Drop a stone into a calm pond. Ripples spread outward from the splash in circles, one inside the other. A leaf floats on the water some distance away. When the ripples reach the leaf, the leaf only rises and falls. It does not travel outward with the ripples.
At a cricket match at Eden Gardens, the crowd makes a wave of cheering fans. The pattern of people standing up races round the stadium. Yet every person stays in his or her own seat.
Tie one end of a long rope to a doorknob and flick the other end once. A bump runs along the rope towards the door. Each small piece of the rope only moves up and then down again.
Three examples, one phenomenon
Look again at the three examples. In each one, something travels a long way, and something else hardly moves at all. The ripple crosses the pond, but the water under the leaf stays where it was. The cheering pattern goes round the stadium, but the people stay in their seats. The bump runs the length of the rope, but each piece of rope stays at its own place along the rope.
The thing that travels is a disturbance. A disturbance is a change from the undisturbed state of the material. Water is raised above its level, a person stands instead of sitting, a piece of rope is lifted off its straight line. The material that carries the disturbance is called the medium. Water, the crowd and the rope are the three media here.
Fig 1.1 puts the rope on the page. The hand moves up and down once, and the rope near the hand follows it. That piece of rope then pulls its neighbour up, and the neighbour pulls the next piece. So the bump is passed from piece to piece along the rope. The readout gives the position of the pulse, which grows from \(1.9\,\text{m}\) at \(t = 1.4\,\text{s}\) to \(8.4\,\text{m}\) at \(t = 4\,\text{s}\). The painted piece is at \(5.0\,\text{m}\) the whole time.
A ripple passes a leaf floating on a pond. Where is the leaf after the ripple has gone by?
A wave is a propagating disturbance. It transports energy and momentum from one point to another without any net transport of matter. The medium through which the wave travels oscillates, but it does not move permanently with the wave.
Consider the energy first. The leaf was at rest before the ripple came. Then it moved up and down, so it had kinetic energy, and it rose, so it gained potential energy. That energy came from the stone, which fell into the water far away. No water travelled from the splash to the leaf. So the energy of the stone must have travelled across the pond in the ripple.
A wave carries momentum too. When sea waves strike a wall, they push on it. On Marine Drive in Mumbai, large concrete tetrapods lie along the sea front for this reason. Each monsoon wave that hits them delivers a push. The water that hits the tetrapods did not travel from far out at sea. The push travelled there in the wave.
Now consider the matter. When the wave has passed, each part of the medium is back where it started. The water under the leaf, the fan in the stadium and the painted piece of rope all end at their starting places. Their net displacement is zero. This is what "no net transport of matter" means. Each part of the medium oscillates, that is, it moves to and fro about its own place, while the wave passes.
Do not say that a wave carries the medium along with it. A wave carries energy and momentum. The medium stays behind. The water of a river does flow downstream, but that flow is a current. A current carries water; a wave does not.
A wave travels along a rope from one end to the other. Which of these does the wave carry from one end to the other?
A wave of cheering fans goes round a stadium. What travels round the stadium?
A stadium wave moves about \(20\) seats a second, and one seat is about \(0.6\,\text{m}\) wide. The ring of seats round the stadium is about \(600\,\text{m}\) long. Estimate the time the wave takes to go once round, in seconds.
Three physicists studied video recordings of stadium waves at large football matches and described them in the journal Nature in 2002. They treated the crowd as a medium in which each fan responds to the neighbours. They found that the wave usually travels at about \(12\,\text{m/s}\), about \(20\) seats a second. Its width is about \(6\) to \(12\,\text{m}\). A few dozen fans standing up together were enough to start one. Their paper treats the crowd exactly as this lesson does: the pattern travels, and every fan stays in the same seat.
Why the rope comes first
This chapter discusses some basic properties of waves. Of the three examples, the wave on the rope is the simplest to analyse. The rope lies along one line, so one coordinate \(x\) gives the place of each piece. One more number, the displacement \(y\) of that piece from the straight line, describes the disturbance. The wave on the rope therefore needs only one dimension for its analysis.
The ripples on the pond spread over a surface, so they need two dimensions. Sound spreads through the air in every direction, so it needs three. So the chapter builds its understanding step by step on the rope. Later chapters then carry the same ideas to the other waves.
Why does this chapter build its description of waves on a rope rather than on ripples on a pond?
A fishing boat is anchored far from the place where a large ship passes. A few minutes later the boat starts to rock. How did energy reach the boat, when no water flowed from the ship to the boat?
The passing ship disturbed the water near it. Each part of the water pushed and pulled on the water next to it, so the disturbance was passed from place to place across the sea. The disturbance is a wave. The wave carried energy and momentum to the boat, while each part of the water only moved about its own place. When the wave reached the boat, it made the boat rock.
On 26 December 2004, an earthquake under the sea near Sumatra lifted a large area of the sea floor. The disturbance travelled across the Bay of Bengal as a wave, a tsunami. In the deep ocean, the water rose and fell by less than a metre, and a ship there would hardly have noticed it. The wave still carried an enormous amount of energy. It reached the coast of Tamil Nadu about two hours later and did great damage. The water near Sumatra did not travel to Chennai. The energy did.
Return now to the leaf, the crowd and the rope. In all three, the disturbance travels: the ripple, the pattern of standing fans, the bump. In all three, the medium stays behind: the water, the fans and the pieces of rope end where they began. The ripple and the bump carry energy that can make something far away move. That is why one word, wave, fits all three.
- Saying that the water of a ripple moves outward with the ripple.The disturbance moves outward. The water under it only rises and falls about its own place.
- Thinking that a wave carries nothing, since the medium stays behind.A wave carries energy and momentum. Only matter is not carried.
- Confusing a wave with a current.A current carries the material itself, as a river carries water. A wave carries a disturbance through material that stays in place.