Why is sq rt of -1 needed in wave equations

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SUMMARY

The square root of -1, represented as the imaginary unit 'i', is not strictly necessary for wave equations but significantly simplifies their analysis. The use of complex numbers allows for easier mathematical manipulation of sinusoidal oscillations by representing them as circular motion in a two-dimensional plane. This approach enables the transformation of wave equations into a more manageable form, where the real oscillation is derived from the projection of the complex oscillation onto the real axis.

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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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