You’re reading these words because light waves are bouncing off the letters on the page and into your eyes. The sounds of the rustling paper or beeps of your computer reach your ear via compression waves travelling through the air. Waves race across the surface of our seas and oceans and earthquakes send waves coursing through the fabric of the Earth.
As different as they all seem, all of these waves have something in common – they are all oscillations that carry energy from one place to another. The physical manifestation of a wave is familiar – a material (water, metal, air etc) deforms back and forth around a fixed point.
Think of the ripples on the surface of a pond when you throw in a stone. Looking from above, circular waves radiate out from the point where the stone hits the water, as the energy of the collision makes water molecules around it move up and down in unison. The resulting wave is called “transverse” because it travels out from the point the stone sank, while the molecules themselves move in the perpendicular direction. A vertical cross-section of the wave would look like a familiar sine curve.
Sound waves are known as “longitudinal” because the medium in which they travel – air, water or whatever else – vibrates in the same direction as the wave itself. Loudspeakers, for example, move air molecules back and forth in the same direction as the vibration of the speaker cone.
In both cases, the water or air molecules remain, largely, in the same place as they started, as the wave travels through the material. They are not shifted, en masse, in the direction of the wave.
The one-dimensional wave equation (pictured) describes how much any material is displaced, over time, as the wave proceeds. The curly “d” symbols scattered through the equation are mathematical functions known as partial differentials, a way to measure the rate of change of a specific property of the system with respect to another.
On the left is the expression for how fast the material is deforming (y) in space (x) at any given instant; on the right is a description for how fast the material is changing in time (t) at that same instant. Also on the right is the velocity of the wave (v). For a wave moving across the surface of a sea, the equation relates how fast a tiny piece of water is physically deforming, at any particular instant, in space (on the left) and time (on the right).
The wave equation had a long genesis, with scientists from many fields circling around its mathematics across the centuries. Among many others, Daniel Bernoulli, Jean le Rond d’Alembert, Leonhard Euler, and Joseph-Louis Lagrange realised that there was a similarity in the maths of how to describe waves in strings, across surfaces and through solids and fluids.
Bernoulli, a Swiss mathematician, began by trying to understand how a violin string made sound. In the 1720s, he worked out the maths of a string as it vibrated by imagining the string was composed of a huge number of tiny masses, all connected with springs. Applying Isaac Newton’s laws of motion for the individual masses showed him that the simplest shape for vibrating violin string, fixed at each end, would be the gentle arc of a single sine curve. A violin string (or a string on any instrument, for that matter) vibrates in transverse waves along its length, which creates longitudinal waves in the surrounding air, which our ears interpret as sound.
Some decades later, mathematician Jean Le Rond d’Alembert generalised the string problem to write down the wave equation, in which he found that the acceleration of any segment of the string was proportional to the tension acting on it. The waves created by different tensions of the string produce different notes – think of how the sound from a plucked string can be changed as it is tightened or loosened.
The wave equation started off describing movement of physical stuff but it is much more powerful than that. Mathematically, it can also describe, for example, the movement of heat or electrical potential, by changing “y” from describing the deformation of a substance to the change in the energy of a system.
Not all waves need to travel through a material. By 1864, the physicist James Clerk Maxwell had derived his four famous equations for the interactions of the electric and magnetic fields in a vacuum around charged particles. He noticed that the expressions could be combined to form wave equations featuring the strength of the electric or magnetic fields in the place of “y”. And the speed of these waves (the “v” term in the equation) was equal to the speed of light.
This simple mathematical re-arrangement was one of the most significant discoveries in the history of physics, showing that light must be an electromagnetic wave that travelled in the vacuum.
Electromagnetic waves, then, are transverse oscillations of the electric and magnetic fields. Discovering their wave-like nature led to the prediction that there must be light of different wavelengths, the distance between successive peaks and troughs of the sine curve. It was soon discovered that wavelengths longer than visible light include microwaves, infrared and radio waves; shorter wavelengths include ultraviolet light, X-rays and gamma rays.
The wave equation has also proved useful in understanding one of the strangest, but most important, physical ideas in the past century: quantum mechanics. In this description of the world at the level of atoms and smaller, particles of matter can be described as waves using Erwin Schrödinger’s eponymous equation.
His adaptation of the wave equation describes electrons, for example, not as a well-defined object in space but as quantum waves for which it is only possible to describe probabilities for position, momentum or other basic properties. Using the Schrödinger wave equation, interactions between fundamental particles can be modelled as if they were waves that interfere with each other, instead of the classical description of fundamental particles, which has them hitting each other like billiard balls.
Everything that happens in our world, happens because energy moves from one place to another. The wave equation is a mathematical way to describe how that energy flows.
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