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

[curtis osterhoudt <flutzpah@yahoo.com>]

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. 


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________________________________
From: John Denker <jsd@av8n.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Wed, February 24, 2010 12:21:18 PM
Subject: Re: [Phys-l] sound waves and beam flexures

On 02/24/2010 09:31 AM, Moses Fayngold wrote:

> I do not see how and why the geometric shape of the wave-front can
> affect the dispersion.

Eppur si muove.

> Whether a wave (more accvurately, its
> propagation) is dispersive or not is determined by the properties of
> the medium and, if we also take into account the non-linear terms, by
> the wave amplitude, 

OK ....

> but not by geometry. 

Really?  How do you know?  How sure are you of that?

Have you done the calculation?  Have you done the experiment?

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