why is sq rt of -1 needed in wave equations
It's not needed or necessary. However, the introduction of complex numbers into the analysis of wave equations can very substantially reduce the effort required to solve and understand them.
The way I like to think of this is that when you have something oscillating sinusoidally, it's easier to do the math if you think of it as going around in a circle where one of the two dimensions the circle needs is imaginary.
cos(theta) = Re(exp(i*theta)) = Re ( cos(theta) + i*sin(theta) )
The real oscillation is just the projection of the complex oscillation onto the real axis.
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