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

Animation of the additive synthesis of a triangle wave with an increasing number of harmonics. See

Fourier Analysis for a mathematical description.

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:

- ${\begin{aligned}x_{\mathrm {triangle} }(t)&{}=\sum _{i=0}^{N}(-1)^{i}n^{-2}\left(\sin[nt]\right)\end{aligned}}$

where *N* is the number of harmonics to include in the approximation, *t* is the independent variable (e.g. time for sound waves), and *i* is the harmonic label which is related to its mode number by $n=2i+1$.

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 *a* is:

- $x(t)={\frac {2}{a}}\left(t-a\left\lfloor {\frac {t}{a}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {t}{a}}+{\frac {1}{2}}\right\rfloor }$
- where the symbol $\scriptstyle \lfloor n\rfloor$ represent the floor function of
*n*.

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

- $x(t)=\left2\left({t \over a}-\left\lfloor {t \over a}+{1 \over 2}\right\rfloor \right)\right$

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

- $x(t)=2\left2\left({t \over a}-\left\lfloor {t \over a}+{1 \over 2}\right\rfloor \right)-1\right-1$

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

- $\int \operatorname {sgn}(\sin(x))\,dx\,$

A simple equation with a period of 4, with $y(0)=1$. 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:

- $y(x)=x\,{\bmod {\,}}4-2-1$

or, a more complex and complete version of the above equation with a period of $p$, amplitude $a$, and starting with $y(0)=a/2$:

- $y(x)={\frac {2a}{p}}{\Biggl (}{\biggl }\left(x{\bmod {p}}\right)-{\frac {p}{2}}{\biggr }-{\frac {p}{4}}{\Biggr )}$

The function (1) is a specialization of (2), with a=2 and p=4:

- $y(x)={\frac {2\times 2}{4}}{\Biggl (}{\biggl }\left(x{\bmod {4}}\right)-{\frac {4}{2}}{\biggr }-{\frac {4}{4}}{\Biggr )}\Leftrightarrow$

- $y(x)={\Biggl (}{\biggl }\left(x{\bmod {4}}\right)-2{\biggr }-1{\Biggr )}$

An odd version of the function (1) can be made, just shifting by one the input value, which will change the phase of the original function:

- $y(x)=(x-1)\,{\bmod {\,}}4-2-1$

Generalizing the formula (3) to make the function odd for any period and amplitude gives:

- $y(x)={\frac {4a}{p}}{\Biggl (}{\biggl }\left((x-{\frac {p}{4}}){\bmod {p}}\right)-{\frac {p}{2}}{\biggr }-{\frac {p}{4}}{\Biggr )}$

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

- $y(x)={\frac {2a}{\pi }}\arcsin \left(\sin \left({\frac {2\pi }{p}}x\right)\right)$

## Arc Length

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

- $s={\sqrt {(4a)^{2}+p^{2}}}$

## See also

## References