Types of Waves

There are many types of waves in this world. Water waves and sound waves are well known to the average person. The waves that are better known to scientists include seismic waves, structural waves, light and other forms of electromagnetic radiation, quantum mechanic pilot waves, magnetic spin waves, and waves in plasmas. In this posting we hope to give an overview of some of the ways waves are classified. We start with classifying by the number of dimensions, i.e. 1D, 2D, or 3D waves.

Number of dimensions

Waves in one dimension

A wave on a string is an example of a one dimensional wave as shown to the left. Mouse over the image to see the action. These waves travel along a line, i.e. along a one dimensional space. These waves are only a function of one space variable, such as x, in addition to time, t. They satisfy the one dimensional wave equation:

where c is the speed of the wave propagation. These have solutions of the form y(x, t) = f(x ±ct), i.e. d'Ambert's solutions. These solutions include pulses, gaussians, and any shape function that travels to the left or right at the speed c. We are often more interested in sinusoidal and complex exponential solutions, i.e. the sine, cosine or exp forms given by    y(x, t) = Asin(κx − ωt) ,   y(x, t) = Acos(κx − ωt),   or   y(x, t) = Ae^{i(κx − ωt)}  . Remember that the wave number is defined as κ = ω/c .

(A discussion of variables such as ω is found at this link. A discussion of simple waves and wave numbers κis found at this other link.)

Waves in two dimensions

Waves can also travel on a surface that is a two dimensional space, such as the surface of water or in a layer of clouds as shown below. These are examples of two dimensional (2D) waves. While one dimensional waves are easiest to understand and analyze, two dimensional waves are probably the most interesting to see and to animate.

Water waves are the best known examples of waves. They exist on the 2 dimensional surface of water. Physicists sometimes call these "gravity waves" because their restoring force is gravity.	Wind can cause cloud waves to form at the interface of two different densities of air. These waves are often made visible by the wind causing regular variations in the cloud formation at this interface.

In these cases, the wave variable is the height, z, of the wave surface above its equilibrium height. The surface is described by the coordinates x and y. Three animated examples of two dimensional waves are shown below. Mouse over each to see the action.

Wave propagating in the y direction.	Wave propagating at a 30 degree angle from the y axis.

Circular wave originating for the origin.

The above waves obey the two dimensional wave equation:

While the three 2D waves above certainly propagate on a two dimensional surface, some scientists would call these one dimensional. Take the left one, for example. This wave does not propagate in the x direction and can be described as:

which does not depend on the x variable. It is also true that in the other two waves above, that is, by picking the right coordinates, one can make the waves depend only on one dimension, in addition to time. The middle animation requires rotating the axes so that one axis lines up with the wave direction. The right one requires use of cylindrical coordinates, i.e. expressing the wave in terms of the radial distance out from the axis, as in

.

Usually radial waves require use of Bessel functions, but the above equation is approximately correct and gives you the idea of what to expect with radial waves.

You might say of all three of the above waves that they are physically two dimensional, but mathematically one dimensional. This distinction is important when we try to work with the mathematics of waves. Of course if we had picked a more complicated wave on a surface, such as the water wave in the photo above, then it would be two dimensional in both the physical sense and mathematical sense.

One interesting mathematical trick is that we can express the waves in the middle animation as:

    ,

where we now need wavenumbers in both the x direction and also in the y direction to express a wave traveling at an angle to either axis. We can consider these two wave numbers as two components of the wavenumber vector and compactly express the equation as we did in the final expression above. The equivalent complex representation is:

    .

We can substitute this expression into the 2D wave equation above and show that

.

which shows that κ_x and κ_y combine to form κ in the same fashion as components of a vector would. It requires more work to show that the wave front actually propagates in the direction of the κ vector.

Radio waves generated by Marconi and other with their spark gap transmitters in the late 1800's led to modern radio. These old transmitters launched waves into three dimensional space. Click here to see this animation and read more about the operation of spark gap transmitters.

