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Superposition overview

Topic 8 of 9

Diffraction and a single slit

Waves spread through openings and around edges. For a single slit, the first dark minima bound a broad central intensity maximum.

Compare wavelength with the opening

Diffraction is spreading through an aperture or around an edge. Its angular spread depends on wavelength relative to the opening or obstacle dimensions.

In a ripple tank, keep water depth and source frequency fixed so the wavelength stays the same. Compare a wide gap with one whose width is comparable to that wavelength. The narrower gap gives greater angular spread. The wide gap still has spreading at its edges; diffraction has not disappeared.

Compare spreading at the same wavelength

The ripple-tank views share one spatial scale and the same wavelength. Each blue line represents a wavefront of the same phase. Line thickness does not represent amplitude or transmitted power.

Wide gap: width = 5 wavelengths

A wide gap retains a broad central front with spreading at its edgesIncident plane wavefronts travel right and are separated by twenty drawing units, one wavelength. The opening is one hundred drawing units, five wavelengths wide. Outgoing fronts have a straight central portion and curved edges, retaining diffraction rather than pretending the wide gap eliminates spreading. All outgoing fronts remain one wavelength apart in their local normal direction. These are schematic phase fronts, not rays, particle tracks or a calibrated intensity map. The two gap drawings have the same scale and wavelength.IncidentλBarrier and opening

Narrow gap: width = 1 wavelength

A wavelength-scale gap produces strongly curved outgoing frontsIncident plane wavefronts travel right and are separated by twenty drawing units, one wavelength. The opening is twenty drawing units, one wavelength wide. Approximately semicircular outgoing fronts illustrate the large angular spread of a wavelength-scale opening. All outgoing fronts remain one wavelength apart in their local normal direction. These are schematic phase fronts, not rays, particle tracks or a calibrated intensity map. The two gap drawings have the same scale and wavelength.IncidentλBarrier and opening

Sound can spread around an edge

Curved sound wavefronts enter the geometrical shadow behind an edgeAn incident sound wave travels right towards the top edge of an obstacle. Above the edge, the fronts continue approximately straight. Below the edge, successive quarter-circle fronts illustrate spreading into the shaded geometrical shadow. A receiver lies in that shadow. The common wavelength spacing is retained; the curves are phase lines, not paths of individual air particles. This schematic illustrates diffraction, without claiming that every sound heard behind a real obstacle is due only to diffraction rather than reflections or other paths.ReceiverGeometricalshadowSoundObstacleCurves show phase, not particle travel.

A narrow opening gives more angular spreading, not automatically greater transmitted power. A wide opening still shows diffraction at its edges.

The two ripple-gap views share the same wavelength and spatial scale. The wide gap retains edge spreading. The sound view identifies an edge and its geometrical shadow. Wavefronts mark constant phase; they are not particle tracks.

A narrow opening does not automatically transmit more power. Spreading describes the directions occupied by the transmitted wave, not how much energy passes through.

Sound can spread around a corner into a region outside the direct geometrical path. Its wavelength compared with the obstacle size helps determine the spreading. Sound from a loudspeaker also diffracts at its emitting aperture. In a real room, reflections may contribute as well, so sound heard around an obstacle need not have only one cause.

Locate first minima for a uniformly illuminated slit

For a single slit of width b in the far-field model, the first minima satisfy:

b sin θ = λ

There is one first minimum on each side of the central direction. The equation locates minima, unlike the grating equation a sin θ = nλ for principal maxima. The first minima enclose the central maximum; weaker side maxima lie beyond them.

Worked central width

Use slit width b = 0.15 mm, wavelength λ = 600 nm and screen distance D = 2.0 m.

sin θ = λ/b
= (600 × 10-9)/(0.15 × 10-3)
= 0.0040

This is a small angle, so θ ≈ 0.0040 rad and tan θ ≈ θ. The distance from the centre to one first minimum is:

y1 = D tan θ ≈ 2.0 × 0.0040
= 0.0080 m = 8.0 mm
Full central width = 2y1 ≈ 16 mm

The first minima bound the broad central maximum

For b = 0.15 mm, λ = 600 nm and D = 2.0 m, the first minima are approximately 8.0 mm from the centre. This graph is intensity divided by the central intensity, not displacement.

First minima at minus and plus eight millimetres give a sixteen-millimetre central widthA normalised nonnegative intensity profile is plotted against screen coordinate from negative 24 to positive 24 millimetres. Its central peak has value one at zero. First minima at negative eight and positive eight millimetres bound the central maximum, whose full width is sixteen millimetres. Weaker side maxima and further minima are also present. Dashed vertical guides and a width bracket identify the first-minimum pair. The profile uses the small-angle screen-position model; the more exact first-minimum distance is 8.000064 millimetres, consistent with the shown precision. No sinusoidal displacement is being plotted.-24-16-80816240.00.51.016 mmI / I(0)Screen coordinate y / mm

The 8.0 mm distance is one-sided. The central width is 16 mm between the first two minima. Unlike a grating condition, this slit equation locates minima.

The plotted quantity is normalised intensity, not displacement. The first minima at approximately -8.0 and +8.0 mm bound the full central maximum. Normalisation states relative brightness without assigning an absolute transmitted power.

The centre-to-first-minimum distance is 8.0 mm on each side; it is not the full width. Using the inverse sine and then the tangent gives the same 8.0 mm value at the stated precision.

Optional check A single slit of width 0.15 mm is illuminated by 600 nm light. A screen is 2.0 m away. The first minimum is about 8.0 mm from the centre on each side. What is the central maximum's full width?
A single slit of width 0.15 mm is illuminated by 600 nm light. A screen is 2.0 m away. The first minimum is about 8.0 mm from the centre on each side. What is the central maximum's full width?

Predict the effect of changing the slit

At fixed wavelength and screen distance, halving b approximately doubles the small-angle first-minimum distances and central width. The pattern spreads more. This statement about width does not imply that the absolute peak intensity stays fixed as the slit narrows.

When reading a pattern, identify the central direction and the first minima on both sides. Use an intensity-position graph or calibrated image; a sine displacement curve does not represent this light-intensity pattern.