Foundations — Wave equation — derivation for string

This page assumes you have seen nothing. We build every letter, every squiggle, and every picture the parent note Wave Equation derivation silently leans on. Read top to bottom — each item is the ladder rung for the next.

1. The string and its picture: $x$, $y$, and $y(x,t)$

Picture a long horizontal string, like a stretched skipping rope. We lay a ruler along it.

• $x$ = how far along the string you are, measured horizontally. It labels which point of the string we are talking about.
• $y$ = how far up or down that point has been pushed from its resting straight line. Up is positive, down is negative.

Now the key notation: $y(x,t)$.

Why the topic needs it. A wave is a shape that moves, so height depends on both where you look and when you look. One variable is not enough — hence two inputs, $x$ and $t$. Everything else in the derivation is about how this one function behaves.

2. Tension $T$ — the pull inside the string

Picture two hands gripping the rope and pulling apart — the rope is now tense. Pluck it and the pluck races along fast; loosen the grip and the same pluck crawls.

Why the topic needs it. Tension is the restoring pull — it is what drags a displaced piece of string back toward straight. No tension, no wave. See Wave speed on a string — v = sqrt(T over mu).

3. Linear mass density $\mu$ — how heavy per metre

Picture chopping the string into 1-metre chunks and weighing one chunk — that weight (in kg) is $\mu$.

From this comes the mass of a tiny piece:

Why the topic needs it. Newton's law is $F=ma$, and mass here is $dm=\mu dx$. Heavier string ($\mu$ big) is more sluggish → slower wave.

4. The tiny piece: $dx$, $dm$, and "$d$-something"

Everywhere in the derivation you see a little $d$ in front of things: $dx$, $dm$. This is not a new letter multiplying — it is a signal.

Why the topic needs it. We cannot apply $F=ma$ to the whole floppy string at once because different parts do different things. So we zoom in on one tiny piece, get its law, and the tininess ($dx$) lets us use straight-line geometry and derivatives. The magic later: the $dx$ cancels from both sides, so the final law holds for every piece — hence the whole string.

5. Angle $\theta$ and slope — how tilted the string is

At the ends of our tiny piece the string is slightly tilted. Call that tilt angle $\theta$ (Greek "theta"), measured from the horizontal.

Two trig tools show up. Here is why each one, in plain words:

And $\tan \theta$:

Why the topic needs it. These convert geometry (a tilt angle) into calculus (a slope = a derivative), which is the bridge to the whole derivation.

6. Partial derivatives: $\frac{\partial y}{\partial x}$ and $\frac{\partial y}{\partial t}$

Because $y$ depends on two things ($x$ and $t$), when we take a rate of change we must say which one we are wiggling and hold the other still. That is what the curly $\partial$ means.

Why the topic needs it. Slope (space) and transverse velocity (time) are different rates of the same $y$. Ordinary $\frac{d}{dx}$ would be ambiguous; the curly $\partial$ keeps them straight. See Partial derivatives and curvature.

7. The second derivatives: acceleration and curvature

Take a rate of a rate. Two of them matter.

Why the topic needs it. The entire result reads: acceleration ($t$-curvature) is proportional to spatial curvature ($x$-curvature). Both are second derivatives — you cannot read the wave equation without them.

8. Wave speed $v$ and the shape-mover $f(x-vt)$

Why the topic needs it. These are the solutions — the actual waves. The special sine wave $y=A\sin (kx-\omega t)$ (see Travelling wave function y = A sin(kx - omega t)) and its rule $v=\omega /k$ (the Dispersion relation omega = vk) are particular cases of this shape-mover.

9. The scaffolding law: $F=ma$

This is the engine of the whole derivation — see Newton's Second Law. Everything above just tells us how to write down the $F$ (tension geometry) and the $m$ ($dm=\mu dx$) and the $a$ (${\partial }^{2}y/\partial t^2$).

Prerequisite map

Equipment checklist

Hide the right side and test yourself before reading the parent derivation.

Connections

• Wave equation — derivation for string
• Newton's Second Law
• Partial derivatives and curvature
• Wave speed on a string — v = sqrt(T over mu)
• Travelling wave function y = A sin(kx - omega t)
• Dispersion relation omega = vk
• Transverse vs Longitudinal Waves
• Standing waves on a string
• Energy carried by a wave