Question

what is the physical significance of imaginary term in simple harmonic motion / wave equation ? why at all we require imaginary term such as exp i(kx-wt) ?

Answer 1

The major advantage of a complex representation, over the more straightforward real representation, is that the former enables us to combine the amplitude, [$ A$] , and the phase angle, [$ \phi$] , of the wavefunction into a single complex amplitudeThe solution of the wave equation is a simple sinusoidal wave. Which means that the displacement will be following a sin or cos function. We use the imaginary part so that we can use the Euler theorem to write the exponential series as a sinusoidal series.

Complex numbers are often used to represent waves and wavefunctions. All such representations ultimately depend on a fundamental mathematical identity, known as Euler's theorem