The sum of two sine waves with the same frequency is again a sine wave with frequency . This is used for the analysis of linear electrical networks excited by sinusoidal sources with the frequency . In such a network all voltages and currents are sinusoidal. The addition of sine waves is very simple if their complex representation is used.
This Demonstration shows the sum of two sine waves with amplitudes , and phases , , respectively. The complex representation , as vectors in the complex plane, is shown in the left graphics column. The sine function is the imaginary part of the vector, indicated by the projection onto the imaginary axis.