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This is a question regarding the standard introductory treatment ofpower
transmitted by a non-dispersive transverse harmonic wave on a string (soas
please keep responses at that level of treatment!)
I see two different "standard" derivations in the textbooks, summarized
follows:SHM, at
METHOD A:
Note that a small mass element of mass dm of the string is executing
any instant of time its energy isyour
1) dE = 1/2 dm* omega^2 *A^2 in the usual notation for a SHM of a mass
element.
2) one notes that dm = mu* dx (mu = mass per unit length)
3) divide by an increment of time dt and write
P = dE/dt = 1/2 mu* dx/dt *omega^2 *A^2 = 1/2 mu *v *omega^2 *A^2
QED,
it seems a little spurious to necessarily equate dx/dt with the wave
velocity, as it implies you chose your element of length dx to be the
distance the wave travels in an element of time dt. I suppose one can
choose it that way, put it is putting a constraint on how you choose
infinitesimal elements.element
but OTOH what else could dx/dt it mean, particularly since it would seem
illogical to assume that power is being transmitted as a speed down the
string different from the wave speed (only one harmonic component here)?
METHOD B:
Note that instantaneous power exerted by an element on the adjacent
(in direction of wave propagation) is F dot vthe
and
F dot v = F_y *v (where y is the transverse coordinate.)
P = v *F *sin (theta )
where theta is the angle between the slope of the sinusuodial curve and
horizontal (wave propagation direction).places
The texts then invoke the small angle approximation by approximating
sin(theta) with tan(theta).
This strikes me as demonstrably incorrect (in general) as there are
on the sinusuodial curve where theta is not a small angle, indeed it isclasses),
sometimes as large as 45 degrees (Pi /4 radians)
I'd appreciate comments or thoughts on these two methods, or if you do a
third method;
Remember this is for introductory (calculus, or even algebra based
so I prefer comments addressed and pertaining to that level of a class.
Joel Rauber