# Sum of Sines

Sum of Sines

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.

f

0

f

0

f

0

aexp(i(ωt+ϕ))

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.

a

1

a

2

ϕ

1

ϕ

2

aexp(i(ωt+ϕ))