Foundations — Harmonics and overtones — on strings and in pipes

Before you can read the parent note on harmonics, you need a small toolbox of ideas. This page builds every single symbol and word from nothing — no prior notation assumed. Read top to bottom; each block only uses things defined above it.

1. A wave, and what "wavelength" $\lambda$ means

Picture a rope you shake up and down. The bumps travel along it. Freeze that picture in time.

WHY do we need it? Because the whole topic is about which repeat-lengths fit between two ends. If we couldn't measure "one repeat", we couldn't ask "how many repeats fit". Look at the red bracket in the figure: that one span is $\lambda$.

2. Frequency $f$ and period $T_{\text{p}}$ — counting in time

Wavelength lives in space (a frozen photo). Now watch one point on the rope over time: it bobs up and down, up and down.

They are opposites of each other: $f=\frac{1}{T_{\text{p}}}.$

(Note: the parent uses the plain letter $T$ for tension, a force. To avoid a clash, this foundations page writes the period as $T_{\text{p}}$. Watch for that difference.)

3. The master rule $v=f\lambda$

This is the single equation that turns a "shape that fits" into a "pitch you hear". Let's build it, not just state it.

Now the reasoning. In one period $T_{\text{p}}$, the point completes one bob, which means the pattern has slid forward by exactly one wavelength $\lambda$ (look at the arrow in the figure: the whole shape shuffles right by $\lambda$ during one full wiggle). Speed is distance over time: $v=\frac{\text{distance in one period}}{\text{that time}}=\frac{\lambda }{T_{\text{p}}}.$ And since $f=1/T_{\text{p}}$, we swap $1/T_{\text{p}}$ for $f$: $\boxed{v=f\lambda }\Rightarrow f=\frac{v}{\lambda }.$

4. Reflection and superposition → the standing wave

Why should a wave only survive at special lengths at all? Because inside a string or pipe, a wave hits the far end and bounces back, then it overlaps with its own reflection.

When a wave and its reflection add up, most patterns fight and cancel — except a few special ones that lock into a standing wave: a shape that no longer travels, it just wobbles in place.

Two spacing facts you will use everywhere (measure them off the figure):

These two numbers are exactly the "rulers" from Section 1 reappearing — that is why the topic keeps writing $L=\lambda /2$ or $L=\lambda /4$.

5. The boundary conditions

A boundary is an end of the string or pipe. What it forces there is the whole secret.

6. Length $L$, the counting number $n$, and the harmonic ladder

7. Where wave speed $v$ actually comes from

The parent uses two different sources for $v$. You don't derive them here (that's for the linked notes), but you must know the symbols.

For a pipe, $v$ is instead the speed of sound in the gas inside — see Speed of sound in gases.

Prerequisite map

Each box on the left is built on this page; together they feed the parent topic on the right.

Equipment checklist

Connections

• Standing waves and superposition — the mechanism that turns reflections into fixed nodes.
• Reflection of waves at boundaries — why an end sends the wave back to interfere with itself.
• Speed of a wave on a string — where $v=\sqrt{T/\mu }$ comes from.
• Speed of sound in gases — sets $v$ for pipes.
• Beats and resonance — how a driven column locks onto these frequencies.
• Timbre and Fourier synthesis — how the mix of these overtones makes an instrument's sound.