C.3 Wave phenomena

Define a wavefront

A wavefront is a line or surface joining points that are in phase. Adjacent wavefronts are separated by one wavelength. In a uniform medium, the wavefronts are perpendicular to the direction of propagation.

Use rays to show propagation

A ray is a line showing the direction in which the wave transfers energy. Draw rays perpendicular to the local wavefronts; straight, parallel wavefronts give parallel rays, while circular wavefronts from a point source give radial rays.

Read the geometry

When a wavefront diagram changes direction at a boundary, compare the ray direction and the spacing of wavefronts on each side. The wavefront construction helps distinguish a change in speed from a change in frequency.

Common trap

Do not draw rays parallel to wavefronts. A ray follows energy propagation and is normal to the wavefront at each point.

Write the boundary relationship

For a ray crossing from medium 1 to medium 2, Snell’s law is $n_1\sin\theta_1=n_2\sin\theta_2$. Each angle is measured between the ray and the normal, not between the ray and the surface.

Solve the geometry first

Draw or identify the normal at the boundary, label incident and refracted angles, then substitute the refractive indices and sines. If the question asks for speed, combine the result with $n=c/v$.

Check the bend

Entering a higher-index medium decreases speed and bends the ray toward the normal. Entering a lower-index medium increases speed and bends it away from the normal; the frequency remains fixed at a stationary boundary.

Common trap

Do not use the angle between a wavefront and the normal as if it were the ray angle. A wavefront is perpendicular to the ray, so convert the angle when the diagram labels wavefronts.

Define refractive index

The refractive index of a medium is $n=c/v$, where $c$ is the speed of light in vacuum and $v$ is its speed in the medium. A larger $n$ means a lower light speed in that medium.

Relate speed and wavelength

At a stationary boundary the frequency is unchanged. Since $v=f\lambda$, a lower speed means a shorter wavelength. For two media, $n_2/n_1=\lambda_1/\lambda_2$ when the frequency is common.

Use a graph or measurement

If a graph’s gradient represents $n$, state that interpretation before reading the value. For uncertainty, use the spread from suitable maximum and minimum lines or the specified uncertainty method.

Common trap

Do not use $n=v/c$, and do not assume wavelength stays fixed when light enters a different medium. Frequency is the quantity that remains fixed at a stationary boundary.

Check the two conditions

Total internal reflection can occur only when a wave travels from a higher-index medium to a lower-index medium, and the incidence angle is greater than the critical angle. At the critical angle, the refracted ray travels along the boundary: $\theta_2=90^\circ$.

Calculate the critical angle

From Snell’s law, $\sin\theta_c=n_2/n_1$ for $n_1>n_2$. For a dense medium to air, $n_2\approx1$, so $\sin\theta_c=1/n_1$.

Use the boundary picture

For incidence below $\theta_c$, there is a refracted ray. At $\theta_c$, it grazes the boundary. Above $\theta_c$, no refracted ray propagates into the lower-index medium and all the light is reflected back into the denser medium.

Common trap

Do not use the critical-angle equation when light travels from lower to higher refractive index, and do not measure the critical angle from the surface rather than the normal.

Add overlapping displacements

When waves overlap, the resultant displacement at a point is the algebraic sum of the individual displacements: $y_{\mathrm{resultant}}=y_1+y_2$. The waves then continue propagating after the overlap.

Keep the signs

Displacements on the same side of equilibrium add; opposite displacements partially or completely cancel. Equal opposite pulses can produce zero displacement at an instant without destroying either wave.

Connect to interference

Repeated superposition of coherent waves can create stable maxima and minima. A diffraction pattern extending beyond a geometrical shadow is evidence that wave overlap and interference are involved.

Common trap

Do not add amplitudes as positive magnitudes only, and do not treat destructive interference as permanent disappearance of the waves.

Define coherence

Two waves are coherent if they have the same frequency and a constant phase difference. The phase relationship does not drift with time.

Connect coherence to a pattern

When coherent waves overlap, the locations of constructive and destructive interference remain fixed, producing a stable interference pattern. An ordinary pair of independent light sources usually has a changing phase relationship and does not produce a stable pattern.

