📂Functions

Half-Wave Symmetric Function

Definition

A periodic function $f$ with a period of $2L$ is said to have half-wave symmetry when it satisfies the following equation for all $t$.

$$f(t)=-f(t+L)$$

Description

Expanding on the definition above, it means ‘when a wave is on the $xy$ plane, the pattern of the wave alternates symmetrically around the $y$ axis based on the midpoint of the period.’

Example

As the name implies, it means being symmetrical for half of it, with simple examples being the sine and cosine functions. Consider $y=\sin x$ and $y=\cos x$. The pattern of these progresses symmetrically around the $y$ axis based on the midpoint of the period. Below are examples of graphs with half-wave symmetry.