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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
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.
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)