Waves are a fascinating natural phenomenon, worth studying in their own right: for the variety and beauty of the phenomena they give rise to, for the elegance and ingenuity of the mathematical and experimental methods developed over the centuries to describe them, and because waves pervade a great many aspects of our lives through the most diverse technological applications.

What is a wave

Before being a scientific term, the word wave (Italian onda) is a common word with a very ancient history. It comes from the Latin word unda, which, as in Italian, refers to the ripples on the surface of water. But the link with water is much deeper. The Latin term has the same origin as the Greek root hyd- found in the word hydor, which means precisely "water", and which survives in all the words beginning with "hydr-", such as hydraulic, hydrant, etc. The root, however, is even older, and goes back to Indo-European, that is, to a time when almost all European languages (and some Asian ones) were a single language. The Indo-European root is vud- or vad-, and it is still recognisable today in a great many ancient and modern languages, referring to water or closely related concepts (Sanskrit udnah, Hittite watar, Gothic wato, English water, German Wasser, Flemish water, Lithuanian vanduo, Gaelic uisge, Russian voda, and similar forms in many Slavic languages). Even though waves in water are actually just one particular type of wave (see for example transverse and longitudinal waves and the wave equation in water), I like to think that such an ancestral word is linked to a concept that is equally widespread, and equally important, in physics.

But an exact science must define its terms precisely, sometimes even giving certain words a meaning different from the everyday one.

So what does wave mean in physics?

a wave is a disturbance that propagates through space and can carry energy from one point to another. The disturbance consists of the variation of any physical quantity (e.g. a variation of pressure, temperature, electric field strength, position, etc.)

Waves: instructions for use

Here are just a few examples:

