Note: Graphs of trigonometric functions in this section use units of radians. If you're not familiar with or comfortable with radians, it's a simple conversion: π radians = 180˚.
Waves are periodic disturbances in some medium, like water waves in water, vibrations of a string or wire (e.g. guitar), sound waves in air, or electromagnetic waves in the electromagnetic field. Those kinds of waves are covered in other sections. This section is about the mathematics we use to model wave behavior.
Waves are well-modeled using the fundamental trigonometric functions, sine and cosine.
In this section, we'll review the basic anatomy and transformations of a sine wave. It's the same for the cosine function; recall that the functions are the same except for a shift of π/2 along the time axis.
The graph below shows two cycles of a sine function. Time is along the horizontal axis, and f(t) = sin(t) is plotted along the vertical axis. We need to become familiar with several terms that describe waves.
Waves have peaks or crests, the high points, and troughs, the low points. The amplitude of a wave is the measure of the height of a peak (or the depth of a trough) from the center line, or the line of zero displacement. The points at which the amplitude of a wave is zero are called nodes. If you've studied p-orbitals and d-orbitals in chemistry, you've learned about nodes of 3-dimensional waves.
Wavelength is the length, in units of length (e.g. meters), of one complete cycle or period of a wave. It's convenient to measure wavelength from node-to-node, peak-to-peak or trough-to-trough, but it can be anywhere, as long as the chunk of time represents one complete cycle of the wave. Wavelength is usually given the Greek symbol lambda, λ.
In later sections on this page, we'll develop other concepts such as frequency and phase of waves, then talk about the mathematical model of a wave.
Imagine for a moment that your eyes are at the surface of the water of a smooth pool when a pebble is dropped in nearby. Waves will pass (say left to right) in time, and they'll pass at a speed that is limited by the medium — water in this case. From our experience, water waves move at about 1-2 meters per second.
Each medium has its own characteristic speed for waves passing through it. Sound waves travel at about 340 ms-1 in dry air, but they actually travel much faster through most solid materials, like steel or aluminum.
Light waves don't actually need a medium in which to travel. They are disturbances in the electromagnetic field, but more on that later. Light waves have the highest speed, 2.99792458 × 108 m·s-1 (exactly).
The number of waves that pass by your fixed point of view in some unit of time (usually a second) is called the frequency of the wave. Think of it as the measure of how frequently you'll see a wave crest. The units of frequency are "reciprocal seconds" (s-1), and the reciprocal second is often called Hertz (Hz). A 240 Hz wave has a frequency of 240 per second (240 s-1= 240 Hz), and 240 waves pass a fixed point each second. (That would be some water wave).
The symbol for frequency is the Greek lower-case letter nu, ν. It's basically a lower-case "v" that looks like the wind is blowing it over from the right.
The unit of frequency is the reciprocal second,
Unit of ν is 1/s
and the unit of wavelength is a unit of length. We'll use the meter. So when we multiply frequency by wavelength, the units are the unit of speed:
λ·ν = m(1/s) = m/s = ms-1
Thus, the product of wavelength and frequency is the speed of the wave.
λ·ν = speed
Here are some speeds of sound in various materials. Notice that the speed of sound in solid materials can be much larger than the speed of sound in air.
The unit of wavelength is the meter, and the unit of frequency is Hertz (Hz). 1 Hz = 1 s-1
The wavelength ( λ ) of a wave multiplied by its frequency ( ν ) is its speed. In a given medium, an increase in wavelength means a decrease in frequency, and a decrease in wavelength means an increase in frequency.
