Phasors

A graphical method that helps in the understanding waves and oscillations, and also helps with calculations, such as wave addition, is called "phasor diagram". Sadly enough, this has nothing to do with Star Trek or the "phasor" weapons used in science fiction movies, although phasor diagrams would undoubtably used in their design when that time should come, since phasor diagrams play a central role in the of understanding of lasers.

If you have trouble understanding this diagram, the animation below may help.

The oscillations and vibrations really only involve motion along one direction or with regard to one parmameter, however in this method we artificially add a second dimension just for the sake of understanding. We imagine a rigid rotor or vector moving around in circles around the origin, as illustrated in the diagram to the right. It is constructed so that the projection or shadow of the tip of the vector moves back and forth exactly like the oscillation we are studying. The angle that the vector makes will the x-axis is made equal to ωt + ϕ. Because of the t, or time, factor that increases as time increases, the angle will constantly increase, making the vector rotate at a constant velocity. The projection that the vector makes on the horizontal or x-axis will be

x = r cos (ωt + ϕ),

so that this process does give us the cosine wavefunction in the end. The angle ϕ is the angle the vector make with the x-axis initially, i.e. at time t = 0.

Phasor diagrams are similar to an analog clock face. In such a clock face we use the rotating hands to keep track of time. Time, however, has nothing rotating or circular about it. To the average man or woman who is not an astronomer, time is a one dimensional progression, more like a time line. Of course time also has a repeating nature to it, like a time line that keeps replaying. But, for mechanical convenience, we use a circular clock face to keep track of time, because repeating motions are easiest to construct with rotating circular devices. Similarly, for computational ease we will use these circular phasors to track oscillations and waves, because these diagrams make computational manipulations of oscillations and waves easier.

To better understand this, study the animation below.

Animated phasor showing shadow (or projection) oscillating back and forth, simulating an oscillation. The angle θ = ωt + ϕ is shown by the angle numbers around the edge (read with the red vector as a pointer.) The position of the tip of the shadow on the ruler is equal to x = r cos (ωt + ϕ). Mouse over the image to see the action and mouse off to stop it. You need a flash player installed on your computer to see this animation.

Presented with a real oscillation to simulate, such as AC power or the pressure oscillations in a musical note, the phasor method instructs us to imagine a rotating vector, whose projection (or shadow) is the observed oscillation. The oscillation shown here has an amplitude of 1, i.e. r = 1. Of course, we can change the vector length for any amplitude. It is perhaps more common to make the projection (or shadow) on the x-axis as is shown in the previous diagram, instead of below the circle as is shown in this animation. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

Another phasor animation:

Animated phasor relating oscillations of a physical object, of our beautiful bungy jumper in this case, to a phasor diagram and a graph of her oscillations (taken at her "center of mass".) For these purposes, the phasor diagram has been rotated 90 degrees and projected horizontally. Also, we assume that the elastic rope is always under some tension, in order to insure cosine dependence.

Mouse over the image to see the action. When the animation is finished, mouse off and on again to replay it. Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

Adding two waves in a phasor diagram

Ok, a phasor diagram is suppose to make it easier to add waves. But, exactly how is that done?

Someone who is very familiar with vector addition may have recognized the equations we laboriously derived for adding cosine waves above. I'll repeat them here:

and

Notice that all the terms are of the form A₁cosϕ or A₁sinϕ . These are just the x and y components (or projections) of the vector A₁ plotted on the phasor diagram to the right. Note that the bold symbol A₃ stands for the whole vector, while the italized A₃ stands only for its amplitude. That is to say:

and

making our equations become:

.

These are just the equations for adding two vectors: we add their x components together to find the x component of the sum, call it x₃, and add their two y components to find the y component of the sum, called y₃. Then the magnitude and tangent of the angle of this the sum vector should be (as we have):

Thus if we add two phasor vectors together, we will get the phasor vector for the correct wavefunction according to the complicated cosine addition formulas.

Below we have an animation for the process of adding two oscillations using phasor diagrams. We use the oscillations of two people on the ends of bungy cords, however the oscillations could also be the oscillations of electrical voltages in AC power or the oscillations of the air pressure as a sound wave passes through. Read the caption below the animation for details on it.

An animated phasor diagram showing phasor addition of two oscillations, that of the woman and that of the man suspended on bungy cords. Like before, the animation shows the relation between the phasor diagram, the actual oscillation, and a graph of the oscillations.

