Foundations — Mechanical waves — transverse and longitudinal

This page assumes nothing. Every letter and squiggle the parent parent note throws at you gets built here, brick by brick, in an order where each brick rests on the one before it.

0. What is "displacement"? (the very first picture)

Before any formula, we need the idea of a particle leaving its resting spot and coming back.

Picture a single dot that likes to sit at one height — its equilibrium (rest) position. Poke it, and it moves away, then a restoring pull brings it back, it overshoots, comes back again… it oscillates. This is exactly Simple Harmonic Motion, the engine hiding inside every wave.

1. Position and time — the symbols $x$ and $t$

A displacement is not enough on its own. We must say which particle and when.

Because $y$ depends on both where you look and when you look, we write it as $y(x,t)$ — read "y as a function of x and t". The brackets just mean "y depends on these."

2. Amplitude — the symbol $A$

3. The sine function — why this shape?

The parent note suddenly writes $\sin$. Where did that come from?

The angle $\theta$ that we feed into $\sin$ is called the phase — it says how far round the circle (how far into the dance) the particle currently is.

4. The clock of oscillation — $T$, $f$, and $\omega$

Sine repeats. We need words for how fast it repeats. Three symbols, all saying the same thing three ways.

So the source particle's motion, $y(0,t)=A\sin (\omega t)$, now reads in plain words: "height = amplitude times the sine of (how far round the dance we are by time $t$)."

5. Wavelength and wave number — $\lambda$ and $k$

Freeze time (the photograph slice). The shape repeats in space too.

6. Wave speed — the symbol $v$

7. The tools that describe how the medium fights back

The parent uses $T$ (tension), $\mu$, $\rho$, $Y$, $B$. Each is one half of a tug-of-war: something that pulls the particle back (elasticity) versus something that resists changing its motion (inertia).

8. The rate-of-change symbols — $\frac{\partial }{\partial t}$ and slope

The parent writes $v_p=\partial y/\partial t$ and talks about "slope." Two ideas of steepness.

The prerequisite map

Equipment checklist

Cover the right side. If you can state each without peeking, you are ready for the parent note.