Waves in three dimensions

Many very important waves propagate in a three dimensional space. These include sound waves, radio waves, light and other electromagnetic waves. It is rather hard to draw three dimensional waves, other than in cartoonish ways, as is done at at the left, or by plotting only a two dimensional slice of space, which we will do soon, in which case they resemble a 2D wave. Three dimensional waves obey a 3D wave equation,

,

where p is the wave variable similar to z in the 2D case. Since the waves propagate in all three coordinates, x, y, and z, we need to use an additional variable. I have picked p, the time varying acoustical pressure in a sound wave as a typical 3D wave. One of the simpler sound waves, a "plane wave", can be expressed as:

    ,

or as:

   

similar to the expression for the oblique 2D wave above.

Transverse and longitudinal waves

Transverse waves

Separate from categorizing waves as to their number of dimensions, we can also categorize them as to they're being transverse or longitudinal. The first animation above, with the monkey waving a rope, is a good example of a transverse wave. A water wave is another example of a transverse wave. A transverse wave is a wave in which the motion of the media is perpendicular to the direction the wave is propagating. What this means is that the monkey's rope moves back and forth in the vertical direction, in a direction perpendicular to the direction the wave is traveling, which is horizontally. Similarly, in a water wave, the wave causes the water surface to move vertically, up and down, while the wave propagates in a direction along the water surface.

A very technologically important type of transverse wave is electromagnetic radiation. Depending on the frequency of this radiation, it is known as radio waves, microwaves, infrared radiation, light, xrays, or gamma rays. Electromagnetic radiation, complicates the mathematical expressions for waves by requiring the use of vectors for the wave variables. Thus, in addition to the wave number being a vector, the varying quantity, or quantities in this case, are vectors. These vectors are the electric and magnetic fields in the wave. Below, we see an animation of the interplay of these vectors with time. Mouse over it to see the action (and mouse off to suspend it). The curved arrows and buttons can be used to change the viewing angle. The animation only shows a short length of the wave, which in reality would go very large distances, even to distant galaxies. I should stress that the animation shows the electric and magnetic fields only along one line, such as for an extremely thin light beam, like that of a laser pointer. Most electromagnetic waves are broader and the fields are simultaneously present throughout a three dimensional volume. All the vectors in a broad beam are extremely hard to show in a two dimensional picture or screen, so we stay with our thin beam animation.

In the animation, the x axis is red, the y axis is blue, and the z axis is gray. The wave is traveling along the z axis. The electric field vectors are shown in red and are pointing in the x direction. The magnetic field vectors are shown in blue and are pointing in the y direction. Note that both types of fields are perpendicular to the direction of propagation and also perpendicular to each other. This is sometimes called a transverse electromagnetic wave, or TEM wave, to distinguish it from electromagnetic waves inside a waveguide or optical fiber which often have electric and magnetic fields that are not perpendicular to the direction of propagation. We can write the equations for the fields in the animation in the various forms as:

    ,

    ,

where E is the electric field vector and B is the magnetic field vector.

I might point out that even though I use red and blue in the animation, this does not indicate the color of the electromagnetic radiation, that there is not necessarily red and blue light involved here. These colors are simply used to distinguish the parts of an electromagnetic wave. The wave itself could be carrying red light, blue light, green light, or be radio waves or x-rays that are invisible to the human eye.

Electromagnetic magnetic waves are certainly more complicated than the other wave types shown above. It took great minds many centuries to understand them and the discovery process makes quite a tale. If we were to try to illustrate the phasor diagrams with these waves, the result would be an extremely complicated tangle. Most people simple rely on the math at this point. They know in their heads that each component can be illustrated with a phasor diagram, like the ones we showed in previous postings. For example, to illustrate the electric field vector, which is pointing in the x direction, we would use the y direction to graph the imaginary part of the x directed electric field, ignoring the existence of the magnetic field. The resulting plot would look the same as that shown here.

