Introduction
In the first chapter of his extraordinary Lectures on Physics, Richard Feynman presents his belief that the sentence "All things are made of atoms" contains more information per word than any other. I believe that the word atoms should be replaced with the word waves. The wave is ubiquitous in the applied sciences, and an argument could be made that it is the most fundamental concept in physics. I hope to convince you of the importance of waves and to help you better understand wave-related fields of science.
The importance of waves to applied science
The study of sound waves in air is useful to musicians, musical equipment manufacturers (both acoustic and electric), architects, and biologists and psychologists who wish to understand human perception. Sound waves in liquid or solid (in which case they are called phonons) media are important to scientists who study electronic devices, the communication of dolphins, SONAR, etc.
Water waves (which aren't too dissimilar from sound waves) are very important in the fields of oceanography and coastal engineering. Seismic waves (essentially sound waves in the earth) are obviously of considerable interest to geologists.
The study of electromagnetic waves is vital to scientists in such fields as optical communication, photolithography, satellite communication, portable electronics, lasers, optical lenses, radar, perception, x-ray imaging, and x-ray crystallography (I could continue for a long time).
I also want to note that the wave-nature of matter has become important to chemists, materials scientists, electrical engineers, and generally anyone who works in an area in which atomic-level understanding is required. There are many other types of waves with which I am less familiar.
The importance of waves to fundamental physics
As far as we know, everything in the universe--matter and energy--is a wave. Recently someone submitted a writeup on gravity waves (a predicted but still unobserved phenomenon) which might be of interest to you. It is often said that the universe has a wave/particle duality, meaning that everything in the universe has both wave and particle characteristics. While this is true, it is interesting to note that a particle (i.e. a precisely localized object) can be visualized as an infinite sum (integral) of waves. In mathematical terms, the Fourier transform of the Dirac delta function* is just 1. This fact implies that the delta function is a sum of equally-weighted sinusoids of all frequencies.
* The Dirac delta function is the limit of a set of functions normalized to 1 as their bandwidth approaches 0. Think of a bell curve (Gaussian) centered around the point X as an example. Since the bell curve represents probability density, its integral over all space is 1. Imagine squeezing the bell curve so that its spread gets smaller and smaller, and its peak gets higher and higher to preserve probability. In the limit of squeezing, the peak is "infinity" and the spread about X is 0. This is the delta function. A function (actually to mathematicians this isn't a true function) defined in such a limit is called a generalized function.
What is a wave?
This is a most difficult section for me! Waves are present in so many forms and contexts that it is difficult to isolate the underlying property that makes them "waves." I will define a wave in a rather technical way. It gets easier from here. A wave is a function of space and time that can be written as a superposition of basis functions of the form
Akcos(kz - w(k)t + φ(k)).
φ(k) is called the phase of a basis wave. k is called a wavevector (though we will work in one-dimension so we might call it a wavenumber). w is known as the angular frequency of a basis wave. t is time and z is a direction in space. Ak is just a weighting coefficient.
Let's look at the term in the parentheses--(kz - w(k)t + φ(k)). At a particular time t and a particular point z the term has some value which we can call C. Notice that If we increase t a little bit (i.e. we let time elapse), we can increase z a little bit to keep the value of the term at C. How much do we have to increase z to keep a constant C? Well, kz - w(k)t + φ(k) = C. So kz - w(k)t is a constant D. That means z = D + (w(k)/k)t and dz/dt = w(k)/k. This is the speed of the wave! Every point on the basis wave travels at this same speed since C was arbitrary.
Usually waves with different angular frequencies w move with the same speed, which we'll call c. However, waves can move in any direction. In the one-dimensional case that we'll consider, this means that dz/dt = +/- c. If dz/dt = +c, the wave is moving rightward along the z-axis. If dz/dt = -c, the wave is moving leftward. We see that w(k)/k = +/- c implies that w(k) = +/- ck--it's just a linear function of k. The reader should realize that dispersion, an effect in which waves with different angular frequencies travel with different speeds, is very critical in many fields. For example, the amount of information that can be transmitted through optical fibers is governed by dispersion.
w(k) has the units of radians/second. 2π radians = 360 degrees. If we look at what happens at a single point z, we see a full oscillation every time w(k)Δt = 2π. Δt is known as the period of the wave. It is equivalent to 2π/w. Notice that there are 1/Δt oscillations per second. The number of oscillations per second is called the frequency (different from angular frequency). The frequency and angular frequency are related by f = w/2π.
There are some other things to note. I assumed that all waves propagate along the z-axis. This is fine for the purposes of understanding basic wave behavior, but one should know that in three dimensions, k is a vector and z is replaced by the position vector r. The polarization of some waves (such as electromagnetic waves) is not critical to understanding waves, but it can be included by making the coefficients Ak vectors. For quantum mechanical purposes, the coefficients Ak are complex numbers.
Superposition/interference of waves
In general, a wave is a sum of the basis waves described above. This sum can be quite complicated--the resultant wave can look nothing like a sinusoid! It is important to realize that implicitly, we are assuming that waves simply add linearly. In other words, if two waves intersect eachother, we can treat them as one wave which is a simple addition of the two waves. This is almost always a valid assumption, but there are nonlinear phenomena that can have interesting effects. We won't discuss them here but you should be aware that the field of nonlinear optics has become fairly important.
So imagine a "wave" that is a sum of two basis waves that are identical in every respect except that one of the phases is 180 degrees (or π radians) away from the other. From basic trigonometry, a sine wave with a constant phase of (180 + x) degrees is just the negative of a sine wave with phase x. Thus the sum of the two basis waves is zero! This situation is called destructive interference. When two waves that are 180 degrees out-of-phase come together at some point, they disappear. Of course, if the phase difference between the two waves is 0 degrees, then the resultant wave is just the simple sum of the two. This situation is called constructive interference. Everything between complete constructive interference and complete destructive interference is possible. The constructive and destructive interference of waves are responsible for many phenomena such as beats and wavepackets.
More information about waves
Originally I intended to discuss several wave phenomena in this writeup. However, it is fairly long as it is, and to this point it has been a review of the very basics of waves, which might not be useful to all scientists. I will link this node to other nodes in which I plan to add future writeups. Actually, on further thought I'm not going to waste my time making these writeups that nobody reads and nobody respects. One or the other would make it worth my while. And since I'm not interested in milkshakes, tv, or penises, this is adieu.