Drop two stones in a pond and where the ripples cross, something quiet and exact happens: crests add to crests, crests cancel troughs, and a standing lacework of bright and still lines appears. The same arithmetic carries sound, light and the sea. Let's build it from a single bobbing point.
Spin a point steadily around a circle and watch only its height. It rises, falls, rises again — tracing a sine wave. Every wave in this chapter is, underneath, this same circular motion seen side-on.
Change how fast it turns and how far it reaches.
Line up many bobbing points, each a half-beat behind the last, and a shape appears to travel across them. But watch the highlighted dot: it only goes up and down. Nothing actually moves sideways — only the pattern does.
Stretch the wavelength and change the speed.
Pin a wave at both ends — a guitar string, an organ pipe — and the travelling wave and its reflection lock into a standing wave. Certain points, the nodes, never move at all; between them the string heaves. Only special whole-number shapes fit: the harmonics.
Step through them and find the still points.
Let one ripple-maker bob in a sheet of water and the wave spreads in rings that climb and dip the whole surface. Here it is as a field of points, read from the front and a little above — each dot floating at the water's height.
Drag the source, stretch the wavelength, raise the height, and add damping so distant ripples fade.
Now the whole pool as a field of floating points, seen from the front. Two ripple-makers send out their rings; where the rings cross, the points themselves pile up and cancel — interference you can watch heaving across the water. Tap to drop a stone: it lands hard and its rings fade as they travel out.