This has helped but I'm still confused. Even if energy (amplitude) were
constant we would expect energy per area to drop as 1/r^2 because the
sphere surface area is expanding. If the amplitude is also decreasing as
1/r then isn't the energy (amplitude squared) per area decreasing as 1/r^4?
On 06/23/2011 08:46 AM, Kyle Forinash wrote:
> A spherical wave spreads out so obviously the energy per area
(intensity)
> decreases as 1/r2, we expect that just from geometric considerations.
OK.
> But does the amplitude also decrease?
Obviously it must.
> A simple y(r,t)=Asin(kr-wt) with r
> being a polar coordinate says no
That's not a solution of the wave equation.
> (and a plane wave in a perfect medium does not).
Plane waves are different. The spreading argument in the first
sentence above does not apply to plane waves.
> Several wave books (Berkeley physics waves p 372) I have show that the
> Poynting vector of the radiation from an accelerating charge is
> proportional to charge squared, acceleration squared and 1/r^2. But how
> does that relate to amplitude of the wave?
Since the energy of such a wave scales like amplitude squared,
in the far field the amplitude *must* scale like 1/r.
By "far field" I mean r large compared to the wavelength.
In the near field things might be slightly more complicated,
since we need to worry about curvature of the wave fronts.
What was I saying yesterday about scaling arguments being
tremendously powerful?
===================
To anticipate a possible follow-up question:
This gets even more interesting if we consider a spherical wave
in air (not the EM field), and in particular a wave that is not
sinusoidal but rather a wave that starts out as a step-function
traveling wave. Such waves commonly arise from popping a balloon
or setting off an explosion, such that some _extra air_ is released
at the origin and it wants to spread out.
We start by writing down an Ansatz, i.e. a hypothetical wavefunction,
of the form
ψ = (1/r) step(r - vt) [1]
That fails because it doesn't account for the amount of /air/
in the wave. In such a wavefunction, the amplitude would
have to fall off like 1/r^2, in order to conserve air molecules.
But we know it can't fall off like 1/r^2, because that would not
conserve energy.
Therefore the Ansatz [1] is not viable.
In fact what happens is that even though the wave starts out as
a step function, it cannot continue that way. Wiggles develop.
Roughly speaking the envelope of the wiggles falls off like
1/r so as to conserve energy ... and within the envelope there
will be positive _and negative_ values of ψ, such that the
negative regions allow for conservation of air.
This is in fact observed. The timbre of the sound of an
explosion changes as a function of r. Up close the sound is
a sharp SNAP, whereas farther away it turns into a BOOM.
By playing off two scaling arguments against each other,
we know this *must* always happen.
To say the same thing another way: We commonly say that air
is a "non-dispersive" medium for sound waves. Beware that
the dispersion relation for spherical waves *is* dispersive,
even though the medium itself is non-dispersive.
Spherical waves are just different from plane waves.
--
------------------------------------------
"Physics is like sex. Sure, it may give
some practical results, but that's not
why we do it."
R. Feynman (as quoted by Bill Beatty)