Re: [Phys-l] spherical waves

[John Denker <jsd@av8n.com>]

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.