[quote]For all is like an ocean, all flows and connects; touch it in one place and it echoes at the other end of the world.[/quote] -Fyodor Dostoyevsky, [url=https://www.goodreads.com/work/quotes/3393910]The Brothers Karamazov[/url]

Waves are everywhere.  Some we see, most we do not.  Some we hear.  Most we do not hear.  Besides common waves like ocean waves and sound waves, light is - among other things - a wave.  Matter is at its core a wave, "solid" materials will invariably have waves propagating through them (called phonons) and any polyatomic molecules will always oscillate in a wavy fashion even at absolute zero. I used quotes around the word "solid" for a reason. Toward the end of the course when we discuss things you've likely taken for granted - things like solids and particles - we'll see that the terms don't well describe anything in nature. [br][br]In the first part of 2016, the scientific community announced the first confirmed detection of gravitational waves by the LIGO research group.[url=https://www.ligo.caltech.edu/]Caltech News Announcement of Gravitational Waves[br][/url][br]One interesting fact about waves in nature is that where there is a wave there tends to be a corresponding entity we call a particle (or quantum of the wave) for lack of a better description.  For electromagnetic (light) waves, we have quanta of light called photons, for molecular vibrational waves we have phonons, for gravitational waves we presumably have gravitons - although there is no data to prove their existence thus far.  [br][br]The idea of those “particles” or quanta of the various fields, is that one quanta is the smallest possible excitation of the wave.  We will speak more of this in future chapters.[br][br]Right now I want to extend your understanding of waves in general and the associated mathematics to describe and manipulate them. This is important for our basic understanding of nature.

We will start with traveling waves.  A traveling wave is not unlike an ocean wave.  Obviously if we had frozen waves on an ocean, they are not traveling. They'd look just roughly like math textbook sine waves.  [br][br]If we observe waves of all types, we see that all waves encountered in nature travel.  Even when they look like they don't - a phenomena called standing waves - or when we can describe them mathematically as if they are not traveling, they really are.  Standing waves will play a critical role in the context of musical resonance, and in many other resonance phenomena.[br][br]Why waves travel is that they are really traveling disturbances.  Think about splashing in a pond.  You disturb the water with your hand and the disturbance propagates outward.  It is these traveling disturbances that we call waves.     [br][br]The mathematical description of a 1D traveling wave of any type is:[br][center][math]y(x,t)=A \sin(kx-\omega t).[/math][/center]

Our convention for a traveling wave is a wave that will travel in the positive x direction as time progresses.  Its amplitude is [math]A[/math].  What that amplitude represents depends on what type of wave we are describing. [br][br]The value of [math]y[/math] at some point [math](x,t)[/math] gives the local magnitude of the disturbance - regardless of the type of disturbance.  That word disturbance may sound vague, but that's because depending on the type of wave we are dealing with, that disturbance will mean something different.  In the case of sound waves we can think either of the variation in air pressure or of the molecular oscillation amplitude as the disturbance.  In the case of light waves the disturbance is both the electric and magnetic field vectors.  For a wave on a guitar string the disturbance is the amplitude of the vibration of the string perpendicular to its own length.  [br][br]In all of these cases, we describe the traveling waves in the same mathematical fashion, so at first we won't worry much about what type of wave we are dealing with.  So if we return to the equation, it's worth noting that the [math]-\omega t[/math] term causes the wave to propagate in the postive x direction as time increases.  You can see that in the animation of a traveling wave below.[br][br]The [math]k[/math] term is called the [b]wave number [/b]and [math]\omega[/math] is called the [b]angular frequency[/b]. Let's look at how these terms relate to things like wavelength and ordinary frequency since it's always a good idea to relate new ideas to ones with which you are already familiar.[br][br]The whole argument of the sine wave, or [math]kx-\omega t[/math] is called the[b] phase[/b] of the wave. We will use the symbol [math]\phi[/math] to describe the total phase. Let me remind you that it is generally true of sine waves or cosine waves that adding (or subtracting) [math]2\pi[/math] phase leads to the wave repeating itself. This just means that mathematically [math]sin(\phi)=sin(\phi \pm 2\pi)[/math]. In the case of our traveling wave equation, we see that: [br][center][math]y(x,t)=A\sin(kx-\omega t)=A\sin(kx-\omega t\pm2\pi).[/math][/center]

