# Triangle wave

A **triangle wave** is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).

## Harmonics

It is possible to approximate a triangle wave with additive synthesis by summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by π) and multiplying the amplitude of the harmonics by one over the square of their mode number, *n*, (which is equivalent to one over the square of their relative frequency to the fundamental).

The above can be summarised mathematically as follows:

where *N* is the number of harmonics to include in the approximation, *t* is the independent variable (e.g. time for sound waves), is the fundamental frequency, and *i* is the harmonic label which is related to its mode number by .

This infinite Fourier series converges to the triangle wave as *N* tends to infinity, as shown in the animation.

## Definitions

Another definition of the triangle wave, with range from -1 to 1 and period is:

where the symbol is the floor function of *n*.

Also, the triangle wave can be the absolute value of the sawtooth wave:

or, for a range from -1 to +1:

The triangle wave can also be expressed as the integral of the square wave:

Here is a simple equation with a period of 4 and initial value :

As this only uses the modulo operation and absolute value, this can be used to simply implement a triangle wave on hardware electronics with less CPU power. The previous equation can be generalized for a period of amplitude and initial value :

The former function is a specialization of the latter for a=2 and p=4:

An odd version of the first function can be made, just shifting by one the input value, which will change the phase of the original function:

Generalizing this to make the function odd for any period and amplitude gives:

In terms of sine and arcsine with period *p* and amplitude *a*:

## Arc length

The arc length per period "s" for a triangle wave, given the amplitude "a" and period length "p":