Start simple: the pattern travels, the stuff stays put
Drop a pebble into a still pond and a ring spreads outward — but a leaf floating on the surface just bobs up and down; it never rides the ring to shore. That is the secret of every wave: the pattern moves, while the material it moves through mostly stays where it is. Shake one end of a long rope and a hump runs to the far end, yet the rope itself goes nowhere. A wave carries energy and shape, not matter.
Build it up: three numbers and one rule
A repeating wave is described by three quantities. The amplitude $A$ is how tall the bumps are (in metres). The wavelength $\lambda$ is the distance from one crest to the next. The frequency $f$ is how many crests pass you each second (in hertz). They lock together in one tidy rule:
Wave Speed
$$v = f\lambda$$
Quick number: if crests are $3\text{ m}$ apart and two of them pass you every second, the wave travels at $v = 2 \times 3 = 6\text{ m/s}$. The period $T = 1/f = 0.5\text{ s}$ is simply how long one full wobble takes.
Go deeper (AP / intro-college): the travelling wave and two different speeds
The full shape of a moving wave is $y(x,t) = A\sin(kx - \omega t + \varphi)$, where the angular frequency $\omega = 2\pi f$ and the wave number $k = 2\pi/\lambda$, so that $v_w = \omega/k = f\lambda$. Here is the subtle part students miss: there are two speeds. The wave speed $v_w$ is how fast the pattern marches along — set only by the medium, $v_w = \sqrt{T_s/\mu}$ on a string (tension over mass-per-length). The particle speed $v_p = \partial y/\partial t = -A\omega\cos(kx-\omega t)$ is how fast a single bit of string flicks up and down. They are unrelated: a slow-moving wave can still have fast-flicking particles if $A$ and $f$ are large. The sliders map straight onto the symbols — $A$ is amplitude, $f$ is frequency, $v$ is the wave speed (so $\lambda = v/f$ follows automatically), $\gamma$ damps the amplitude as $A_0 e^{-\gamma t}$, and $\varphi$ shifts the starting phase.
Try this in the sim above
First, set damping $\gamma = 0$ and watch the wave run forever; then nudge $\gamma$ up and see the crests shrink as $A_0 e^{-\gamma t}$. Next, hold $f$ fixed and slide the wave speed $v$ up — the wavelength visibly stretches because $\lambda = v/f$, even though the bobbing rate never changes. Finally, open Superposition and set $\Delta f$ small (about $0.5\text{ Hz}$): the two tones drift in and out of step, producing a slow beat at $|f_1-f_2|$ — exactly how musicians tune an instrument by ear.
Phase vs particle speed. Wave speed $v_w=f\lambda$ is how fast the pattern moves. Particle speed $v_p=\partial y/\partial t=-A\omega\cos(kx-\omega t)$ is how fast individual medium elements move. These are entirely different — a string particle is always perpendicular to $v_w$.
2
Standing waves. Two identical waves travelling in opposite directions superpose: $y_1+y_2=2A\cos(kx)\sin(\omega t)$. The $\cos(kx)$ factor creates fixed nodes at $kx=n\pi$ — points that never move. Antinodes at $kx=(n+\frac{1}{2})\pi$ oscillate with amplitude $2A$.
3
Superposition & beats. Adding $f_1$ and $f_2$ gives beat envelope at $|f_1-f_2|$ and carrier at $(f_1+f_2)/2$: $y=2A\cos(\pi\Delta f\cdot t)\sin(2\pi\bar{f}\cdot t)$. Beats are used to tune instruments — frequency is correct when beat rate → 0.
4
Damping. Real waves lose energy: $A(t)=A_0 e^{-\gamma t}$. Power loss: $P\propto A^2\propto e^{-2\gamma t}$. Damped waves are ubiquitous — seismic waves, sound in air, oscillations in circuits.
Wave speed $v_w=f\lambda$ is the speed at which the waveform pattern propagates — determined entirely by the medium (tension, density). Particle speed $v_p=\partial y/\partial t$ is the transverse velocity of any individual medium element — proportional to amplitude and frequency. A very low-frequency, high-amplitude wave can have slow wave speed but fast particle motion.
Key takeaway: $v_w=f\lambda$ (pattern speed) and $v_p=-A\omega\cos(kx-\omega t)$ (particle speed) are unrelated.
Seismic P-waves (longitudinal, travel through Earth's core), S-waves (transverse, only through solids), guitar strings (standing waves set pitch), ultrasound imaging (2–20 MHz), MRI gradient pulses, sonar (submarine detection), ocean waves (actually surface waves with both components), bridge oscillations (Tacoma Narrows), and noise-cancellation headphones.
Key takeaway: Waves govern music, seismology, medical imaging, and structural engineering.
At a node, displacement is always zero but slope $\partial y/\partial x$ is maximum — the string is tilted there. The potential energy (stored in slope) is maximum at nodes; the kinetic energy (stored in particle velocity) is maximum at antinodes. Energy oscillates back and forth between adjacent antinodes, passing through the nodes.
Key takeaway: Nodes: max potential energy (slope), zero kinetic energy. Antinodes: max kinetic, zero potential.
For string element $dm=\mu\,dx$ under tension $T$: net transverse force $dF=T\sin\theta_2-T\sin\theta_1\approx T(\partial^2y/\partial x^2)dx$. Newton: $dm\cdot\partial^2y/\partial t^2=dF$ → $\mu\,\partial^2y/\partial t^2=T\,\partial^2y/\partial x^2$. This wave equation has solution $y=f(x\pm v_wt)$ with $v_w=\sqrt{T/\mu}$.
Key takeaway: Wave equation: $\partial^2y/\partial t^2=v_w^2\,\partial^2y/\partial x^2$ with $v_w=\sqrt{T/\mu}$.
Resources: Khan Academy; HyperPhysics (hyperphysics.phy-astr.gsu.edu); MIT OCW.
Section 05
Common Misconceptions
❌ Wave speed depends on the amplitude of oscillation.
✅ Wave speed in a linear medium depends only on the medium properties: $v_w=\sqrt{T/\mu}$ for strings, $v_w=\sqrt{B/\rho}$ for sound. Amplitude has no effect on wave speed in linear media. (In nonlinear media like large-amplitude ocean waves, speed can depend weakly on amplitude.)
📖 HRW 10th Ed., §16-2.
❌ Particles in a transverse wave move in the direction of propagation.
✅ Particles in a transverse wave move perpendicularly to the direction of propagation. In a wave on a string travelling along $x$, each particle moves only in the $y$ direction. Only in a longitudinal wave do particles move along the propagation direction.
📖 HRW 10th Ed., §16-1.
❌ A node in a standing wave is a point of zero energy.
✅ A node has zero kinetic energy (particles are stationary) but maximum elastic potential energy (the string is most steeply sloped there). The total energy at a node is not zero — it is all potential. Energy continuously shuttles between kinetic (at antinodes) and potential (at nodes).
📖 HRW 10th Ed., §16-5.
Misconception research: Arons — A Guide to Introductory Physics Teaching; Driver et al. — Making Sense of Secondary Science.