By definition, waves repeat themselves [b]spatially[/b] (or in space) after a distance of one wavelength [math]\lambda[/math].  This means we should expect that [center][math]y(x,t)=y(x+\lambda,t)[/math].[/center]Putting this together with the fact that we can shift a sine wave by [math]2\pi[/math] phase and have it repeat itself, means we can write: [math]A\sin(k(x+\lambda)-\omega t)=A\sin(kx-\omega t\pm 2\pi)[/math].  Therefore [math]k\lambda=\pm 2\pi[/math].  The convention is to use the positive solution, and to define [br][center][math]k=\frac{2\pi}{\lambda}.[/math]  [/center]The constant [math]k[/math] is called the wave number and has units of [math]\frac{rad}{m}[/math].  This notion of radians per meter seems odd, but it literally means that if we know [math]k[/math] for a wave and wonder how many radians of phase fits in some portion of space of a given length [math]\Delta x[/math], the answer will be phase [math]\Delta\phi = k\Delta x[/math].  If [math]\Delta\phi[/math] happens to be [math]2\pi[/math] this would mean that exactly one wavelength fits in that space.  Try to digest this concept of wave number now, because it will be used a lot in our upcoming examples.  Perhaps a better term than wave number would have been [i]phase density[/i], but that's not what anyone calls it. It is, however, sometimes referred to as the [b]spatial frequency[/b] - a term that differentiates it from ordinary frequency or temporal frequency. So while ordinary (temporal) frequency tells how quickly the wave changes in time, the spatial frequency (wave number) describes how quickly a wave changes in space.

Besides spatially shifting a wave, we can ask what should happen if a wave is time delayed (or advanced), or [b]temporally[/b] shifted.  Waves repeat themselves temporally in one period [math]T[/math].  The period of an ocean wave is just the time between waves hitting the shore.  Mathematically, we should expect that [math]y(x,t)=y(x,t+T)[/math].  [br][br]To find [math]\omega[/math], we do much the same thing as we did with the spatial shift, which with our traveling wave equation gives us [br][center][math]A\sin(kx-\omega(t+T))=A\sin(kx-\omega t\pm 2\pi).[/math]  [/center]Taking the positive solution gives: [br][center][math]\omega = \frac{2\pi}{T}.[/math] [/center]The term [math]\omega[/math] is called the [b]angular frequency[/b] and is measured in [math]\frac{rad}{s}[/math].  Just as the spatial frequency (wave number) gave us the amount of phase that fits in a given space, this angular frequency tells us how much phase "fits" in a given period time, or [math]\Delta\phi = \omega\Delta t[/math].  Another way to understand this (again using ocean waves) is that if you were standing in the water as waves were coming in to the shore, the angular frequency describes how many radians of phase would hit you per second.  So instead of counting waves, you'd count radians of phase... where you certainly recall that [math]2\pi[/math] radians fit in one wavelength.

To find the speed of a traveling wave, it's useful to consider something like a surfer riding an ocean wave.  "Catching" the wave and riding it means that the surfer is at a constant phase [math]kx-\omega t[/math] as they surf.  We know this because their value of y(x,t) would be constant as they surf.  This costant phase means that the time derivative of the phase should be zero.  Therefore [math]kx-\omega t = constant[/math] and [math]\frac{d}{dt}(kx-\omega t)=0[/math].  This leads to [math]kv = \omega[/math], or [center][math]v=\frac{\omega}{k}=\frac{\lambda}{T}.[/math][/center]In case you haven't been exposed to the "ordinary" notion of frequency (as compared with angular frequency) we'll have a brief discussion.  Standing on a beach, the wave frequency would be a count of how many waves strike the beach per unit time.  Unit-wise this frequency [math]f[/math] is measured in [math]\frac{1}{s}\equiv Hz[/math].  This unit is called the[b] hertz[/b].  It should be apparent to you that [br][center][math]f=\frac{1}{T}.[/math][/center]Angular frequency is related to this ordinary frequency by [math]\omega = 2\pi f[/math] since one wave contains [math]2\pi[/math] radians of phase.  So [math]f[/math] counts waves per second and [math]\omega[/math] counts phase per second.  Knowing this means another variation of the wave speed is [br][center][math]v=\lambda f.[/math][/center]If any of these equations ever gets confused in your head, keep in mind that inspecting the units should allow you to get it in the right order.  That's what I do when my memory fails me.

