Sum of Sines

red wave

amplitude

a

1

1

phase

ϕ

1

0

blue wave

amplitude

a

2

0.5

phase

ϕ

2

0

play

The sum of two sine waves with the same frequency

f

0

is again a sine wave with frequency

f

0

. This is used for the analysis of linear electrical networks excited by sinusoidal sources with the frequency

f

0

. In such a network all voltages and currents are sinusoidal. The addition of sine waves is very simple if their complex representation

aexp(i(ωt+ϕ))

is used.

This Demonstration shows the sum of two sine waves with amplitudes

a

1

,

a

2

and phases

ϕ

1

,

ϕ

2

, respectively. The complex representation

aexp(i(ωt+ϕ))

, 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.