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

[Moses Fayngold <moshfarlan@yahoo.com>]

--- On Thu, 2/25/10, John Denker <jsd@av8n.com> wrote:


> 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 was not my bold assertion. It is a TRUE statement, see any sourse on Math. Phys. Here is one quatation, to save your time: "All these waves (cylindrical, spherical etc. - MF) can be constructed as a superposition of plane waves... ...GENERAL expression for a wave in 3-D space (say, a single pulse - MF)... can be represented as a 3-D Fourier integral... For SIMPLE HARMONIC waves Psy =  G*exp(-i*omega*t) where G is a solution of a 3-D Helmholtz equation, the integration must be carried out over the spherical surface in k-vector space, since the length k(=omega/u) is fixed..."
 (M. Morse, H, Feshbach, Methods of Theoretical Physics, Part. II, McGraw Hill, 1953, Sec. 11.3)
   Integration over the spherical surface DOES NOT INCLUDE integration over omega = k*u, where u(omega) is the phase velocity of the wave for given frequency. There cannot be any integration over omega precisely because the frequency is fixed in the Helmholtz equation, and my question was explicitly about the case with a fixed frequency.  
  I think it is now appropriate to reformulate this question in the form of a thought experment (which is, in principle, executable). Consider two trials with acoustic L-wave generation in a homogeneous isotropic medium: first, a uniform vibration of a large flat membrane along the line perpendicular to its plane, say, with frequency 420 Hz. This will generate a monochromatic L-wave which is plane at distances small with respect to the radius of membrane or with respect to its side if it is quadaratic. In the second trial, use a uniform radial vibration of a thin elastic spherical shell with the same frequency 420 Hz and in the same medium. In this case we will generate a spherical wave of the same frequency. Outside the source, both waves are described by the same Helmholtz equation with the same frequency, with the distinction that in the first case we would rather look for its solution in Cartesian coordinates, whereas in the second case - in
 spherical coordinates. As stated above, a monochromatic spherical wave can be represented as a superposition of the plane waves (propageting in different directions, but all with the same frequency!). But J.D. claims that this expansion will contain a range of frequencies. This is a really bold assertion, and it can be tested by our thought experiment. My prediction is that, even regardless of whether the medium is dispersive or not, in the described case, out of set of varioius tuning forks only the one with frequency 420 Hz will respond to both - the plane acoustic wave, AND to the spherical wave, while all the others will remain dormant. Do you disagree with this? Will you claim that in the wave from the spherical source a set of tuning forks with different frequencies will also resonate, apart from the one with 420 Hz?  
  
 
> It is untrue in even in D=1 ... 
 
  Thius statement is totally irrelevant to our case since there cannot be any spherical waves in D=1.

> 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)

Again, this is a false analogy - it attempts to argue for 3-D case in terms of the 1-D case. These cases are totally different. Particularly, in 1-D case, a wave non-monochromatic with respect to k automatically means non-monochromatic with respect to omega, since now there is no flexibility in the relation k = u*omega. In your example, as you yourself emphasize, 
"-- k = +2π on one side of the origin, while
  -- k = -2π on the other side of the origin."
 
This is sufficient to require all range of frequencies.

> If you take the Fourier transform of the whole thing, you
> get contributions from infinitely many different frequencies.
 
In this case - yes, but, again, this case is totally irrelevant to my original question.
 
 
MF,
NJIT  
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