We can also say that the wave as shown above is polarized, meaning that the electric field vector stays in one direction, the x direction in this case, and the magnetic field stays in the y direction. Other forms of polarization are possible. Light from a light bulb or the sun is unpolarized because its electric and magnetic field vectors point in wildly varying directions. More specially, the electric and magnetic fields in sun light stay perpendicular to the propagation direction and to each other, but these two vectors swing around together in the x-y plane erratically. On the other hand, light from many types of lasers is often highly polarized.

Another aspect of electromagnetic waves is that they propagate extremely rapidly, faster than anything else. In air and space, they travel at a rate of 300,000 km/sec or 185,000 miles per second or slightly over a billion kilometers per hour (10⁹). According to Einstein's theory of special relativity, nothing except light can go this fast, not even superman or a rocket. Some very high energy cosmic rays and particles in high energy accelerators go almost this fast, but never just as fast. In media such as water and glass, light slows down a little, however even then it is still going extremely fast. Click here to read more on the speed of light.

Longitudinal waves

Sound is a good example of a longitudinal wave. The air molecules are pushed back and forth in the same direction that the wave propagates. While sound is normally hard to visualize, we have made an animation to make it easier to see sound in motion. Mouse over the animation below to see the action, mouse off to stop it, and click to restart it.

The animation above shows a rack of squishy balls through which a very slow sound wave is passing. The elephant is causing the wave, which propagates to the right and ends up pushing the man back and forth. The wave causes the balls to move back and forth horizontally, at the same time they are being stretched and compressed. The balls are filled with special gas that changes color when compressed (red) or expanded (blue), making it especially easy for you to follow the action of the waves. This allows you to see the zones of compression and rarefraction (stretching) that are steadily progressing to the right. We have ignored the process of reflection at the right side and assumed that these waves do not reflect...perhaps because the man is offering just the correct resistance to avoid reflection, so we have a pure traveling (and longitudinal) wave. We will discuss reflections in a later posting. Also, normally sound travels much, much faster than the animation shows. In air it usually travels at about 340 m/s or 1000 ft/s.

If you click on "show graph", a graph will appear to plot the acoustical pressure in the wave as a function of position. Note that the red pressure maximums line up with the wave crests in the graph. If you click on "show phasors" each ball will have a little phasor vector inside it, indicating the direction of the phasor at that point. (Read other postings on this web site for detailed discussions of phasors.) On the little phasor diagrams, the real axis is horizontal, pointing to the right, while the imaginary axis is vertical with positive being up. The phasors are indicating the phase of the acoustical pressure wave. Note that the phasors in the red portion of the wave are pointing to the right, at which position the real part is maximum which agrees with the pressure being maximum there. Likewise, in the blue parts of the wave the phasors are pointing to the left, at which angle the real part of the phasor is most negative corresponding to this negative part of the pressure cycle. Note that we are considering only the changing part of the pressure, call the acoustical pressure, which oscillates positive and negative about the average atmospheric pressure.

One of the main points of the above animation is that each point in space through which a wave is traveling has a phasor associated with it. The phasors at different positions have different angles, and can also have different magnitudes. Click on "attenuation" to see the same animation with attenuation, a decrease in sound amplitude as the sound progresses through the balls. In real waves, attenuation is almost always present and results in conversion of sound into heat. Note that in the attenuating case, the lengths of the phasor vectors decrease with larger distance from the source. Similarly, the amplitude on the graph also decreases with distance. Clicking any of the buttons a second time will turn off the feature of that button.

