Force That ClicksBonus: Waves and Force ① (trilogy, part 1)

In the main series we peeled force all the way apart. In the bonus we chase the deep kinship between force and "waves"

From Force, a Wave Is Born A wave is not some separate thing ── it's a restoring force (the spring of Episode 3) and inertia (Episode 1) lined up side by side across space, holding hands.
A wave is force playing a game of "telephone." Its speed is set by √(restoring force / inertia).

Tools you'll need: inertia from Episode 1, the spring from Episode 3, √ v = √(tension / linear density) = √(restoring force / inertia)

Over the 12 episodes of the main series, we peeled the true nature of force back to relationship, geometry, gauge, and curvature. In the bonus we shift our gaze a little and, over three installments, chase the kinship between force and "waves." Waves ── a vibrating string, sound, the water's surface, light ── feel as if there's some independent "wave stuff" out there. But there isn't. The first installment takes the most naive direction: how, exactly, is a wave born from force? The answer is almost anticlimactically simple, and it's built entirely from the tools we made in the main series (spring = restoring force, and inertia). A wave turned out to be a game of "telephone" in force ── each neighbor pushing the next.

01Spring (restoring force) + inertia = the seed of a wave

String a lot of little weights (masses) together with springs, like beads on a chain. Flick one weight sideways ── what happens? As we saw in Episode 3, the spring pulls its neighbor with a restoring force that tries to snap things back. The neighbor, being pulled, doesn't move right away ── because of the inertia from Episode 1, it starts moving a little late. Once it moves, it pulls the next neighbor in turn ── and this chain of "pull (force) → move a beat later (inertia) → pull the next" travels from end to end. That is a wave.

What a wave really is ── a game of telephone in force

Restoring force (the force pulling a neighbor back, Episode 3) + inertia (can't move instantly, Episode 1).
Line these two up across space, and a single wiggle gets passed along with a delay, "to the neighbor, and the next."
A wave is not an independent entity; it's the phenomenon of a chain of force and inertia propagating.

02Wave speed = √(restoring force / inertia)

So how fast does the message travel? Two tugs-of-war decide it ── the stronger the restoring force, the quicker it yanks the neighbor, so faster. The larger the inertia (weight), the more sluggish the start, so slower. For a string, the restoring force is the tension \(T\) and the inertia is the linear density \(\mu\) (the weight per meter). Written out, it's just \(F=ma\) (Episode 1) applied to the chain, and it comes out like this.

Wave speed $$v=\sqrt{\frac{T}{\mu}}=\sqrt{\frac{\text{strength of the restoring force}}{\text{inertia}}}$$

Tighten a guitar string (large \(T\)) and the pitch rises (the wave is faster). A thick string (large \(\mu\)) is low (the wave is slower) ── exactly as this formula says. The speed of a wave is set by the ratio of force to inertia inside the medium. In the figure below, change the tension and watch how far the wave travels in the same amount of time.

Figure: weights strung together with springs, flicked at the left end. Strengthen the tension (restoring force) and the wave travels faster in the same time. v=√(T/μ)
A connecting voice ── and the sister series' c·t = constant "Wave speed = √(restoring force / inertia)" is a form common to every wave. For sound, the restoring force is pressure (the elasticity of air) and the inertia is density. And the speed of light \(c\) can be written as the ratio of the electric and magnetic restoring force and inertia of that "medium" called the vacuum: \(c=1/\sqrt{\varepsilon_0\mu_0}\) ── exactly the same skeleton. This quietly connects here with the sister series "Cosmology That Clicks," where \(c\) took the leading role. Wave speed is nothing but the force properties of the medium itself.
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03What peeling it back revealed ── a wave is "how force travels"

The conclusion of this first bonus installment. A wave is not something other than force; it's the "way force travels" when a restoring force and inertia are lined up across space. A single wiggle is carried from end to end by the chain of a force pulling the neighbor and inertia making it move a beat late. So asking "how fast is the wave?" is the same as asking "how are force and inertia balanced inside this medium?" The backbone of the main series, "force is a relationship," here takes the form of "a wave is the figure of force ── a relationship ── traveling through space."

The honest line

Here we didn't write the wave equation itself; we described the "feel" of \(v=\sqrt{T/\mu}\) (the tug-of-war of restoring force ÷ inertia). Strictly, applying \(F=ma\) to the chain and taking the continuum limit gives the wave equation \(\partial^2 y/\partial t^2 = v^2\,\partial^2 y/\partial x^2\). Also, \(v=\sqrt{T/\mu}\) is for small displacements (the linear regime); for large amplitudes, or for media where the speed varies with wavelength (dispersion), it gets corrected.

The figure is a schematic of a transverse pulse; it is not a numerical solution of the actual motion of a spring chain (it only reproduces the point that the speed is proportional to √T).

Practice problems
  1. Explain in one line how a wave propagates, without invoking the wave equation.
    See the answer
    A chain of the restoring force pulling the neighbor (the spring) and inertia that can't move instantly. A single wiggle propagating as "pull → move a beat later → pull the next" is a wave.
  2. Why does tightening a guitar string raise the pitch (make the wave faster)?
    See the answer
    Because the tension T (the restoring force) increases. In v=√(T/μ), the stronger the restoring force, the more quickly it yanks the neighbor and the faster the wave travels. A thick string (large μ) is slow and low.
  3. Can the speeds of sound and light be written in the same form?
    See the answer
    Yes. Sound is v=√(pressure elasticity/density), and light is c=1/√(ε₀μ₀). All are the square root of "restoring force / inertia." Wave speed is nothing but the force properties of the medium.

SummaryA wave was a game of telephone in force

Line a restoring force (the spring of Episode 3) and inertia (Episode 1) up across space, and a single wiggle propagates through the chain of "pull the neighbor → move a beat late → pull the next." That is a wave. A wave is not an independent entity; it's the figure of force and inertia traveling through space. Its speed is decided by a tug-of-war, \(v=\sqrt{T/\mu}=\sqrt{\text{restoring force}/\text{inertia}}\). A guitar string, sound, and even light can all be written with the same skeleton.

The main series' "force is a relationship" became, here, "a wave = the propagation of force, a relationship." Next time we go the other way ── the story that force itself was a wave. The field of the electromagnetic force is a nearby force when still, and a wave (light) that flies far away when you shake it. We'll see the two faces of one and the same field.

This document is installment ① of "Waves and Force," a bonus of the "Force That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. That the speed of a transverse wave on a continuous elastic body (a string with tension T and linear density μ) is \(v=\sqrt{T/\mu}\); that in general the wave speed is set by the square root of "restoring force / inertia"; that applying \(F=ma\) to a mass-and-spring chain and taking the continuum limit derives the wave equation \(\partial_t^2 y=v^2\partial_x^2 y\); and that the sound speed \(v=\sqrt{K/\rho}\) and the speed of light \(c=1/\sqrt{\varepsilon_0\mu_0}\) share the same form ── all of this is established content. \(v=\sqrt{T/\mu}\) is a small-amplitude (linear) approximation and is corrected under dispersion or nonlinearity. The figure is a schematic of a transverse pulse, a conceptual model reproducing only the √T dependence of the speed. ── To print, use your browser's "Print" and choose "Save as PDF" (in the print version the slider and answers are static and hidden).

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you see the wave travel faster as you strengthen the tension. "See the answer" opens each solution.