Waves

Someone across the room says your name and the sound reaches you, but no air rushed from their mouth to your ear. Drop a stone in a pond and rings spread outward, yet a leaf floating on the surface just bobs up and down and stays put. A stadium wave races around the seats while every person only stands and sits. Something is clearly traveling. It just is not the stuff.

What travels is the disturbance itself - a bump, a squeeze, a nudge passed from one bit of the medium to the next. Each bit does a little dance in place and hands the motion on, like the stadium crowd. This lesson is about that handed-along pattern, which we call a wave.

First, a guess

You are floating on your back in the ocean, well past the breakers, when a smooth swell rolls past you toward the beach.

Watch the pattern, not the water

This is a pond seen from above. Turn on one source and rings spread out. Turn on two and watch what happens where the two sets of rings overlap. Look especially for the calm lines that hold still even though everything is moving.

In the single-source mode you can read off a wave's three basic numbers. In the two-source mode you meet the effect that makes waves waves: they pass through each other and add up. Let us name the pieces.

Wavelength, frequency, and one clean rule

Freeze a wave in a photograph and the distance from one crest to the next is the wavelength, written $\lambda$ (lambda). Now stand at one spot and count how many crests pass you each second - that is the frequency, $f$, in hertz. These two, plus the speed, are tied together by one relationship you can derive in your head.

In the time it takes one full cycle to pass - the period $T=1/f$ - the wave moves forward by exactly one wavelength, because the pattern has repeated once. Speed is distance over time, so

That is derived straight from the definitions, and it is the workhorse of the whole subject. Its most useful consequence: for a fixed speed, frequency and wavelength trade off. Double the frequency and the wavelength must halve to keep $v$ the same. You can watch that trade happen on the frequency slider in the widget.

A worked number: sound travels about $343$ meters per second in air. The musical note A is $440$ hertz, so its wavelength is $343/440\approx 0.78$ meters - a bit under an arm's length. High notes have short wavelengths, low notes long ones, all at the same speed.

Two shapes of wave

Waves come in two flavors, told apart by which way the medium jiggles relative to where the wave goes. In a transverse wave, the medium moves side to side across the direction of travel - flick one end of a rope and the hump races along the rope while the rope itself only moves up and down. Light works this way. In a longitudinal wave, the medium is squeezed and stretched along the direction of travel - sound is bunched-up and spread-out air, pushing forward. Both still obey $v=\lambda f$.

Either way, each individual point is just doing the simple harmonic motion from the last lesson - oscillating in place around its resting spot. A wave is a whole line of oscillators, each one a beat behind its neighbor.

When waves meet: superposition

Here is the rule that made those calm lines in the widget. When two waves overlap, the medium's displacement at each point is simply the sum of what each wave would do on its own. That is superposition, and it has two dramatic outcomes.

Where a crest lands on a crest, the two add into a bigger crest - constructive interference. Where a crest lands on a trough, they cancel to flat calm - destructive interference. With two steady sources the spots that always cancel line up into fixed quiet lanes - the still lines you saw. Nothing is broken there; two waves are simply arriving forever out of step and erasing each other.

Check yourself