An equation for the acoustical wave at all the points can be written very compactly as:

   

where z is the direction of the wave propagation and p is the acoustical pressure. This equation is similar to that for the wave the monkey is generating at the start of this posting. The last term in the above equation is the expression for the complex phasor representation of the wave. The above equation is for the case of no attenuation. Attenuation will add an additional term of e^-αz where α is the attenuation constant and is zero in the case of no attenuation and becomes large for large attenuation. Alternately, we can use the attenuation length, l, that equals 1/α and is the distance in which the amplitude will be attenuated to 1/e = 37% of its initial amplitude. The equation for the acoustical wave with attenuation in various forms is:

Note that the term A₀exp(-αz) could be replaced with a decreasing amplitude A(z) which is defined to equal A e^-αz.

Alternately, we could have graphed and given phasors for the ball motions instead of their acoustical pressures. Remember the balls are moving a little to the right and to the left as the wave passes through them. If we have made the graph and phasors for the horizontal ball velocities, the graph and phasors of this velocity would have been "in phase" with the graph and phasors of the pressure. Alternately, if we had used the horizontal positions relative to the equilibrium positions, the graph and phasors would have lagged the pressures and velocities by 90 degrees. The relationship between the pressure and the gas motion is a very important aspect of acoustical waves (and all other waves), called the wave impedance (or acoustical impedance, characteristic impedance, etc depending on the wave type and usage), and will be discussed in more length in a future posting.

Some additional points concerning sound are:

• While the wave propagate steadily from left to right, the media or balls do not, and instead oscillate around fixed equilibrium points. This is true for all waves passing through a material media. The material does not undergo net displacement nor does it follow the wave.
• A wave is a pattern of motion that moves through a media. To track the waves, most people track the wave crests, the compression regions which we have colored red. In the case of sound waves, these represent the acoustical pressure maximums that arrive at your ear and force your eardrum to move back and forth, in the same fashion as the man in the animation above is pushed back and forth. With real sound waves hitting your ear, the pressure cycles come much, much faster, too fast for the eye to follow.
• The waves shown here are longitudinal because the media moves back and forth in the same direction as the waves propagate.
• A transverse wave, on the other hand, has its media move perpendicular to the direction of wave propagation, such as is the case of the wave on a rope that the monkey is demonstrating above.
• Longitudinal waves are generally harder to visualize, because the media motion is mixed up in the same direction as the wave propagation direction. When we want a wave which is truly easy to visualize, we show a one dimensional transverse wave, such as waves on a string, or a cross section of water waves.

The animations in this posting can be downloaded free from George Mason University Archival Repository. Please read the fair use policy for this work. © P. Ceperley, 2007.

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NEXT: Superposition and standing waves PREVIOUS TOPIC: Waves using complex phasors

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Good references on WAVES

Good general references on resonators, waves, and fields

Wikipedia on waves. Zona Land on waves, lots of animations. The physics classroom on waves. Thinkquest on waves. Dr. Dan Russell, Kettering University Applied Physics, nice animations. All sorts of demonstration setups and animations from the Univ. of California at Berkeley. And many more sites.... search on waves and traveling waves. Scroll down farther for a list of the various related topics covered in postings on this blog.

Think Quest -- Wave Express technorati.com The Physics Classroom HyperPhysics intutor.com Wikepedia

True Waves

Jean le Rond d'Alembert (1717-1783).

So far we have discussed oscillations. Examples of these are oscillating voltages in AC power or in many electronic signals. They may be pressure oscillations at a point due to a sound wave passing through, or electric field oscillations at a point due to an electromagnetic wave passing through. It turns out that we can do better than just expressing oscillations at a point due to a passing wave, we can actually express the whole wave mathematically.

The mathematical trick for working with waves can be attributed to Jean le Rond d'Alembert (1717-1783). He found that the general mathematical solution to the differential equation governing simple waves traveling in the x direction are functions of the form:

y(x,t) = f(x - ct)

where c is the velocity of the wave.

A Gaussian Function. Note it is centered on x = 0.

To understand this concept, consider the peaked exponential function shown at the right, called a gaussian or gaussian function. The equation describing it is:

Another notation for this same equation is y = exp(−x²). This is a very useful function which is perhaps best known for describing the statistical distribution of random events, but it is also useful for several purposes in oscillations and waves. This is a totally real function, and not directly related to the complex exponential that Euler invented.

