The Physics
Opus in profectus

The Nature of Waves

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The basics


A wave is a disturbance that propagates through a medium. There are three words in that definition that may need unpacking: disturbance, propagate, and medium.

Let's list a few key examples of wave phenomena and then connect them to this definition. The first example that comes to mind when most people hear the word wave are the kinds of waves that one sees on the surface of a body of water: deep water waves in the ocean or ripples in a puddle. The most important kinds of waves for humans are the waves we use to sense the world around us: sound and light.

Imagine a calm pool. The surface is flat and smooth. Drop a rock into it. Kerploop. The surface is now disturbed. It is higher than normal in some places and lower than normal in others. The disturbed water at the point of impact disturbs the water next to it, which in turn disturbs the water next to it, which disturbs the water next to it, and so on. The disturbance spreads outward, transmits, or propagates. The medium through which this disturbance propagates is the surface of the water.

Imagine a quiet room. The air inside is still. Drop a book onto a table in that room. Thwap. The air between the book and the table is squeezed out in a fraction of a second. The air pressure in that rapidly decreasing gap rises above normal and then rebounds. The rise and fall of pressure is like the rise and fall of the surface of the pool in the previous example. The air under the book bumps the air on the edges of the book, which bumps the air next to it, which bumps the air next to it, and so on. The medium through which this disturbance propagates is the air.

Those were the easy examples. Water waves and sound waves are examples of mechanical waves — waves that propagate through a material medium. Light is not so easy to understand as a wave, which is why there are multiple sections of this book devoted to it. Still, I am going to try to describe it briefly.

Imagine a dark cavern, deep within the Earth. The electric and magnetic fields inside are relatively static and unchanging. Strike a match. Skeerach. The atoms of carbon in the wood of the matchstick combine with the atoms of oxygen in the air releasing heat. The heat agitates the atoms of the combustion products resulting in the phenomenon known as fire. The electrons bound to the rapidly vibrating atoms disturb the electric and magnetic fields in the space surrounding them. These fields are "elastic" in a sense. A wiggle in the fields in one place causes a wiggle in the fields nearby, which causes a wiggle in the fields nearby, and so on. These wiggles eventually make it to your eye, which you perceive as light. The electric and magnetic fields that fill all of space are the medium.

essential property

Waves transfer energy, momentum, and information, but not mass.

A naive description of a wave is that it has something to do with motion. But the motion of a wave on the water is not the same as the motion of the water from a hose. When waves move over the surface of the ocean, where does the ocean go? Nowhere. When waves reach the shore, does the water accumulate into great heaps? No. The water moves in and out, and the ocean stays behind. Even when huge tsunamis strike, the wall of water deposited on the land eventually drains back into the sea. In this case, no net transfer of mass has occurred.

Compare this to the water from a hose. The water comes pouring out the open end and stays where it lands forming a puddle or drains away to some other location. It most certainly does not jump back into the hose. In this case, mass has been transferred from one location to another.

Any sensible person who owns waterfront property should be familiar with the word erosion. Ocean waves (or waves on the Great Lakes for that matter) break on the shore, beating the rock and soil into submission and pulling it away. This material will never return. (If there were no plate tectonic forces lifting the land up in some places or volcanoes creating new land in others, the Earth would be covered in a global sea of uniform depth.) A force (F) has been exerted and mass has been displaced (s). Work has been done (W = Fs). The ability to do work is one definition of energy (W = ∆E). Thus waves transfer energy.

Sticking with the example of ocean waves, anyone who surfs knows that waves transfer momentum. I have less to say on this subject.

Sound and light are the two primary examples of the way we gather information around us as humans. We have specialized sensory organs called ears and eyes for doing just that.


Here's a list of some phenomena or activities that satisfy the definition of a wave given above.

not examples

Just because the word wave is used doesn't mean the thing being described is a wave in the sense used in this book.

