sinusoid

A sinusoid or sine wave is a periodic wave whose shape is the trigonometric sine function.

Sine wave arise naturally from circular motions:

A sinusoid has the following form:

where

• is the amplitude
• is the independent variable (often representing time in seconds)
• is the frequency, measured in hertz (Hz), representing the number of oscillations per second
• is the angular frequency, in units of radians per second
• is the phase (in radians), specifying the starting point of the oscillation at ¹

Properties

• the sum of same frequency sinusoids is another sinusoid of that frequency

Intuition

Let be the angle on circular motion over time. Then we have

on the circle. There will be some minimal time such that returns to its initial position (the fundamental period). We can translate the period into a frequency , measuring the number of cycles completed per second.

If we assume move at constant speed, then , so that after second,

which is a full cycle. And if we substitute into and , we get

²

Related

• Euler’s formula - Transform between the polar form of a complex number into sinusoids
• Fourier series - Decompose functions into linear combination of sinusoids

Footnotes

1. Sine wave - Wikipedia ↩

2. 1.2. Periodicity and waves — Digital Signals Theory ↩