A Shifted Gaussian. Note that the peak of the curve is now at x = 2.

We first need to understand that if we replace the x with x −2 that the function will shift over by 2 as shown in the graph at the left. The equation is now

. This is just a prescription for shifting something along the x axis by 2. It is sort of a shifter equation.

We can make this more general and replace x with (x − x₀), giving an equation looking like:

.

An animation showing the effect of various x0 values on the gaussian written as a function of (x − x0). Click on the "x0 =" button to see a new x0 value appear. The leprechaun locates this value on the x axis (with his rainbow) and this is where the new peak occurs.

The animation to the right shows the effect of a variety of x₀ values.

To get the peak to move with time we substitute ct in for for x₀, where c is the speed of the shifting. Thus, if c were equal to 5m/s, then at 1 second, x₀ = ct would equal 5m. At 2 seconds, it would equal 10m. At 3 seconds, it would equal 15m, and so on. The point is that x₀ would keep on increasing and making the peak of the curve shift continuously to the right. The animation below demonstrates this continuous shifting at a variety of velocities.

An animation showing the time dependence of the gaussian function written as a function of (x − ct ). Mouse over the animation to see the action. Mouse off to stop it and back on to restart it. Clicking on it will restart it. Every time it is restarted, a new wave velocity c is generated. This velocity, the time, and the offset x0 = ct are displayed at the bottom. Note how the offset increases as time increases. Notice also that when c is greater, the wave moves faster. This type of gaussian wave is a type of solitary wave.

To summarize, we have demonstrated the continuous shifting of the gaussian function by d'Alembert's method. As we have seen, d'Alembert's function shifts the gaussian curve over to the right more and more as time progresses, just like a wave does. It turns out that it works on any function, not just the gaussian. Start with any function f(x) and substitute (x − ct) in for x and you will have a function that plotted versus x, shifts over to the right as time increases. And, as we stated before, all these shifting versions of functions happen to be the solutions to the differential equation governing simple waves.

Shifting cosine functions

Simple cosine function representing a typical oscillation, written as a function of x instead of time.

Most of us think of water waves when we think of waves: those long regular humps of water moving in towards the shore. Except for their final moment of breaking at the beach, these can be pretty accurately described in terms of cosine functions, similar to the oscillations we discussed above. We repeat one of the earlier graphs to the right which shows the function

.

The argument of the cosine function needs to be in units of radians. On the other hand, with waves, x usually means a distance in space, in meters, for example. In order to use x here we need to first multiply it by the constant κ, the wave number, to convert the meters into radians.

.

The wave number is in units of radians per meter and is a measure of how tightly bunched the peaks of the wave are in the x direction. It is similar to ω, the angular frequency, which is the radians per second in an oscillation. The wave number works on x in the "spatial domain", while the angular frequency works on time, in the "time" domain.

Now let's convert the static cosine function into a dynamic moving wave using d'Alembert's method, replacing the x with x−ct, i.e.

.

This is usually simplified by using the relation between wavenumber and angular frequency: κc = ω

.

Because the cosine function is symmetric around the y axis, i.e. cos(−x) = cosx, the above equation can also be written with an inverted argument

,

without changing its value or its graph. One of the oddities of this science is that physicists tend to use the first way of writing the equation, whereas electric engineers use the second. Mathematically they are equivalent.

Cosine function representing a typical wave, written as a function of x shown at three times: t = 0s, 1s, and 2s. We assume the wavenumber κ = 2rad/m and the angular frequency, ω = 0.8rad/s. This results in a wave velocity given by: c = ω/κ = 0.4m/s.