Classification of waves

by medium

Waves can be classified according to the medium through which they propagate.

mechanical waves
…require a material medium. Sound is the most important example of a mechanical wave. Sound waves cannot travel through a vacuum.
electromagnetic waves
…propagate through the electric and magnetic fields that are everywhere in space. Light is the most important example of an electromagnetic wave for humans. Electromagnetic waves can propagate through transparent materials and can also propagate through empty space. The basic electromagnetic spectrum in order of increasing frequency is radio waves, microwaves, infrared, visible light, ultraviolet, x-rays, and gamma rays.
gravitational waves
…propagate through the gravitational field that is everywhere in space. Similar to electromagnetic waves, gravitational waves can propagate through matter or empty space. Gravitational waves are a consequence of Einstein's theory of general relativity and should not be confused with gravity waves, which is a name for the kind of waves found in deep water (a.k.a. ocean waves). Gravitational waves were predicted to exist in 1916 (or 1918) but were not confirmed with direct observation until 2015.
matter waves
…refer to the quantum mechanical description of fundamental particles like electrons and quarks as a wave. Each particle type has a wave equation that describes a particle field which can then be used to generate wave functions for the individual particles of that type. Squaring a wave function results in a probability distribution that describes the likelihood of finding a particle at a specific location. In a sense, mater waves are probability waves.

by the type of disturbance

Waves can be classified according to the type of disturbance — meaning its relative direction or shap. There is a lot that can be said about this organizational scheme. I'm starting this part of this section with a quick summary in table form followed by a rather detailed follow up.

Waves classed by the type of disturbance
name disturbance is… examples
perpendicular to propagation light and all electromagnetic waves, gravitational waves, matter waves, nerve impulses, peristalsis, secondary seismic waves (S waves a.k.a. shear waves), locomotion in snakes and eels, stringed instruments, drums
parallel to propagation sound, shock waves, traffic jams, primary seismic waves (P waves a.k.a. pressure waves), locomotion in worms and slugs, wind instruments, density waves in galaxies
circular or elliptical ocean waves (gravity waves), ripples (capillary waves), tsunamis (tidal waves), Rayleigh seismic waves (R waves a.k.a. ground roll)
a twist bridges, skyscrapers, airplane wings, wires
hard to categorize dominoes, Rube Goldberg devices, popping a soap bubble, some forms of structural collapse, locomotion in centipedes and caterpillars

transverse waves

A transverse wave is one in which the direction of the disturbance is perpendicular to the direction of propagation. The word transverse describes something pointing in a sideways or lateral direction. As dynamic phenomena, waves are better represented with animations than with static images. Click on the static image below to see a transverse wave in action.

A grid of dots, shifted as if a transverse wave was passing throughAnimate

A cartoon representation of this kind of wave is your classic wiggly line. People with a bit of math knowledge will tell you they drew a sine curve. Those with a little bit more math knowledge will say they drew a sine or cosine curve.

A sine curve with the high parts labeled crests and the low parts labeled troughsMagnify

The high parts on a curve like this are called crests. The low parts are called troughs. Since directions like up and down don't always make sense for waves, what the parts really represent are the maximal changes. The points labeled crests are points corresponding to a maximal increase of the changing quantity in a whatever direction is decided to be positive. The points labeled troughs are the points corresponding to the maximal change in the opposite direction.

Pronouncing words ending in -ough in English is often a mystery. The word trough rhymes with off. A trough is what one uses to provide food and water to livestock and other domesticated animals — typically a long, narrow open container that an animal would dip its head down into. The word crest rhymes with best. A crest is something at the top of something. Many birds, usually male birds, have crests. Hills and mountains are are sometimes said to have crests. The crest on a men's sports jacket or a school uniform gets its name from the heraldic crests that were originally worn on knight's helmets above the visor. Crests are up high. Troughs are down low.

The most important example of a transverse wave for humans is light. Most of what I am about to say in the following bullets will really be discussed elsewhere in this book.

Lots of musical instruments make use of transverse waves to generate their characteristic sound.

Some animals propel themselves by sending transverse waves down the length of their bodies.

longitudinal waves

A grid of dots, shifted as if a longitudinal wave was passing throughAnimate


complex waves

classed by orientation of change

A surface wave on the boundary between air and waterAnimate

torsional waves

A row of connected parallel bars, twisted as if a torsional wave was passing throughAnimate

by duration

classified by duration


by appearance

Waves can be classified according to what they appear to be doing.

traveling waves
…are waves that appear to be propagating. This might seem like a distinction made by the Department of Redundancy Department, since propagation is a key part of the definition of a wave, but there actually are waves that do not appear to be going anywhere.
standing waves
…are waves that do not appear to be propagating. (They are also called stationary waves.) This appearance is actually an illusion that arises when a wave meets its own reflection under the right circumstances. A more complete discussion of this phenomenon appears in a in a separate section of this book.

Now look at these pretty, moving pictures.

Animation of a traveling waveA traveling wave in action

Animation of a standingling waveA standing wave in action