Foundations — Interference — constructive, destructive conditions

This page assumes you have seen none of the symbols in the parent note. We build every one — $y$, $a$, $\omega$, $t$, $\varphi$, $\lambda$, $\Delta x$, $I$ — from a picture, in an order where each depends only on the ones before it. By the end, every line of the parent note should read like plain English.

1. A wave as a moving up-and-down graph

Before any symbol, picture a rope you shake. Each little piece of rope bobs up and down; the bump travels sideways along the rope. A wave is that shape moving.

2. Making the wave move: $t$, $T$, and $\omega$

A single point on the rope does the same up-down trip over and over. To describe that we need three time-words.

Now — why does the parent note write $\sin (\omega t)$ and not just $\sin (t)$? Because $\sin$ only "resets" after its input grows by $2\pi$. We want our wave to reset after one period $T$, not after $2\pi$ seconds. So we need a conversion factor that turns "seconds" into "radians of the sine." That factor is $\omega$.

So $y=a\sin (\omega t)$ reads: "height = amplitude × (sine of how far around the circle we've turned)." Every symbol now earned.

3. Wavelength $\lambda$ — the space version of the period

$T$ was one full cycle in time (freeze one point, watch the clock). Now freeze the clock and look along the rope: the shape repeats in space too.

4. Path difference $\Delta x$ — why two waves can be out of step

Here two waves finally meet. Suppose they leave two sources and travel to the same listening point $P$, but along paths of different length. The longer-travelled wave arrives "behind."

5. Phase difference $\varphi$ — the same idea in angle-language

We now have two ways to say "how out of step are they?":

• as a distance: $\Delta x$ (measured in metres),
• as an angle: $\varphi$ (measured in radians), how far apart the two circle-dots point.

The parent's master link just converts one language to the other. One whole wavelength of extra path ($\Delta x=\lambda$) is one whole cycle of lag ($\varphi =2\pi$). Scaling that:

6. Intensity $I$ — why brightness is amplitude squared

The last symbol. Your eye and ear don't sense amplitude directly; they sense power delivered — brightness, loudness.

See Energy in Waves for the full $I\propto a^2$ story.

7. The trig fact the derivation borrows

The parent collapses $a\sin \omega t+a\sin (\omega t+\varphi )$ using one identity: $\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$ You don't need to prove it here — just trust it does one job: it turns a sum of two sines into a single sine times a constant factor. That constant factor, $2a\cos \frac{\varphi }{2}$, is the resultant amplitude. This is why "adding two waves gives one wave of the same frequency, just resized."

How these foundations feed the topic

Equipment checklist

Cover the right side and self-test. If any answer is a surprise, re-read that section.

Connections

• Principle of Superposition — the "just add the displacements" rule these symbols serve.
• Coherence and Path Difference — why a steady $\varphi$ (hence steady $\Delta x$) is required.
• Energy in Waves — the full reason $I\propto a^2$.
• Young's Double Slit Experiment — where $\Delta x$ gets geometry.
• Standing Waves — same superposition, opposite-travelling waves.
• Beats — superposition when the two $\omega$ differ.