Why is sq rt of -1 needed in wave equations

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The square root of -1, represented as an imaginary number, is not strictly necessary for wave equations but simplifies their analysis significantly. Using complex numbers allows for easier mathematical manipulation, particularly in representing sinusoidal oscillations as circular motion in the complex plane. The real oscillation can be viewed as the projection of this complex representation onto the real axis. This approach streamlines calculations and enhances understanding of wave behavior. Overall, the use of complex numbers is a valuable tool in the study of wave equations.
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why is sq rt of -1 needed in wave equations
 
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gaminin gunasekera said:
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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