This is a question regarding the standard introductory treatment of power
transmitted by a non-dispersive transverse harmonic wave on a string (so
please keep responses at that level of treatment!)
I see two different "standard" derivations in the textbooks, summarized as
follows:
METHOD A:
Note that a small mass element of mass dm of the string is executing SHM, at
any instant of time its energy is
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 your
infinitesimal elements.
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 element
(in direction of wave propagation) is F dot v
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 the
horizontal (wave propagation direction).
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 places
on the sinusuodial curve where theta is not a small angle, indeed it is
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 classes),
so I prefer comments addressed and pertaining to that level of a class.