Whereas in our earlier discussions on oscillations, our functions were only of one variable, namely t or time, we now have a function of two variables, x and t. It is difficult to graph the function of two variables as simply as we do a function of one variable, but we can try anyway. One way to do this is as shown to the right, to graph the function versus x for various times. If we read the graph carefully, we can see that the function is moving to the right as time progresses. That is, the waveforms for greater times are shifted to the right.

Using animation however, we can do much better and graph the function versus x as time progresses on a continuous basis. This is shown below.

An animation showing our cos wave as a function of x with increasing time. Mouse over the animation to see the action. Mouse off to stop it and back on to continue it. Clicking on it will restart it. Every time it is restarted, a new set of parameters κ, ω, and c (consistent with κ and ω) are generated. These are shown at the bottom of the animation. Notice also that when c is greater, the wave moves faster. This type of wave, that I have been calling a "true wave" is know as a traveling wave.

A more general wave

To describe a real wave, we need to add a few more constants to our equation, making it look like

.

An animation showing the general cosine wave as a function of x with increasing time. Mouse over the animation to see the action. Mouse off to stop it and back on to continue it. Clicking on any of the yellow buttons will restart it with a new value of the particular constant labeled on the button. The numbers over the green dots are just the results of those in the yellow buttons (see the table and associated text below for more on that.)

The constants and variables in this equation are:

• A, the amplitude (in units of whatever the wave is related to, e.g. in a pressure wave A will have units of pressure).

  It is basically the size of the wave, or more specifically the maximum height minus the average level of the wave. It is also half the vertical distance between the wave crests and the wave troughs.

• ω, the angular frequency (omega) (in radians per second.... 6.28 radians make a complete wave cycle).

  How fast the wave oscillates (goes up and down) at any fixed point as the wave sweeps past you. It is another form of the frequency (see below).

• t, time (seconds).

  You don't have any control over time. It just keeps going on.

• κ, the wave number (kappa) (in radians per meter.)

  How tightly packed the peaks of the wave are in a snap shot of the wave (freeze it). It is inversely related to the wavelength (see below).

• x, position (in meters)

  A measure of where you are along the x axis.

• φ, the phase offset (phi) (in radians).

  What phase of the wave is lined up with x = 0 at zero time, (t = 0). If φ is positive, the wave initially will appear to be shifted to the left, and the reverse if φ is negative. The floating red apple (or dot) on the wave crest of the animation is at the point on the wave where the argument of the cos, all except for the φ, is zero. Just at the starting time (or time of resetting) the red dot will be at the y axis if φ = 0, to the left of the axis if φ is positive and to the right if φ is negative. If you are generally confused about phase, see the discussion at the end of this posting on phase.

• y₀, the average wave height (in the same units as A).

  The height offset of the wave. If the wave is centered around zero in the vertical direction, then y₀ = 0. However y₀ can be greater or less than zero to raise or lower the average value of the wave.

The animation to the right and above will allow you to vary these constants to better understand the effect they each have on the wave.

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Equations linking various parameters to each other. The frequency, f, is similar to the angular frequency, ω, except that f has units of cycles per second, instead of radians per second. One of the equations above shows how to convert one into the other. A cycle per second is officially called a Hertz, abbreviated Hz. The wavelength, λ, is the length of one cycle (in the x direction in our examples) and has units of meters. Alternately, we can use as units meters per cycle to emphasize that is it the length of just one cycle. It is the inverse of the wavenumber times 2π as shown in the equation above.

Relationship between wave constants

A number of the above constants are related to each other. To the left we present a table of these relationships. You can derive these with little thought experiments, or you can consider them as sort of a units change exercise. The right hand column of the table shows this unit change logic for each equation. I've found this unit change method very convenient for quickly rederiving (or checking) the relationships when needed. You can find more on simple waves on Wikipedia.

More on phase

Phase can be a confusing topic to anyone new into the wave and oscillations field. Just what do we mean by "the phase" of a wave?

