--- title: "Phasor Addition and the Mathematics of Interference" description: "Phasors represent coherent waves as arrows. Their vector sum gives the resultant amplitude, and intensity is proportional to its square. This explains the two-slit intensity profile and the narrowing " 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Phasor Addition and the Mathematics of Interference Phasors represent coherent waves as arrows. Their vector sum gives the resultant amplitude, and intensity is proportional to its square. This explains the two-slit intensity profile and the narrowing principal maxima of equally spaced multiple slits. 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 Recall two slits: whole-wavelength path differences preserve an in-phase start. For coherent waves of equal frequency and matching polarisation, a phasor is an arrow of length A (amplitude) and angle phi (phase). Its components are A cos(phi) and A sin(phi). Sum horizontal components to X and vertical ones to Y; the resultant amplitude is R = √(X² + Y²). Head to tail, the resultant joins the first tail to the last head. Use one angle unit throughout. For two equal phasors separated by phase phi, choose an axis halfway between them. Components perpendicular to that axis cancel. Components along it add to 2A cos(phi/2), giving resultant length |2A cos(phi/2)|. In the same medium, light intensity is proportional to amplitude squared. If either wave alone gives intensity I1, their sum gives I = 4I1 cos²(phi/2). Doubling amplitude multiplies intensity by four. *(drawing: These are three samples of I/I1, not a continuous intensity curve or a phasor diagram.)* For initially in-phase slits separated by d, the far-screen path difference is approximately d sin(theta). Theta is measured from the central direction; lambda is wavelength in the medium. The phase gap is phi = 2pi d sin(theta)/lambda radians. For equal contributions, each giving intensity I1 alone, I(theta) = 4I1 cos²(pi d sin(theta)/lambda). Maxima have d sin(theta) = m lambda for integer m. Equal amplitudes cancel at odd half-wavelength path differences. To find intensity from a path difference delta, first convert distance to phase: phi = 2pi delta/lambda radians. For initially in-phase waves, this is their arriving phase gap. With equal contributions, each giving intensity I1 alone, halve phi and use I = 4I1 cos²(phi/2). Substitute I1 and the cosine-squared value, keeping the intensity units. Adding the two separate intensities alone misses the phase-dependent interference term. **Example.** N equal phasors with successive phase steps of 360/N degrees close a regular polygon and sum to zero. Four phasors at 0°, 90°, 180° and 270° cancel in opposite pairs. If those four waves instead arrive in phase, their resultant amplitude is 4A and intensity is 16I1. More generally, N equal in-phase waves give amplitude NA and intensity N²I1. For N equally spaced slits with equal coherent contributions, all phasors align at principal maxima. The nearest minima occur at adjacent phase gaps ±360/N degrees from alignment. Increasing N reduces these gaps. At fixed wavelength and slit spacing, those minima move closer to each principal maximum, narrowing it. Secondary maxima can remain between principal maxima; the intensity is not zero everywhere between them. Interference changes where energy arrives. In a lossless arrangement, a dark fringe does not destroy energy. Reduced intensity in some regions is balanced by enhanced intensity elsewhere when the full energy flow is counted. A finite screen need not collect that whole flow. Opening more slits can also increase transmitted power, so comparisons must state what illumination is held fixed. **Recap.** Add coherent phasors to find amplitude, then square it for intensity. Two equal waves give I = 4I1 cos²(phi/2). More equally spaced slits narrow principal maxima. *The lesson, practice questions and flashcards were written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Coherence and Young's Double-Slit Fringes](https://lightmysky.com/learn/science/coherence-and-youngs-double-slit-fringes-mt_mONvXhPmwq) ## Optional connections - [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) ## 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)