Foundations — Huygens' principle — wavefront propagation

This page assumes nothing. If the parent note Huygens' Principle — Wavefront Propagation used a symbol, we build it here from the ground up.

1. A wave = a wobble that travels

Start with the most basic object. Imagine a long rope. You flick one end up and down. A hump travels down the rope, but each bit of rope only moves up and down — it does not travel along with the hump. That travelling pattern is a wave.

Why we need this first: Huygens' principle is a rule about how the travelling pattern advances. If you don't picture "the material stays, the pattern moves", every later statement ("the wave moved distance $v\Delta t$") is meaningless.

2. Oscillation and phase — "how far along the wobble am I?"

Pick one single bit of rope and watch only it. It goes up → back to middle → down → back to middle → up… over and over. One full trip (up, down, back to start) is called one cycle.

Phase answers the question: at this instant, how far through its cycle is this point?

Look at figure s02: the yellow dots are all at the top of their wobble at this instant — same phase. The red dot is at the bottom — a different phase.

Why we need phase: the parent note defines a wavefront as points of equal phase, and says phase "controls interference". Phase is the single most important word on this whole topic — everything downstream is bookkeeping of phase.

See Phase and Path Difference for how phase differences become path differences.

3. Wavefront — the surface of equal phase

Now stop watching one point and look at the whole rope (or whole pond) at one frozen instant. Mark every point that is currently in phase — say, every point sitting exactly at a crest.

• On a pond from a dropped stone, all the crest-points at one instant form a circle (a ring).
• Far away, or from a wide flat source, those equal-phase points line up flat.

Why we need it: Huygens' construction takes a wavefront now and builds the wavefront next. It is the object the whole principle operates on.

4. The ray — the arrow perpendicular to the wavefront

The wavefront tells you where equal phase is; the ray tells you which way the energy is heading.

5. Speed $v$, time $\Delta t$, and the distance a wavelet travels

Two symbols the parent note uses constantly: $v$ and $\Delta t$.

Why this matters: step 2 of Huygens' recipe draws a circle of radius $\rho =v\Delta t$ around each point. That radius is this formula. Same $v$ and same $\Delta t$ ⇒ every wavelet is the same size, which is exactly why a plane wave stays plane.

6. Secondary wavelet and the envelope (common tangent)

In figure s04, each blue point on the old wavefront grew a green circle of radius $\rho =v\Delta t$. The yellow line that grazes every green circle is the new wavefront. We keep the forward envelope and throw away the backward one (the obliquity factor forces the backward amplitude to zero — the parent note explains this).

Why we need this pair of words: "secondary wavelet" and "envelope" are literally the two nouns in the statement of Huygens' principle. Without them the principle cannot even be spoken.

7. The sine of an angle — a ratio of two lengths

Before we can measure how a wavefront tilts at a surface, we need one small piece of trigonometry. We introduce it now, before we use it.

Why sine and not something else? When a slanted wavefront meets a surface, the piece that "sticks up" (the opposite side) compared to the slanted length (the hypotenuse) is exactly what encodes the tilt angle. Sine is the tool that converts "how far one edge travelled" into "what angle the wavefront makes" — which is precisely the job in Snell's law below.

8. Angle from the normal — $i$ and $r$

When a wavefront hits a surface (a mirror or glass), we measure angles. But measured from what line?

Why measure from the normal, not the surface? Because the physics laws come out clean: $\angle i=\angle r$ for a mirror, and $\frac{\sin i}{\sin r}=\frac{v_1}{v_2}$ for refraction. Measuring from the surface would just add clutter (you'd write $90^{\circ }-i$ everywhere).

To see why the sines appear, look at the labelled triangle in figure s06. A slanted wavefront $AB$ reaches the surface: end $A$ touches first at the surface point $A$, while end $B$ still has to travel down to the surface point $C$.

• In time $t$, corner $B$ covers $BC=v_{1}t$ in medium 1 — this length sits opposite the incidence angle $i$.
• In that same time the wavelet from $A$ reaches into medium 2 a distance $AD=v_{2}t$ — this sits opposite the refraction angle $r$.
• Both right triangles share the same slanted line $AC$ as hypotenuse.

Using the sine definition from section 7 on each triangle: $\sin i=\frac{BC}{AC}=\frac{v_{1}t}{AC},\sin r=\frac{AD}{AC}=\frac{v_{2}t}{AC}$ Divide, and the shared $AC$ (and the $t$) cancel: $\frac{\sin i}{\sin r}=\frac{v_1}{v_2}.$

See Laws of Reflection and Snell's Law and Refractive Index for where these angles lead.

9. Refractive index $n$

Feeding $n=c/v$ into the ratio above turns the speed form of Snell's law into the familiar index form $n_1\sin i=n_2\sin r$ (worked out in the parent note). See Snell's Law and Refractive Index.

How the foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer out loud.

Connections

• Huygens' Principle — Wavefront Propagation (the parent this page prepares you for)
• Phase and Path Difference (phase turned into measurable path lengths)
• Wave Optics — Interference (equal-phase surfaces overlapping)
• Young's Double Slit Experiment (each slit as a secondary source)
• Diffraction (wavelets bending around edges)
• Snell's Law and Refractive Index
• Laws of Reflection
• Fresnel–Kirchhoff Diffraction