Think of the phase or phases of the moon. A short list of these are new moon, first quarter, full moon, and last quarter, all of which repeat over and over again, just like waves and oscillations do. The phase of the moon identifies what part of the lunar cycle you are considering at the moment. An oscillation or wave similarly goes through cycles. For a cosine function (oscillation or wave), the four "quarters", to use the lunar analogy, are maximum positive value (pressure, voltage, temperature, whatever the oscillation entails), zero, negative value, and zero, as shown in the graph below.

You might noticed that this graph is plotted with angle as the arguement, in degrees in our graph, however we might have used radians just as well. When we discuss phase, we always think in terms an angle: the angle which is the arguement associated with the part of the oscillation we are focused on at the moment. If we are focused on the maximum positive part of the cycle, then we say the phase there is zero degrees (look at the graph). If we are focused on the maximum negative (or minimum) part, then the phase is 180 degrees or π radians. If we are between these extremes then we can make a guess based on remembering the graph and just how far we are from the extremes. If a scientist is discussion the phase of 45 degrees (π/4 radians) then he means the wave is decreasing from the maximum and has yet to hit the first zero in its cycle.

You might also be aware that "phase" can also be used to mean slightly different, but related things and to understand the exact meaning you have to be aware of the context (just as with many, many other terms our English language). Sometimes a person will use "phase" to mean the difference in the phases of two oscillations which are going on simultaneously. The term "phase difference" or "phase shift" would be more appropriate, but that is two words instead of one. Another person might use "phase" in place of "phase offset" to mean what is the phase of the oscillation at time equals zero (and perhaps x=0 also for a true wave) as we discussed above.

© P. Ceperley, 2007.

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NEXT: Waves using complex phasors PREVIOUS TOPIC: The algebra of complex phasors

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Good references on WAVES

Good general references on resonators, waves, and fields

Wikipedia on waves. Zona Land on waves, lots of animations. The physics classroom on waves. Thinkquest on waves. Dr. Dan Russell, Kettering University Applied Physics, nice animations. All sorts of demonstration setups and animations from the Univ. of California at Berkeley. And many more sites.... search on waves and traveling waves. Scroll down farther for a list of the various related topics covered in postings on this blog.

Think Quest -- Wave Express technorati.com The Physics Classroom HyperPhysics intutor.com Wikepedia

Waves using complex phasors

This is perhaps the most famous pieces of art featuring a wave, "The Great Wave off Kanagawa", a woodblock printing by the Japanese artist Hokusai published in 1832.

Previously, we discovered that we can write an equation for a complete wave, not just for an oscillation, but for an entity that changes with both position and time. The appropriate cosine equation for a simple wave is

. (1)

In this section we use complex phasors to represent waves. As we did for oscillations, we replace the cosine with a complex exponential or rotor:

. (2)

As discussed before, one does not usually write the Re[ ] operator, but instead simply remembers that it is there. Instead they write:

. (3)

People who use complex phasors a lot view the complex phasor as having a life of its own. They don't even think about the real part until all the calculations are done and they need to connect their result to the real world. To them the complex form is a "real" wave, i.e. a real complex wave, perhaps as a doctor might view the "real you" has your outer self and well as your inner parts. To specialists in waves, the complex form represent the internal workings of the wave, while the real projection is the outer or visible part.

We can add extra constants as we did before, to vary the amplitude, phase, and offset:

    . (4)

We use the tilde to emphasize that the constant, A, can, in general, be complex. Also, y₀ can be complex, and of course y is complex.

Lets focus on Equation (3) for a moment, because it is simpler. One way to view this equation, is that it is a complex rotor, e^iωt, with a phase offset that varies with the x position. It is like we have a different phasor for every position x, each with a slightly different phase. In the following animation, we try to illustrate this idea.