1. Many natural phenomena are described in terms of waves
   • Waves in water are both the prototype of a wave, as illustrated in the previous section, and a rather special case, since they mainly involve the boundary surface between two media (air and water), and therefore have less general features than waves travelling in a homogeneous medium (see transverse and longitudinal waves, wave equation in water).
   • The question of the nature of light is a typical example of a case that is harder to settle. Opposing theories supporting the corpuscular and the wave nature of light have followed one another since antiquity, and found an original solution in contemporary physics, according to which both radiation and matter have features common to both particles and waves when observed at a microscopic level (see wave ray, uncertainty).
   • In the case of electromagnetic waves, because of the complexity of their interaction with matter, waves of the same physical nature can have very different effects depending on their frequency, ranging from radio waves used in telecommunications, to visible light, to dangerous ionising radiation such as X-rays and gamma rays.
   • Sound waves, besides being one of the physiological means of human communication, are actually a particular type of mechanical wave, to which the whole section Sound and its perception is devoted.
   • Mechanical waves, moreover, concern many aspects of our everyday life, from the most obvious small-scale ones involving oscillations and vibrations (see resonance phenomena in mechanical systems), to less evident but no less important large-scale ones, such as planetary waves, which arise in the fluids of the ocean and the atmosphere because of the rotation of the Earth.
2. If the way a wave propagates in a certain medium is known, the wave itself becomes a powerful tool for investigating the properties of that medium. For example:
   • measuring the properties of seismic waves, such as amplitude, direction of oscillation and arrival times at the seismograph, not only shows where the epicentre of an earthquake is, but can give valuable information about the internal structure of the Earth.
   • Most properties of matter at the molecular or atomic level are measured by irradiating a sample with waves (usually electromagnetic, such as visible light, ultraviolet light, X-rays, gamma rays, but also beams of electrons or neutrons) and observing the response of the sample.
   • The same principle underlies many modern diagnostic techniques in medicine: X-rays (radiography) or ultrasound (ultrasonography) are used to "see" parts of the body that cannot be seen from outside with ordinary visible light.
   • The science of conservation and restoration of works of art also uses techniques based on suitable spectroscopies (fluorescence, absorption, transmission in the optical, infrared, ultraviolet and X-ray bands)
   • Forensic science is today a fundamental tool for law enforcement: based on a set of physical, chemical and biological techniques, it makes it possible to gather evidence for reconstructing a crime scene, which is often decisive for the investigation. Here too, the use of electromagnetic waves of various kinds to reveal the presence of various types of traces, or to analyse them, has become common practice.
   • Illuminating certain substances with waves of a certain type often reveals features that would otherwise remain hidden: in mineralogy it is common practice to use Wood's light (ultraviolet) to check the luminescence of minerals, and police forensic departments use similar methods to reveal organic traces that would otherwise be invisible at a crime scene.
   • Measuring the change of direction of a light wave when it passes from air to glass tells us the value of the refractive index of glass relative to that of air (see the section refraction), but in general the colour or transparency, the reflectivity, and all the optical properties of a material result from the interaction (which can be very complex) of that material with electromagnetic waves.
   • Observing how a wave is reflected or transmitted at the interface between two different substances shows directly how much the media oppose the transmission of energy by the wave (see the section refraction in one dimension, in Italian), and allows us to optimise the coupling between different media (such as a hi-fi amplifier and a microphone, or loudspeakers) to obtain maximum efficiency
   • Waves make it possible to study phenomena that are distant in space and time: astronomy and astrophysics base their observations on the study of the spectrum of electromagnetic radiation coming from celestial bodies, interstellar matter and deep space. The radiation studied covers the entire electromagnetic spectrum, and comes from such distances that its origin can date back to times as remote as the first moments of the universe. Moreover, Doppler frequency shifts give us information about the evolution of the universe.
   • Finally, the general theory of relativity predicts that gravitational waves, consisting of a fluctuation in the curvature of spacetime, can be generated in certain cases (for example by a binary system, or in a supernova explosion) and propagate through spacetime.
3. Waves carry energy and momentum. In this case we speak of radiant energy, or radiation.
   • No science-fiction film lacks extremely powerful and lethal, if imaginary, "ray guns", and soon after electromagnetism was put in order at the end of the 19th century, scientists were looking for ways to use electromagnetic waves to transfer energy (see for example this patent for a wireless system by Tesla, from 1900). This shows how strongly the idea of transmitting energy at a distance affects the imagination. A wide range of applications makes the transport and use of energy by means of waves more efficient, from the laser to the microwave oven. However, the first and simplest device for radiating and directing the energy produced by a source is the antenna (see also antennas and interference).
4. The transmitted energy can become information, and waves become the most effective and fastest means of communication.
   • Waves lend themselves to carrying information through a modulation of their properties. For example, in radio communications one distinguishes amplitude modulation (AM), frequency modulation (FM), phase modulation (PM), etc.
   • Sound waves are the natural medium of vocal communication, while electromagnetic waves are the main means of long-distance communication on our planet.
   • Much of the science of telecommunications has been devoted to the problem of how to generate signals suitable for transmitting different types of information, and how to transmit them over the greatest possible distance while minimising the noise in the transmission channel. A large part of analogue and digital electronics deals with the generation, conditioning, transmission and reception of signals.

As you can see, some phenomena and applications are very well known and widespread, even though actually understanding exactly what happens in each case can be complex. The good news is that waves have some very general features that allow the main aspects of their behaviour to be understood simply. The pages of this site are devoted to presenting these aspects.

Main features of waves

Let us now try to grasp some features of waves on the basis of a few simple examples

What vibrates in a wave?

As already said: a wave is a disturbance that propagates through space and can carry energy from one point to another. The disturbance consists of the variation of any physical quantity (e.g. a variation of pressure, temperature, electric field strength, position, etc.) around its equilibrium value (which corresponds to the absence of disturbance).

• The most important point emerging from the definition is that a wave is a disturbance that travels: at first sight, for this to happen, a medium seems to be necessary which, as the wave passes, undergoes some form of local oscillation without transport. That is, the particles of the medium do not travel with the wave; only the disturbance they undergo travels. This is what happens in so-called mechanical waves. So the answer to the question in the title of this section is that what vibrates are the particles of the medium in which the wave propagates

