dealing with the motion of a planar membrane
(i.e. a drum head) where we take the distortion field to be the local
transverse (i.e. vertical) displacement of the height of the membrane
we note that (absent any relevant gravitational field) the potential
energy of the membrane is independent of the overall height of that
membrane. The elastic membrane only cares about how the relative
height of the membrane *changes* from place to place along the
membrane itself. Thus the transverse elastic potential energy of the
membrane depends locally on only the spatial derivatives of the
transverse displacement field. Consider its dependence on the lowest
order spatial derivative, i.e. the gradient of that field. We think
of this expansion as a power series in powers of the gradient of the
distortion field. Now the zero order term in the expansion can,
without loss of generality, be taken to vanish because we can choose
the zero level of the potential energy (density) to be any value we
wish. Changing the zero value of a potential has no effect on the
forces involved or the equation of motion. The first order term in
the gradient of the distortion field also vanishes because the medium
is taken to be spatially isotropic and their is no background vector
field or direction that could be used to contract (i.e. take the dot
product with) with the gradient vector of the distortion field to
make the scalar linear term in the potential energy.

This means that the first nontrivial term in the expansion of the
potential (free) energy density is the 2nd order term in the gradient
(first derivative) of the distortion field. We can make a scalar
potential energy density out of the dot product of the gradient of
the distortion field with itself. Since we assume that the medium--
when it is distorted--is *nearly* in equilibrium we can neglect
the effects of any and all the higher order terms relative to the
leading 2nd order term as long as we keep to a gentle gradual
distortion limit approximation. Physically, this means that we
only consider those spatial distortions whose wavelength is huge
compared to any internal microscopic material length (such as the
interatomic lattice spacing in a uniform solid, the mean free path in
a gas, or the spring length for a very long chain of masses connected
by springs). So for all practical purposes we can assume the medium
is has a potential energy that is proportional to the square of the
gradient of the disturbance amplitude.

We also ignore any terms in the expansion that are in powers of
spatial derivatives higher than gradients. The reason for this is
that any higher derivative term to first order would require some
background tensor field of higher rank to contract with those
higher spatial derivatives of the distortion, and the assumption
of the isotropy of the medium forbids the existence of such a
tensor background. This means that the higher derivative terms
would themselves have to be contracted among themselves to make a
scalar contribution to the potential energy density. The lowest
such terms would have to be at least second order in these
higher derivatives, and as long as we are looking at the behavior
of the medium in the gradual, relatively long wavelength limit
we can safely ignore any and all such terms.

The upshot of this initial discussion is that the potential energy
density can be considered as being proportional to the dot product of
the gradient of the distortion field with itself. The appropriate
stiffness modulus is read off from the expansion coefficient for this
squared gradient term (half the appropriate elastic modulus *is* the
expansion coefficient) This procedure is entirely analogous to
writing the potential energy of a spring as (1/2)*k*x^2 where half
the stiffness k is the coefficient of the leading Hookeian term in
the generic expansion of spring's potential (free) energy. The
potential energy density can thus be reasonably be modeled as:

u = (1/2)*(stiffness modulus)*|grad(distortion field)|^2

Besides the elastic response the second important behavioral factor
that the medium has is some sort of inertia associated with changes
in its state of motion. The medium locally has some sort of mass,
mass density, inductance, etc. that measures the tendency of the
medium to resist changes (i.e. accelerations) in the state of motion
of the medium. Such inertia is typically mathematically represented
by some version of a generalized Newton's 2nd law. Here the mass-like
inertia factor times the rate of change of the medium's motion
(acceleration) is locally equal to the total generalized force acting
on the medium. Since the elastic response of the medium dominates we
can understand and model that response entirely in terms of the linear
elastic response coming from the derivative of the potential
energy we previously constructed. The variational derivative of the
potential energy w.r.t. the distortion field yields a generalized
restoring force that is proportional to the Laplacian of the distortion
field. Putting this force term together with the inertial term in the
equation of motion we thus have an equation of motion for the local
behavior of the medium that relates the 2nd temporal derivative of
the distortion field to its Laplacian effectively saying:

(inertia coefficient)*d^2(distortion field)/dt^2 =

= (stiffness modulus)*div(grad(distortion field))


This is a 2nd order partial differential equation (dispersion-free
hyperbolic wave equation) whose solutions have the property of being
wave-like disturbances in the medium. To see how the waves behave
we can look at spatially harmonic modes of fixed wavelength (whose
wave number is k). (This is equivalent to taking a spatial Fourier
transform of the equation of motion.) For each such mode the
equation of motion boils down to:

(inertia coefficient)*d^2(distortion mode)/dt^2 =

= - k*k*(stiffness modulus)*(distortion mode)

We see that the second derivative of the disturbance of each mode is
proportional to the negative of the amount of the disturbance itself.
Such an equation has a solution given in terms of harmonic oscillations
(because sine waves have the property that their second derivative is
proportional to their own negation). If we consider a solution to the
above generalized equation for each Fourier mode involving a harmonic
response in time as some sort of sine wave or complex exponential
with angular frequency [omega], the equation of motion for that mode
is solved as long as the condition:

(inertia coefficient)*[omega]^2 = (k^2)*(stiffness modulus)

holds. The (phase) speed c of the waves is given by c = |[omega]/k|.
(This is equivalent to saying the wave speed is the wavelength
[lambda] times the frequency f because f = [omega]/(2*[pi]) and
[lambda] = 2*[pi]/k.) Solving the above condition on the elementary
harmonic waves gives:

c = |[omega]/k| = sqrt((stiffness modulus)/(inertia coefficient)).

We thus arrive at the result that the wave speed is typically
given as the square root of the quotient of the medium's
generalized stiffness (appropriate elastic modulus) divided by its
generalized local inertia (appropriate mass density).

In the special case of the transverse vibrations of the 2-d
membrane we have c = sqrt((surface tension)/(areal mass density)).
In the case of 1-d transverse wave motions on a string we have
c = sqrt((tension)/(mass per unit length)). In the case of
longitudinal vibrations along an elastic rod the wave speed is
c = sqrt((Young's modulus)/(mass density)). In the case of ordinary
sound the wave speed is

c_sound = sqrt((bulk modulus)/(mass density)).

It needs to be kept in mind that the appropriate bulk modulus for
acoustic vibrations is the *adiabatic* bulk modulus since such
fluctuations happen so fast in the medium that heat doesn't have
enough time to flow between the compressed regions and the
adjacent rarified ones before they switch places. The wavelike
response is a quite adiabatic process. In the special case of
sound through air the gas can be well approximated by the
ideal gas approximation and thus the adiabatic bulk modulus is
[gamma]*P where P is the pressure and [gamma] = 1.4 is the
specific heat ratio for a diatomic gas. If we recall that for
an ideal gas the mass density = (average particle mass)*P/(k*T)
(where k is Boltzmann's constant and T is the absolute temperature).
We see that substitution causes the pressure factors to cancel,
and we are left with only a temperature dependence, i.e.

c_sound = sqrt(1.4*k*T/(avg. particle mass)) .

I hope this explanation helps show why in general typical
wave motions tend to have a wave speed c that is essentially

c = sqrt((generalized stiffness coef.)/(generalized inertia coef.))

Even for electromagnetic waves we can think of 1/[epsilon] (inverse
permitivity) as the electromagnetic stiffness against dielectric
polarization of the medium, and we can think of [mu] as the
generalized inertia of the medium coming from its magnetic
inductance. The wave speed is thus c = sqrt(1/[epsilon]*[mu]).

David Bowman
David_Bowman@georgetowncollege.edu