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The wavefunction I'm considering is $$\xi(x,t)=A sin (k x \pm \omega t +\psi)$$

The minus or plus are for progressive or regressive waves. The indipendent parameters are 4: ##A##, ##k##, ##\omega##, ##\psi##.

The first doubt I have is: can ##A## be negative or is it defined as positive in any case? (In the following examples I willl consider a progressive wave, so the wavefunction is ##\xi(x,t)=A sin (xk - \omega t +\psi)## ).

- Consider a oscillating rope: all I know is that one end of the rope moves according to the function ##f(t)=B sin(\gamma t)##, with ##B>0## and ##\gamma>0##. What I would do here is impose that ##\xi(0,t)=f(t)##. Since there is a minus in front of ##\omega##, before imposing ##\xi(0,t)=f(t)## I think I must rewrite ##\xi(0,t)=A sin (-\omega t+\psi)=-A sin(\omega t-\psi)=f(t)##, from which I can conclude ##A=-B## and ##\gamma=\omega## (and then also ##\psi=0##). So ##A## would be negative.
- Another example of ambiguity on the sign of ##A## is when I'm given ##k## and ##\omega## and the two conditions ##\xi(\bar{x},\bar{t})=s_0## and ##\frac{\partial \xi}{\partial t} |_{(\bar{x},\bar{t})}=v_0## . That means ##A sin (k \bar{x} -\omega \bar{t} +\psi)=s_0## and ##A \omega cos(k \bar{x} -\omega \bar{t}+\psi)=v_0## ##\implies## ##A^2=s_0^2+(\frac{v_0}{\omega})^2##. So I get ##A^2## but how to choose the sign of ##A## in this case?

I really appreciate any suggestion regarding these two doubts!