Click on "together" to see the action. When the animation is finished, click on this or other buttons to see this or related animations. Move the mouse off the animation to suspend it and back on to restart it.

The "together" animation shows the pair oscillating in unison or "in phase" with a phase difference of 0 degrees. The gray sum vector represents the sum of the two oscillations and for zero phase difference has an amplitude equal to the sum of the amplitudes of the two people. Zero phase difference results in the largest possible sum oscillations.

One use of summing the two oscillations is to find the motion of the center of mass of the man and woman. The center of mass motion is simply the sum of the motions divided by two. It is shown as a green cross and moves up and down with the motion.

The "in opposition" button delays the release of the woman to put the pair oscillating with 180 degree phase shift between them, or in opposition with each other. When the man is down, the woman is up, etc. This phase results in a sum amplitude equal to the difference between the two amplitudes. In this case the sum is the smallest possible for all possible phases. Note that the center of mass hardly oscillates at all. If the oscillations of the two people were of exactly the same amplitude, then the sum, as well, as the center of mass oscillations would have zero amplitude in this case of 180 degrees phase difference. That is to say that with 180 degree phase difference the two oscillations would cancel in a summing process or would destructively interfere.

You should compare the relative directions of the phasor arrows in the two cases (after the woman has been released. In the first case, with 0 degrees phase shift, the arrows are together, cause the maximum possible sum. In the second case, the arrows are spaced 180 degrees apart, causing the minimum possible sum.

The "random phase" button delays the woman's release a variable amount so that a different phase shift will occur each time this button is clicked. The resulting sum oscillation is the vector sum of the man's and woman's phasors. The summing parallelogram is seen in light gray in the phasor diagram to the left. The sum oscillations have an amplitude some where between the amplitudes of the two extremes cases, that of 0 and that of 180 degree phase difference. In general there will be a phase difference between the sum and either person's oscillations.

The "difference" button shows the difference between the man's and woman's oscillations. One application for a difference is to calculate the distance between the two people as a function of time. We illustrate this length by the length of the blue ribbon stretched between their hands. You can see that its length varies in an oscillatory fashion. This button also adds a random phase shift between the man's and woman's oscillations.

Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

All the above "buttons" show cases in which the frequency of the two oscillations are the same. However the phasor method also works for adding or subtracting oscillations of differing frequencies. We show this next:

An animated phasor diagram showing phasor addition of two oscillations when the frequencies of the two oscillations are different. This allows one of the rotating phasor vectors to rotate faster than the other, continuously changing the phase angle difference between the two. At times, the two phasors will add up in the maximum way, demonstrating constructive interference. At other times, they will subtract, demonstrating destructive interference. The result is that the amplitude of the sum varies with time in a repetitive way. This phenomena is called "beating" and is commonly encountered in the addition of two sound waves of only slightly different frequency or tone. The resulting sound "warbles", i.e. rapidly varies in amplitude. It turns out that the frequency of the amplitude variations, i.e. of the beating, is equal to the difference between the two initial frequencies.

Click on either button to see the action. Mousing over the other button can be used to speed up or slow down the action without restarting it. Mouse off the animation to suspend it and mouse on to continue it. Click on a button to restart it. Clicking will restart the action whether or not it has finished. Each time the action is restarted, a new frequency difference will be used, changing the rate of beating. In these animations, we neglect damping, which will normally reduce the oscillations as time goes by. We will discuss damping in future lessons.

We needed to compress the horizontal axis to show many more oscillations than in the previous animation in order to be able to see the beating phenomena. Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

Options:

• If you are having trouble with your flash player, you can see
  • a still image by clicking here
  • or a animated gif file (2.5MB) of part of the action by clicking here.
    
  
  
• To hear examples of acoustical beating, click on the captions below the following flute players.

Why do phasor diagrams work?