[u][b][i]Human Hearing Example[/i][/b][br][br][/u][i][b]These wave equations work for all types of waves.  For instance, humans can typically hear sound between frequencies of [/b][/i][math]20 Hz[/math][i][b] to [/b][/i][math]20kHz[/math][i][b].  Sound travels through air at around [/b][/i][math]340 m/s[/math][i][b] (the speed depends on temperature).  What are the equivalent wavelengths of these frequencies of sound?[br][br]Using the equation above [/b][/i][math]\lambda = v/f = \frac{340m/s}{20Hz} = 17m[/math][i][b], and [/b][/i][math]\frac{340m/s}{20kHz} = 0.017m[/math][i][b].  What this means is that if you could see the high pressure waves of sound coming from a sub woofer (a speaker designed to play low bass tones), they'd be [/b][/i][math]17m[/math][i][b] apart and racing toward you at [/b][/i][math]340m/s[/math][i][b].  On the other hand, a high frequency tweeter (as they call speakers that make only high pitched sound) would send pressure waves toward you at the same speed but spaced as close as [/b][/i][math]17mm[/math][i][b] apart.[/b][/i]

Since we have just spent time developing the definition of traveling waves and some of the mathematics, and since we use the term [i]traveling[/i], we should pause to ask this question: What, if anything, is traveling in a traveling wave?  At first you might think sound waves are made of traveling air, or that ocean waves are made of traveling water.  If that were true, standing in front of a speaker while listening to your favorite music would feel more like standing in front of a fan, and bobbing up and down in ocean waves would be more like being carried by water down river rapids.[br][br]If you think about a wave on a guitar string, it is clear that the string is not going anywhere, but is rather just oscillating about some equilibrium.  The same is true of light waves.  The electric and magnetic fields don't themselves travel.  Nor do you travel while floating in the ocean as waves pass - unless there is also a little current in addition to the periodic waves.  In sound waves the air molecules don't travel long range at all, but rather just oscillate in place.  So it should be clear that the medium is not the thing that is traveling.[br][br]So what, if anything is actually traveling?  It turns out that the whole role of a wave is to pass energy and momentum from one place to another - and sometimes angular momentum as well.  In this sense, for example, in a sound wave, one air molecule pushes and pulls on its neighbor which subsequently pushes and pulls on its neighbor, in an ongoing process much like passing a baton.  In this case it's a baton of energy and momentum rather than a metal stick.  [br][br]With that in mind, here is a simple rule:  [b]The rate of energy transport (or power delivered) is always proportional to the square of the disturbance amplitude, or [/b][math]\text{Energy delivered}\propto A^2[/math].  The momentum [math]p[/math] transferred is the energy transferred divided by the speed of the wave, or [math]p=E/v[/math].  In order to write the energy proportionality as an equality, it takes knowledge of the specific system.  We will look at those specifics a little later on.

Waves are often either [b]transverse[/b] or [b]longitudinal[/b].  In a transverse wave the disturbance is perpendicular to the wave propagation.  In a longitudinal wave the disturbance is parallel to the propagation.[br][br]Consider sound waves in air.  A speaker cone pushes and pulls on air along the same line as the sound wave propagates.  This constitutes a longitudinal wave - one in which the air molecules being disturbed are oscillating parallel to the sound wave.  [br][br]On the other hand, a guitarist strums strings on an instrument, and the waves on the strings propagate along the string's length while the plucking (or disturbance) is perpendicular to the string's length. These string waves are transverse.

