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.

External Links

Complex Addition of Harmonic Motions and the Phenomenon of Beats

Derivatives of the Sum of Two Sines

Sums of Sine Waves with Several Step Sizes

Permanent Citation

Carsten Roppel

"Sum of Sines"
http://demonstrations.wolfram.com/SumOfSines/
Wolfram Demonstrations Project
Published: December 1, 2011