• This apparently obvious answer was considered unconditionally valid until 1905 (the year in which Einstein wrote the paper "On the electrodynamics of moving bodies", introducing the special theory of relativity): after all, it was thought, if something propagates there must be a medium that transmits the propagation. In reality electromagnetic waves, whose existence was predicted theoretically by the Scottish physicist J. C. Maxwell in 1865, have the extraordinary property of propagating in vacuum (if by vacuum we mean the complete absence of matter). What "vibrates" in this case is an immaterial entity called the electromagnetic field: despite being immaterial, this field produces real effects familiar to everyone (for example, modern telecommunications rely on the propagation of such fields, or, even more simply, light is nothing but a particular electromagnetic wave)

How the disturbance travels

• The classic example is the "Mexican wave" in a stadium, in which "fans" are seen standing up and sitting down while always staying in their own seats, without following the disturbance that moves along the stands
• Another example is a flute, in which the air is set into oscillation by the musician's breath
• How do the two examples differ? In the first case it is easy to see that what oscillates (the little people) does so in the direction perpendicular to the direction of propagation of the disturbance (we speak of transverse waves), while in the second what oscillates (the individual particles) does so in the direction parallel to the direction of propagation of the disturbance (we then speak of longitudinal waves)

longitudinal wave	transverse wave
The wave consists of the propagation of the "compression" disturbance, which corresponds to the orange colour on the connections between the masses.	The wave consists of the propagation of the "vertical displacement" disturbance, which is associated with compressions and expansions in the upper and lower surfaces of the medium.

There are actually other types of waves in which the relation between the direction of propagation and the direction of oscillation is more complicated. For an animated overview, visit the page transverse and longitudinal waves

an original example of a longitudinal "Mexican wave"	traditional "Mexican wave"
© 1992 Studio Ghibli	© D. Helbing, I. Farkas, T. Vicsek

"Collisions" and interactions between waves

• The most characteristic trait of waves, beyond their peculiar physical description, emerges when two waves arrive at the same point in space at the same instant. How do they "interact"? Do they collide as, for example, two carts would, crumpling up after the impact?

The answer is surprising: a picture is worth a thousand words!!

"Constructive" interference	"Destructive" interference
two perfectly linear pulses "collide", superpose, and continue their journey undisturbed	two pulses have opposite sign: there is a moment when their sum is zero, but immediately afterwards they re-emerge and continue undisturbed

Waves "pass through" each other undisturbed and, when they occupy the same position in space, their amplitudes add or subtract. Physicists say that waves obey the superposition principle and that, at the moment of the "collision" (the term really no longer seems appropriate!), they interfere. Actually, the superposition principle does not hold unconditionally: however, even in situations where it does not hold, waves undergo an interaction that makes their amplitudes reinforce or cancel each other (although no longer exactly as the sum or difference of the amplitudes).

How a wave is described

In everyday perception, when we observe a wave we immediately notice some features that make it complex to describe: the wave extends through space and propagates, changing its "profile" over time. This change often happens repetitively, giving us a sense of regularity that we often associate with pleasant sensations (waves can rock us, we can listen to music, we can get a tan...). At other times, instead, the wave appears as a single threatening pulse advancing relentlessly (think of a tsunami). The phenomena involving waves are surprisingly varied. We will try to describe many of them qualitatively; of course, to reach a more advanced quantitative treatment we will be "forced" to introduce physical and mathematical concepts (e.g. period, frequency, wavelength, delay time, etc.) If you want to explore these aspects further, visit the page How is a wave described?

Further reading and links

• If you want to see animations of wave phenomena in all their "spectacle", you cannot miss our virtual wave laboratory. For now, have fun "playing" with the app, selecting oscillating sources, changing frequencies, building walls. Do not ask yourself too many questions: observe and get curious. We recommend the 3D view!!

• Every type of wave is subject to phenomena that are, so to speak, its trademark: in the sections devoted to reflection, refraction, interference, diffraction and the superposition principle you will find plenty of further detail. Reading these pages, even without specific mathematical training, you will fully appreciate the breadth and complexity of the phenomena that can be described by the unifying concept of a wave.

• If you are a musician, you cannot miss the section devoted to sound waves: there you will find the bare essentials (there is not even a single formula!) for understanding the particular way a wave is described when it represents a sound phenomenon. Then you will not be able to resist visiting the section From sounds to music; there you will "play at home", but you will still find surprises: the musical language you know so well will appear to you in a new light.

• If you are interested in understanding how musical instruments work, visit the relevant page