Re: Energy Transmission on a string.

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

John,

Thanks for the response,

> IMO Method A is total nonsense.  It gets the correct answer
> primarily because, on dimensional grounds, the correct answer is
> essentially the only answer.  Method B is fine.

While I'm not sure its "total" nonsense, it does involve more hand-waving
than I am comfortable with.  I do like the idea of carrying over results of
SHM, typically covered in the previous chapter.


> With regard to your concern, I think you are simply mistaking the
> mathematical representation of a sine function with the sinusoidal
> shape of the physical waveform.

The derivations are assuming a single harmonic component, where the
wave-form is truly sinsoidal (spelling by Encarta, this time).  Tipler pg
507 or 4th edition Hallicay Resnick & Krane page 427.  They are presenting
what appears to be an attempt at a "rigorous" mathematical derivation, at
least by physics intro standards.

Given the constraint that it is truly sinsoidal, I don't see a way to avoid
the fact that some mass elements along the curve have slopes of 45 degrees,
which IMO would invalidate the use of the small angle approximation (at
least for that region of the string).

At best, IMO, the "standard" deriviation can only apply to instantaneous
regions of the string where the small angle approximation is valid, and says
nothing about the other regions.

However, the average power transmitted is the usual 1/2 mu* v* omega^2 *A^2
form.  And one could argue, I suppose that on average the same average
energy per unit time must make it pass the non-small angle regions at more
or less the same amount of time and conclude the average power transmitted
everywhere is the standard result.

(how does that statement work for others?)

> Often, (usually?) the small angle
> approximation is completely reasonable.  Consider waves on a
> guitar string.

> Long before you would ever see a 45 degree angle,
> you'd see a very broken string (or guitar!)

I interpret what you say above as implying that a real guitar string would
never have a single harmonic sinusoid as its wave-form.  In which case the
derivation B doesn't apply either as they assume such a wave-form.


> John Mallinckrodt      mailto:ajm@csupomona.edu
> Cal Poly Pomona          http://www.csupomona.edu/~ajm