Use one source when needed

A single source split into two paths can provide a common frequency and phase relationship. The resulting secondary sources can then act coherently for a double-source interference experiment.

Common trap

Same frequency alone is not enough. The phase difference must also remain constant; otherwise bright and dark locations move or wash out over time.

Define path difference

Path difference is the difference between the distances travelled by two waves from their sources to the same observation point. For in-phase coherent sources, it determines whether the waves arrive in phase or out of phase.

Apply the conditions

Constructive interference occurs when path difference $=n\lambda$. Destructive interference occurs when path difference $=(n+\tfrac12)\lambda$, where $n$ is a whole number.

Count fringes carefully

Start from the central bright fringe when the path difference is zero. Each additional bright fringe changes the path difference by $\lambda$; dark fringes lie halfway between adjacent bright conditions.

Common trap

Do not assign $\lambda/2$ to every dark point. The first dark condition is $\lambda/2$, then $3\lambda/2$, $5\lambda/2$, and so on.

Set up two-source interference

Two coherent sources emit waves with the same frequency and a constant phase difference. At each observation point, compare the two source-to-point distances to find the path difference.

Map bright and dark regions

For in-phase sources, path difference $n\lambda$ gives constructive interference and a bright or high-amplitude region. Path difference $(n+\tfrac12)\lambda$ gives destructive interference and a dark or low-amplitude region.

Read the pattern

Points equidistant from the two sources have zero path difference and form a central constructive line. Further maxima and minima occur where the path difference changes by half-wavelength steps.

Common trap

Do not use source-to-source separation as the path difference. It is the difference between the two travel distances to the same observation point.

Relate fringe spacing to the apparatus

For Young’s double-slit interference, fringe separation is $s=\lambda D/d$, where $\lambda$ is wavelength, $D$ is slit-to-screen distance and $d$ is slit separation.

Rearrange before substituting

Use $\lambda=sd/D$, $d=\lambda D/s$, or $D=sd/\lambda$ as needed. Measure the separation between adjacent bright or dark fringe centres; if several fringes are measured, divide the total width by the number of intervals.

Check the trends

Fringes spread farther apart when wavelength or screen distance increases, and become closer when slit separation increases. The small-angle model assumes $D\gg d$ and approximately plane wavefronts normal to the slits.

Common trap

Do not use the total width across several fringes as s without dividing by the number of fringe spacings, and do not confuse slit separation d with screen distance D.

A boundary can split wave energy

When a travelling wave reaches a boundary, part of its energy may be reflected back into the first medium and part may be transmitted into the second. The transmitted wave is refracted when its speed changes and it crosses the boundary at a non-zero angle.

Read rays and wavefronts together

Angles are measured from the normal. Rays show energy-transfer direction and remain perpendicular to wavefronts. Frequency stays fixed at a stationary boundary; a change in speed therefore changes wavelength and wavefront spacing.

Boundary cases

At normal incidence the transmitted ray does not bend, although its speed and wavelength may change. Unless absorption is stated, reflected and transmitted energy together account for the incident energy; their amplitudes do not generally add directly.

Diffraction is wave spreading

A wave diffracts when its wavefront bends into the region behind an obstacle or spreads after passing through an aperture. The wave remains in the same medium, so diffraction itself does not require a change of speed or frequency.

Compare wavelength with the opening or body

Spreading is most noticeable when the aperture width or obstacle size is comparable to the wavelength. An aperture much wider than the wavelength gives a broad central region that travels nearly straight with limited edge spreading; narrowing the aperture increases the angular spread.

Wavefront–ray representation

Before a straight aperture, incident wavefronts are parallel. Beyond a narrow aperture, draw curved outgoing wavefronts; rays stay perpendicular to them and fan outward. Around a body, wavefronts curve into the geometrical shadow.

Do not confuse mechanisms

Refraction is a direction change caused by a speed change at a boundary. Diffraction is spreading caused by an edge or aperture, even when the medium is unchanged.