Physics · Ch 14 — Waves
Introduction
Introduction
14.1 Introduction
From Isolated Oscillations to Waves
The previous chapter dealt with objects oscillating in isolation — a mass on a spring, a simple pendulum. But what happens when many such oscillators are connected together? A material medium is exactly that: a collection of particles bound by elastic forces, where the motion of one particle affects its neighbours.
Drop a small pebble into still water. The surface is disturbed, and that disturbance does not stay at one point — it spreads outward in expanding circles. If you keep dropping pebbles, you see circles moving rapidly outward from the point of disturbance. It looks as though the water itself is flowing outward. But place a piece of cork on the surface: it bobs up and down but does not move away from the centre. The water mass does not travel outward with the circles. What travels is the disturbance itself.
The same happens when you speak. Sound moves outward from you, but there is no bulk flow of air from your mouth to the listener's ear. The disturbances in air are invisible — only our ears or a microphone can detect them.
These patterns, which move without the actual physical transfer of matter as a whole, are called waves.
A wave is a disturbance that propagates through a medium (or through empty space) transporting energy and information, without transporting the medium itself.
Waves carry energy, and the pattern of disturbance carries information that propagates from one point to another. All communication — speech, radio, television, the internet — depends on transmitting signals through waves. Speech is the production of sound waves in air; hearing is their detection. Often, one kind of wave is converted into another: sound waves become an electric current signal, which generates an electromagnetic wave transmitted by cable or satellite, and detection reverses the process.
Do All Waves Need a Medium?
No. Light waves travel through vacuum. Starlight reaches us across hundreds of light-years of nearly empty space. The most familiar waves — waves on a string, water waves, sound waves, seismic waves — are mechanical waves. They require a material medium for propagation and cannot travel through vacuum. They involve oscillations of constituent particles and depend on the elastic properties of the medium.
Electromagnetic waves (which you will study in Class XII) do not require a medium. Light, radio waves, X-rays — all are electromagnetic waves. In vacuum, every electromagnetic wave travels at the same speed :
A third kind, matter waves, are associated with the constituents of matter itself — electrons, protons, neutrons, atoms, molecules. They arise in the quantum mechanical description of nature. Though more abstract, they have practical applications: the electron microscope uses matter waves associated with electrons.
This chapter deals only with mechanical waves — those that require a material medium.
Historical Context
The scientific analysis of wave motion began in the seventeenth century, with contributions from Christiaan Huygens, Robert Hooke, and Isaac Newton. The physics of waves grew directly from the physics of oscillations — masses on springs and simple pendulums. Waves in elastic media are intimately connected with harmonic oscillations.
The Spring Model of Wave Propagation
Consider a collection of springs connected end to end, as in Fig. 14.1 of the textbook. If the spring at one end is pulled suddenly and released, the disturbance travels to the other end. What happens?
The first spring is stretched from its equilibrium length. Because the second spring is connected to it, it too is stretched or compressed, and so on. The disturbance moves from one end to the other, but each spring only executes small oscillations about its own equilibrium position.
Think of a stationary train at a railway station. The bogies are coupled by spring couplings. When an engine pushes the first bogie, the push is transmitted from one bogie to the next — the entire train is not bodily displaced, but the disturbance travels along it.
Sound Waves in Air
As a sound wave passes through air, it compresses or expands a small region. This changes the density of that region by an amount , which in turn induces a change in pressure in that region. Pressure is force per unit area, so there is a restoring force proportional to the disturbance — exactly like a spring.
The quantity analogous to the extension or compression of a spring is the change in density. If a region is compressed, molecules are packed together and tend to move into the adjoining region, compressing it in turn. The original region then becomes rarefied (less dense). If a region is rarefied, surrounding air rushes in, making the rarefaction move to the adjoining region. Thus compression and rarefaction move from one region to another, enabling the propagation of a disturbance through air.
Waves in Solids
In a crystalline solid, atoms (or groups of atoms) are arranged in a periodic lattice. Each atom is in equilibrium under forces from its neighbours. Displacing one atom (while holding the others fixed) creates restoring forces — again, exactly like a spring. We can think of the atoms as points connected by springs between every pair.
Do not confuse the motion of the medium's particles with the motion of the wave itself. In a mechanical wave, each particle oscillates about its equilibrium position; the wave moves forward, but the particles do not travel with it.
The subsequent sections of this chapter will discuss the characteristic properties of waves in detail.
Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.
Fig. 14.1 is a simple but powerful sketch. It shows a horizontal chain of three coil springs. The leftmost end of the chain is labelled A, and a leftward-pointing arrow is drawn at A to indicate that you pull it. The rightmost end of the chain is fixed to a hatched rigid wall — that wall does not move.
The figure is not a graph; it has no axes. It is a physical diagram meant to make one idea concrete: when you disturb one part of a connected system, that disturbance travels. You pull end A to the left, stretching the first spring. That stretch pulls on the second spring, which then pulls on the third, and finally the wall feels a tug. The disturbance — the stretching — has moved from A to the wall, even though no single piece of spring has moved all the way across. Each coil only shifted a little relative to its neighbour.
This is the core of wave motion: a disturbance propagates through a medium without the medium itself flowing along with it. The textbook makes this point by contrasting the spring chain with the earlier example of a pebble in a pond. In both cases, the material (water or spring coils) moves locally — up and down, or back and forth — but the pattern of disturbance moves outward. The cork pieces on the pond do not travel with the ripples; the spring coils do not travel from A to the wall.
The key formula that emerges from this picture is the wave speed for a transverse wave on a string (or a longitudinal wave in a spring, depending on how you pull). For a string or spring under tension, the speed of the wave is:
Here is the tension in the string (or spring) — the force you apply when you pull end A — and is the mass per unit length of the string (or the linear mass density of the spring). The formula tells you that a stiffer pull (larger ) makes the disturbance travel faster, while a heavier string (larger ) slows it down.
The figure also sets up the idea of a pulse. The single pull at A creates one hump of disturbance that travels to the right. If you kept pulling and releasing A rhythmically, you would generate a continuous train of pulses — a wave. The spring chain is a one-dimensional model, but the same principle applies to two-dimensional surfaces (like a water surface) and three-dimensional media (like air for sound).
A common mistake is to think that the spring coils themselves move from A to the wall. They do not. Each coil oscillates about its own equilibrium position. What moves is the shape of the disturbance — the region where the coils are closer together or farther apart. The figure makes this visible: the first coil stretches, then the second, then the third, but no single coil leaves its original spot.
In short, Fig. 14.1 is a visual definition of a wave: a disturbance that travels through a medium, carrying energy and information, while the medium itself stays put. The formula is the first quantitative handle on how fast that disturbance moves.