Complex numbers can be used to represent waves and calculate their behavior. The simplest example of complex waves would be a simple mass, m, attached to a spring of stiffness S.
The natural frequency of oscillation for this system is computed as $ \omega_0 ^2 = \frac{S}{m} $.
The behavior of this spring-mass system can then be modeled using complex numbers.
$ Displacement=A_1*e^{j * \omega_0 * t} + A_2*e^{-j * \omega_0 * t} $
$ Velocity=A_1*\omega_0*e^{j * \omega_0 * t} + A_2*\omega_0*e^{-j * \omega_0 * t} $
$ Acceleration=A*\omega_0^2*e^{j * \omega_0 * t} + B*\omega_0^2*e^{-j * \omega_0 * t} $
Where A and B are constants and t is time.
