---
title: "Phasor Addition and the Mathematics of Interference"
description: "Representing each contributing wave as a rotating arrow turns interference into vector addition, so intensity comes from the length of the resultant. Two-slit, multi-slit and single-slit patterns all "
canonical: https://lightmysky.com/learn/science/phasor-addition-and-the-mathematics-of-interference-mt_dxh7HHbIUw
source: https://lightmysky.com/learn/science/phasor-addition-and-the-mathematics-of-interference-mt_dxh7HHbIUw.md
retrieved: 2026-09-12
---

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# Phasor Addition and the Mathematics of Interference

Representing each contributing wave as a rotating arrow turns interference into vector addition, so intensity comes from the length of the resultant. Two-slit, multi-slit and single-slit patterns all follow from the same picture.

Subject: Science · Area: Waves, Light & Sound · Ages 19 to 21
Page: https://lightmysky.com/learn/science/phasor-addition-and-the-mathematics-of-interference-mt_dxh7HHbIUw

## Ready when they can

- Adds two or more waves of equal frequency with a phasor diagram and reads off the resultant amplitude
- Derives the intensity pattern of a two-slit interference figure
- Explains how the pattern sharpens as the number of slits increases

## Lesson: Adding waves with arrows

Give each wave an arrow called a phasor. Its length is the wave amplitude and its angle is the wave phase. To add waves of equal frequency, lay the arrows head to tail: the arrow from the first tail to the last head is the resultant, and its length is the total amplitude.

Detectors feel energy, which grows as amplitude squared, so brightness goes as the squared length of the resultant. For two slits spaced d apart, the paths to a screen point differ by d times sin theta. Bright fringes sit where that difference is a whole number m of wavelengths, and the fringe spacing on a far screen is wavelength times screen distance over slit spacing.

**Example.** Try three equal waves whose phases step by 120 degrees. Their three arrows close into a triangle, so the resultant is zero and the screen goes dark there. Put the same three waves in step instead, and the amplitude triples while the brightness grows ninefold.

**Tip.** With many slits the arrows line up fully only at the main maxima, and elsewhere they curl toward zero, so the bright fringes get sharper and narrower. No energy is lost in the dark zones: it is all piled into the bright ones.

**Recap.** Draw each wave as an arrow, add the arrows, and square the resultant length to get the brightness.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Combining a Sine and a Cosine into a Single Wave](https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq)
- [Coherence and Young's Double-Slit Fringes](https://lightmysky.com/learn/science/coherence-and-youngs-double-slit-fringes-mt_mONvXhPmwq)

## Opens up

- [Thin-Film Interference and Anti-Reflection Coatings](https://lightmysky.com/learn/science/thin-film-interference-and-anti-reflection-coatings-mt_3ugW-ZijzH)
- [Intensity in Single-Slit and Double-Slit Diffraction](https://lightmysky.com/learn/science/intensity-in-single-slit-and-double-slit-diffraction-mt_HURhzZdIkM)