Whip crack

Why does the crack of a whip come from its tip rather than from the hand that swung it?

A whip is swung by an arm moving at walking pace, perhaps a few metres per second. Nothing strikes anything. There is no impact, no explosive charge, no rigid object to ring like a bell. And yet a sound comes out of it sharp enough to be heard across a field — and it comes from the far end, not from the hand. Where can a bang come from, if nothing collides?

Take the second oddity first, because it is the useful one: the sound is louder than anything the arm is doing, and it arrives from the thin end. That already tells us the whip is not merely transmitting the arm's motion. Something along its length is amplifying it — energy put in slowly at one end coming out fast at the other.

What could a rope do to accomplish that? Consider what the arm actually gives it. The hand accelerates and then decelerates, and what it leaves behind is a bend — a loop of doubled-back rope travelling away down the length. The rope as a whole is not moving at the loop's speed; the loop is a travelling disturbance, in the way a wave crossing a pond is not the water travelling.

Now look at the whip's shape. It is not a rope of uniform thickness: a bullwhip is heavy at the handle and tapers steadily to a thin fall, ending in a light cracker. So ask what happens as a travelling loop moves into thinner and thinner material.

The loop carries a certain amount of motion with it. When it reaches a section holding less mass, the same motion must be expressed by a smaller quantity of rope — and a smaller mass carrying the same energy has to be moving faster. So the loop accelerates simply by advancing into thinner rope, and the taper means it never stops doing so. That is the amplifier: a wave running down a decreasing-mass line speeds up along the way, and everything the arm did is progressively concentrated into less and less material.

Which gives us a prediction we can test. If the tip goes fast enough, it will exceed the speed at which air can get out of its way — about 343 metres per second in ordinary conditions. Beyond that, the air cannot smoothly part; the pressure disturbance piles up into a shock front, and a shock front reaching an ear is heard as a bang. The whip crack, on this account, is a small sonic boom.

Is that what actually happens, or merely a satisfying story? It was checked. High-speed shadowgraph imaging in the late 1990s photographed the shock wave leaving the tip, and Goriely and McMillen's 2002 analysis of the whip's shape tied the tip's motion to the emission of the crack. Their work also corrected the folk version of the mechanism: the simple "momentum is conserved so the thinner part goes faster" argument is not sufficient on its own, because the tension in the whip is doing work all the while. The acceleration comes from the taper and from the loop tightening as it runs.

One more finding from that work overturns the natural picture. You might imagine the tip cracking at the instant it reaches the end of its travel, like the last carriage of a train being snapped. In fact the tip passes the speed of sound well before the loop arrives at the end — the whip is already supersonic while there is still rope to run through, so by the moment the crack is emitted the tip is travelling considerably faster than sound.

It helps to be clear about what is moving and what is not. The whip's material does not travel from handle to tip; the loop does, and only the last small length of the whip ever reaches great speed. Nearly all of the whip is doing something quite slow while a very short segment of it is going supersonic.

It is also worth resisting an over-tidy version of the energy argument. Saying "the same energy in less mass means higher speed" gets the direction and the intuition right, but a whip is not a closed system while the crack is developing — the hand and the whip's own tension continue to feed it, and the loop's radius is shrinking at the same time. The honest summary is that the taper is the essential feature, that the acceleration is real and measured, and that the full accounting needed the careful treatment Goriely and McMillen gave it rather than a one-line conservation argument.

Finally, the sound's character. An aircraft's sonic boom is a low double thud because the object is large and the shock persists; the whip's tip is tiny and the event lasts a fraction of a millisecond, so the same physics arrives at the ear as a crack rather than a rumble. A difference of scale, not of kind.

• What taper profile maximises tip speed for a given whip length.
• How the same amplification appears in tsunamis reaching a coast and in a horn concentrating sound.

Every term the collection defines is gathered in the glossary.