Re: [Phys-l] sound waves and beam flexures

[John Denker <jsd@av8n.com>]

On 02/25/2010 02:35 PM, curtis osterhoudt wrote:
> Denker is correct. Waves act fundamentally differently in
> odd-dimensional geometries than they do in even-dimensional
> geometries. For example, Huygens' principle can be used with the same
> general physical conclusions in dimensions 1, 3, 5, ... (though for
> technical reasons one usually says it's not used in dimension 1 --
> see Farlow's "Partial Differential Equations for Scientists and
> Engineers, Lesson 24), but it leads to fundamentally different
> physical behavior in odd-dimensional geometries.
>
> The technical term is "wake formation". Morse and Feshbach's Section
> 7.3 gives all the math (in terms of Green's functions) one would ever
> want.
>
> The *physics* results are that in 1D systems, and 3D systems, there
> is no wake formed (plucked strings return to their pre-plucked
> positions; sound waves pass us by and we don't continue to hear their
> ringing). The shape of the wave _may_ be unchanged (barring
> dispersion); in 1D systems a given displacement (with zero initial
> velocity) will be unchanged in shape. In 3D systems, a given pulse
> shape is unchanged if the initial *displacement* is zero and the
> initial *velocity* is imposed on the system. No wakes are formed. In
> 2D systems (membranes), there is a wake formed, no matter if the
> medium is dispersive or dissipative or not. If you were to "hear" a
> cylindrical pulse on a large drumhead, an initial impact would be
> followed by distorted copies for a long time, until dissipation
> killed them.

That's all true.

Going farther down that road, it is easy to show that for
a radially-symmetric wave in 3D, the shape of the waveform 
_must_ change as it goes along.  If φ is the wavefunction,
then (rφ) obeys a simple wave equation, just like the one-
dimensional equation for φ ... but (rφ) is not φ, and as r
changes φ must change.

That's the math.  There is a nifty physics argument that 
leads to the same qualitative result.  Consider a pulse that
is everywhere positive.  For sound, that would mean everywhere
compressive, such as you might get from popping a balloon or
setting off a firecracker.  The amount (mass) of air in the
pulse scales like r^(D-1) φ so to conserve mass, φ must go like 
1/r^(D-1).  Meanwhile, the energy scales like r^(D-1) φ^2, 
so to conserve mass φ must scale like sqrt(1/r^(D-1)).  That 
is to say, as the pulse spreads out, you cannot conserve both
mass and energy using the same shape of pulse ... not for any 
dimensionality D greater than 1.

=================

Also BTW, the earlier bold assertion from M.F. that 

> > For each
> > frequency, a spherical wave can be represented as a linear
> > superposition (3-D Fourier transform) of plane waves with the same
> > frequency. 

is completely untrue.  It is untrue in even in D=1 ... and 
equally untrue in all higher dimensions.

If this is not obvious, please take a look at the following
snapshot of a radially-symmetric outbound traveling wave in
one dimension.  
  http://www.av8n.com/physics/img48/radially-symmetric-wave-1d.png
The wave is emitted by the point source at the origin.  The
wavefunction is 
   φ = sin(k|r| - ωt)

In terms of length, "the" wavelength is one unit, but in
terms of the wavevector, 
  -- k = +2π on one side of the origin, while
  -- k = -2π on the other side of the origin.

If you take the Fourier transform of the whole thing, you
get contributions from infinitely many different frequencies.