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 Problem Statement

A string of tension T and linear density ##\rho## is driven at ##x=0## and two travelling wave in positive and negative xdirection had been created. Calculate the force and power applied to create the travelling wave.
I am studying chapter 8 of the book the physics of waves by Howard Georgi and I am quite confused. I read the entire chapter but I still don't quite get how to acquire the travelling wave mode of a system. Can someone check my solution to this problem but also explain how to systematically work out the "travelling wave mode" from conditions provided like this question?
 Relevant Equations

##\frac {\partial^2 \psi} {\partial t^2} = \frac T \rho \frac {\partial^2 \psi} {\partial x^2} ##
##F =  T \frac {\partial \psi} {\partial x}##
Again I am really confused, but I just put the travelling wave as:
##\psi(x,t) = Dcos(kx \omega t)## for positive x
##\psi(x,t) = Dcos(kx+ \omega t)## for positive x
Then I simply differentiated and plugged in ##x=0##
##F(t) =  T D k sin(\omega t)##
and from this
## \langle P \rangle = T D^2 k \omega sin^2(\omega t)##
##\psi(x,t) = Dcos(kx \omega t)## for positive x
##\psi(x,t) = Dcos(kx+ \omega t)## for positive x
Then I simply differentiated and plugged in ##x=0##
##F(t) =  T D k sin(\omega t)##
and from this
## \langle P \rangle = T D^2 k \omega sin^2(\omega t)##
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