An animation showing the exponential representation of a simple wave with increasing time. Mouse over the animation to see the action. Mouse off to stop it and back on to continue it. The dials are the exponential rotors exp(iωt) for various positions, and show that each is advanced a little from its neighbor (90 degrees in the animation). The real part, i.e. real projection or shadow, is shown as a horizontal blue line inside each dial. Note that the real parts vary with time as the rotor goes through its paces. The lengths of each real part is doubled and shown vertically (also in blue) in the graph just below it. The graph shows the real part of these exponential rotors or dials as both a function of x and time, t. Clicking on the graph will restart it, in case you would like to see the cowboy and his dog again.

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The next animation uses an alternate way to show the complex wave representation. It places the rotor dials so that their axes line up with the wave direction, i.e. the x direction. This orientation lets us pack more dials in per unit length along the x axis. Then we can connect the dial pointers together is a smooth curve, which turns out to be a helix. The helix further shows that the pointer of each dial is a little progressed from the previous one at the starting time. Then as time progresses, all the pointers rotate at the same speed, maintaining the helix, but causing it to spin. The actual, real wave that we are representing is the projection of this helix on the surface below, where the snake is in this animation.

An animation showing the exponential representation of a simple wave with increasing time. This shows a different way to position the complex phasor dials. Mouse over the animation to see the action. Mouse off to stop it and back on to continue it. Click the "−" in the +/− button to suppress the cartoon features if they are a distraction, and click the "+" to restart the cartoon features. Clicking on the graph will restart it, in case you would like to see the sexy witch again. As in the previous animation, the red pointers of the dials are the exponential rotors exp(iωt) for various positions, and show that each is advanced a little from its neighbor in the clockwise or negative polar direction. When the dial pointers (or hands) are connected, they make up the blue helix or corkscrew. Note that the helix goes from pointer tip to pointer tip, even as they all rotate. The real part, i.e. real projection or shadow, is the blue sinusoid on the floor below, also illustrated with a snake. If you examine it carefully, you will note that positive displacement of the wave on the floor is towards your right elbow, while negative is away from it. These are set by the real axes of the dials, as indicated in red on the first, or left-most dial. It is amazing that the simple expression contains so much information, enough to describe a whole wave in a very compact expression! In using the complex representation, we need to remember the phasor pointers all rotate at each point along the x axis and to remember the unwritten Re ("real part of") operator. The helix IS the complex wave and is plotted in a three dimensional space. The real and imaginary components of the phasor make up two of the dimensions as labeled by the red axes on the left most phasor dial. The third dimension is that of the x axis which is the direction the wave is propagating. Halloween night and the harvest moon.

Perhaps a overall view or summary is in order: what does this complex helical waves have to do with real waves? Well, suppose we are considering a real wave, such as a sound wave or a radio wave, propagating along, and we need to work with it mathematically. The math might be require so that we can calculate something related to the wave, like how much it will attenuate. We will see some uses of the math involved with waves in the next section. The most straightforward way to mathematically represent the wave is with a cosine function as we did in the last posting. However, the representation that allows the most convenient computation of various wave phenomena is the complex representation as discussed above and as illustrated with the two animation just above. In this representation, the simple cosine wave, shown on the floor of the above animation, is changed into a helix that propagates in space as shown above. Note that one direction is imaginary both in the mathematical sense, and in the sense that this axis is purely an add-on to anything to do with the real world. However, having it does make it easier to follow the progress of the wave, both mathematically and conceptually. This is the key to the phasor representation of waves. © P. Ceperley, 2007.

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Good references on WAVES

Good general references on resonators, waves, and fields

Wikipedia on waves. Zona Land on waves, lots of animations. The physics classroom on waves. Thinkquest on waves. Dr. Dan Russell, Kettering University Applied Physics, nice animations. All sorts of demonstration setups and animations from the Univ. of California at Berkeley. And many more sites.... search on waves and traveling waves. Scroll down farther for a list of the various related topics covered in postings on this blog.

Think Quest -- Wave Express technorati.com The Physics Classroom HyperPhysics intutor.com Wikepedia