When two waves meet, the disturbance is the sum of the two waves.  We call this the [b]principle of superposition[/b], and the term given to two waves meeting is called [b]interference[/b].  I have always thought the term interference is an unfortunate one to use in this context because most times when we hear interference we think of a ball being deflected in some sporting event or someone interfering with some aspect of your life.  The result in the case of the ball is that the path is deflected.  If someone interferes with your affairs, the course of them is altered.  [br][br]Waves don't meet like that.  Ocean waves don't collide, but pass through one another.  While at the same place and time, their amplitudes add, but afterwards it's as if they'd never met.  Their courses are unaltered for having met.  The word superposition is a much better one.  It just means they are placed atop one another which is just what happens as you can see in the animation.

Since waves meet and their effects are simply added, we need to have a mathematical means of adding waves together.  It's not enough to just draw pictures.  We need to do the math to learn the real interesting stuff nature has in store for us.  There are trigonometric identities for simple cases that you've probably memorized in the past, but they will often not be adequate for our purposes since our waves will have varying amplitudes and relative phases.  We will rather use the mathematics of [b]phasors[/b].  The idea of a phasor is similar to that of a vector in the sense that it has a magnitude and a direction, but is different in that phasors rotate in time.  As the name suggests, the purpose of a phasor has much to do with adding waves of different phase.[br][br]Below is an animation of a sine wave and its corresponding phasor to its left.  Note that just as in a trigonometric unit circle, the phase angle increases in the counter-clockwise direction.  In this case, the overall phase of the wave is decreasing in time due to the minus sign before the [math]\omega t[/math] term.  The reason the minus sign is there, is to guarantee that the wave propagates in the +x direction, which is our convention.

In our upcoming examples we will deal with sound waves and light waves primarily.  For light, we will generally use laser light.  This is because laser light has certain properties that normal white light does not have.  Laser light is both [b]monochromatic[/b] and [b]coherent[/b].  Monochromatic means "single color".  To a physicist this means single wavelength or frequency.  Coherent means for all practical purposes "continuous".  We usually measure the coherence length of a laser.  This coherence length indicates how long (spatially) the wave can be expected to be a continuous sine wave.  [br][br]We will soon be discussing situations where two waves meet at the same place and time.  Often one of the waves will be shifted, or have a [b]phase shift[/b] relative to the other wave.  In order to be sure that a phase shift of [math]2\pi[/math] radians actually leads to a self-similar wave, we need the wave to be continuous rather than only piece-wise continuous.  Therefore the coherence length - if expressed as phase - needs to exceed the phase difference between the waves.  The relationship is that the coherence length [math]L[/math] relates to the maximum allowable phase shift [math]\Delta \phi[/math] by [br][center][math]\Delta \phi = kL.[/math][/center]The monochromatic property isn't absolutely required, but speaking of just one wavelength at a time makes our discussions clearer.  The coherence generally need not be long.  A millimeter is often plenty - but it will certainly depend on the experimental setup.

When sine waves of equal wavelength are added, even if they have different amplitudes and phases, the result will be another sine wave.  Phasors allow us to find the resulting amplitude and phase of the sum.  Later when we discuss music, we will see what happens when two waves of differing wavelength are added.[br][br]Below is an animation with two waves and corresponding phasors.  In addition, the sum of the two waves is plotted, as well as the sum of the two phasors.  Play around with that animation until the phasor addition and the corresponding wave addition makes sense to you.

Many interesting phenomenon in nature with regards to waves tend to occur when a single wave splits and the two parts undergo different circumstances and then the parts recombine.  What we measure upon recombination is affected by the relative phase and the amplitudes of the two parts.  In most cases we will stick to only two parts, but once we're comfortable with the topic we'll discuss the situation of a light wave splitting into [i]n[/i] parts where [i]n [/i]will be anywhere from a few parts to uncountably many parts.  That is the topic we'll take up in the next chapter.