A Travelling Wave Is Two Standing Ones

Last timeAny Shape Is a Sum of Simple Ones

The shapes of a clamped string stand still and swell. A disturbance on a long rope travels. Adding two standing shapes a quarter turn apart produces one that moves.

Everything so far has stood still. The shapes of a clamped string swell and flatten

in place, with no part of the string getting ahead of any other.

But flick one end of a long rope and something plainly travels. A hump sets off

down the rope at a definite speed, keeps its form, and arrives at the far end some

time later. That looks like a different phenomenon with a different description.

It is not. It is two of the standing shapes added together.

FIG 1Dragging time forward on a travelling wave

how much time has passed, in cycless = 0

the shape of the rope

Drag the knob and the shape slides to the right without altering in any way. It does not grow, shrink or deform, and at a full cycle it is back where it began. No individual point on the rope goes anywhere: each one simply moves up and down, slightly behind the point to its left, and that steady lag from one point to the next is the whole of what travelling means.

FIG 2A standing shape at three moments, for comparison

at full swingpart way backat full swing the other way

Here nothing slides. The three curves cross the line at exactly the same places, and those crossings never move for as long as the motion lasts. All that changes between the three moments is a single overall multiplier, which is what it means for a shape to keep its shape. A reader looking only at this figure could not say which way the wave was going, because it is not going anywhere.

Adding two standing shapes

Now take two standing shapes of equal size. Shift the second a quarter of a

wavelength along the rope, and start it a quarter of a cycle later. Where the first

is flat, the second is at full swing, and the other way round.

FIG 3Two standing shapes, added

: how tightly the wiggle is packed along the rope, so that a full wave occupies a distance of two pi divided by k
: how fast each point goes up and down, in radians per second
: the combination that decides the displacement. A crest keeps this fixed, so as time advances it must move along the rope, which is the wave travelling
: the travelling result. Its size is the same everywhere and at all times, unlike either of the standing shapes it was built from

Both terms on the left are standing shapes: each is a fixed shape in x multiplied by a number that depends only on t, so each has crossings that never move. Their sum is a single function of the combination of x and t, which is the signature of travelling: hold that combination fixed by moving along as time passes, and the displacement does not change. One identity from trigonometry, and the two pictures are the same picture.

Read the identity in either direction and it says something useful. Left to right:

two standing shapes properly offset make a travelling wave. Right to left: a

travelling wave can always be written as two standing shapes, which is what makes

the earlier lessons apply to it.

The lesson stops here

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A Travelling Wave Is Two Standing Ones

Adding two standing shapes

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