A **triangular wave** or **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).

## Definitions

### Trignometric functions

A triangle wave with period *p* and amplitude *a* can be expressed in terms of sine and arcsine (whose value ranges from -π/2 to π/2):

### 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.

### Floor function

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

where is the floor function.

### Sawtooth wave

Also, the triangle wave is the absolute value of the sawtooth wave:

or, for a range from -1 to 1:

### Square wave

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

### Modulo operation

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:

## Arc length

The arc length per period for a triangle wave, denoted by *s*, is given in terms of the amplitude *a* and period length *p* by

## See also

- List of periodic functions
- Sine wave
- Square wave
- Sawtooth wave
- Pulse wave
- Sound
- Triangle function
- Wave
- Zigzag

## References