λ·ν = speed
The wavelength-frequency equation for this kind of wave (electromagnetic radiation or light) is
where the speed of light is about 3 x 108 ms-1. Rearranging to solve for the frequency and plugging in the given wavelength (632 nm is red light), we get
The frequency is
We should always try to simplify such numbers with our metric prefixes: 103 = Kilo (K); 106 = Mega (M), 109 = Giga (G), and 1012 = Tera (T). so this frequency is best presented in units of Terahertz (THz):
The wavelength-frequency equation is
where the speed of sound (about 760 mi./h) is given. Rearranging to solve for the wavelength gives us:
Now we can find the wavelength of the lower frequency (low-pitch sounds).
and the higher frequency (high pitched sounds):
So the waves most of us can hear have wavelengths between 17 cm and 17 m.
The sine function can be transformed in a number of different ways. The simplest is multiplication of the function by a constant, A, like this:
f(t) = A·sin(t)
Here A corresponds to the amplitude of the sine wave. So when using a sine function to model a wave, we can adjust the A parameter to get the amplitude right.
The graph shows a sine function (it's sin(2t) — more on that below) with an amplitude of 1 and the same function multiplied by 2 (magenta).
Move the slider on the plot of f(t) = A·sin(t) below to see how changing the parameter A affects the graph.
How do we increase or decrease the frequency of a sine wave to match the wave we're trying to model?
It turns out that changing the frequency is just stretching or compressing the function horizontally (along the time axis). It looks like this:
f(t) = sin(ωt)
where ω is the Greek lower-case letter "omega," and is often called the frequency factor when the sine function is associated with waves.
The frequency factor is the number of full cycles of the sine wave that fit between 0 and 2π.
The graph shows two full cycles of sin(t) between 0 and 4π, and four full cycles of sin(2t) (magenta) between 0 and 4π.
Move the slider below to change the frequency of this sine wave. Notice that as ω gets larger the number of cycles per unit time increases.
The phase of a wave is unimportant in some applications, but crucial in others. You can think of a phase difference as the difference in relative "starting points" of two or more waves. Here's a picture. The magenta wave is shifted to the right by π/2 compared to the gray wave.
A phase shift of a sine wave is accomplished mathematically by employing the usual horizontal translation transformation:
f(t) = sin(t + φ)
The Greek letter phi, φ, is often used to denote phase.
Lasers produce what is called coherent light. This means that all waves have the sam phase, or are "in phase." In laser light all peaks and troughs line up and the range of emitted wavelengths is very tight. Compare that to a white light bulb, which produces light across the visible spectrum, and generally of all phases — non-coherent light.
Phase differences lead to all sorts of interesting and useful effects because they can produce constructive and destructive interference. More on that later.
Move the slider on the plot below to see how adjusting φ shifts the wave along the time (horizontal) axis.
The last transformation of the sine function is the vertical translation or vertical shift. In wave mathematics it's sometimes called the vertical offset or DC offset (DC for "direct current").
The transformation is acheived by simply adding or subtracting a constant from the function,
f(t) = sin(t) + k
where k is the offset. In the graph, a normal sine function is elevated by 2 units along the vertical axis by adding 2 to f(x).
Move the slider on the plot of f(t) = sin(t) below to see how the horizontal offset parameter (k) works.
A parameter is an adjustable constant in the definition of a function that is different from the independent variable(s). Parameters are not independent variables. For example, in the quadratic function
f(x) = Ax2 + Bx + C
A, B and C are parameters which change the shape of the graph of the function. x is the independent variable. A, B and C are fixed for any particular version of f(x), but x can range from -∞ to +∞
We can put all four wave transformations together into one equation. It's the wave equation you should know.
The periodic nature of waves allows them to add in interesting ways. Waves can interfere with each other, producing a wave of larger or smaller amplitude than its "parents." When waves add to increase the amplitude, we call it constructive interference. When they add to produce a lower amplitude than the absolute sum, it's destrucive interference. Perfect destructive interference between two waves can completely cancel both.
In this animation, two waves are added, point-by-point, along the t axis
Move the slider on the plot of
f(t) = sin(t) + sin(t - φ)
below to see how changing the phase of one wave gradually diminishes the sum. When the waves are out of phase, that is their phases differ by π = 180˚, they cancel each other.
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