We have already answered this above, at least in part, but it is an important point that could stand repeating and further refinement. Our answer:

• Most simple oscillations, ones having a "linear" restoring force, can be mathematically represented with cosine (or sine) functions, for example: y(t) = A cos(ωt + ϕ).
• This is a result of the type differential equations that these oscillating systems are constrained to follow and has nothing to do with circular motion.
• It just so happens that a cosine function is also, independently of oscillating phenomena, the projection of a vector rotating at constant angular speed around the origin.
• Thus, if we should want, we could choose to have the projection of a rotating vector represent an oscillation. This would have the handy feature that the argument of cosine function in the oscillation which has units of radians, now can be represented as an actual physical angle.
• If we have two vectors rotating around the origin, then their sum vector will also rotate around the origin. This sum vector will have an x-component equal to the sum of the x-components of the two original vectors. Thus the projection of the sum vector is the sum of the two oscillations.
• Whereas there was no simple illustration for adding two cosine functions, with the phasor method there is a simple graphical method for adding two rotating vectors.
• So, given oscillations to add and subtract, we construct the required rotating vectors, called a phasor diagram, and graphically add and calculate the resulting sum and/or difference oscillation.

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NEXT: More on Complex Numbers.

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© P. Ceperley, 2007.

Good references on phasors.

Note that searching under "phasors" will produce links to complex phasors, which we will cover in a later posting. Below are some of the better links to the idea of simple phasors.

• Wikipedia.
• Nice animation from S. Schneider of ITU.
• The complete list of S. Schnieder's physics animations.
• Nice clean animation of phasors and phasor addition.

Complex Phasors

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A complex representation of oscillations - complex phasors

We'll start off by repeating Euler's formula and our cosine formula for a wave:

.

We see that the real part of e^ix has a cosine in it just like the oscillating cosine equation above. To connect the two concepts, we use the fact that the real part of the right side of Euler's formula is the cos x. Thus, we can write the second equation as:

,

where the "Re" in the equation stands for "real part of". Some authors use the symbol ℜ for this instead of Re. The real part of a complex number is the term or terms that do not contain an i. Often the phase shift ϕ is incorporated into the constant A as:

,

where

. We can view this new A, with a tilde on it, as a complex amplitude, containing both the strength or amplitude of the oscillations, and also the phase shift ϕ.

An animated phasor diagram showing temporal behavior of e^iωt. The box in the top left corner shows the frequency, angular frequency, time, and angular position of the vector (both in radians and degrees). Notice how t and ωt both increase as the animation runs. Also notice that ωt corresponds to the polar angle of the vector at any time, i.e. that when ωt = 180 degrees or about 3 radians, the pink vector is pointed left at the 3 radian mark. The projection (or shadow) on the floor represents the real part of e^iωt.

The frequency, f, is the number of complete rotations the vector makes per second. The angular frequency, ω, is just f times 2π. The frequency of oscillation does not necessarily correspond exactly to the frequency of oscillation of the shadow you observe on your computer because of variations from computer to computer.

Mouse over the animation to start it and mouse off to stop it. Click on the animation to restart the time, t, from zero. Every time it is restarted, a different frequency is used. Notice that when the frequency is lower, the rotational speed is lower.

This animation is very similar to one used earlier for phasors. After all, the concept of phasors is very, very similar to these plots on the complex plane of terms with e^iωt in them. The advantage of the complex notation over phasors is that complex notation gives us a powerful algebraic notation for oscillations and waves.

Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

e^iωt is a real show shopper ... the central genius of the complex representation of oscillations.

What about the e^iωt term? A couple of pages ago, we discussed e^{i ϕ} and found that it represented a vector in the complex plane that started at the origin and had a length of one (or 1). It made a polar angle of ϕ with the real (or x) axis.

Now we are replacing the ϕ with ωt. The time factor, t, constantly increases making the product iωt also to constantly increase with time. The iωt is the polar angle in e^iωt, meaning that the polar angle increases with time. Thus the complex point representing e^iωt on the complex plane will be moving around in a circle of radius 1. As above, we usually add a vector from the origin to the point, just to help with the visualization.

We show this in the animation to the right. When ωt is larger than 360 degrees (6.28 radians) then the vector starts a new trip around the circle. After all an angle of 360 degrees is equivalent to 0 degrees in most situations.

Below we show another animation of an oscillating parameter expressed as the complex equation

.

The animation shows the relationship between the rotating complex vector, the oscillating voltage, and the graph of the voltage versus time. Notice here that the horizontal axis indicates time. We might have also labeled the horizontal axis as ωt with numbers ranging from 0 degrees to about 1600 degrees (or the corresponding radians) which is about 4.3 complex cycles.

An animated diagram showing the relationship between the rotating vector on the complex plane, the value of the oscillating parameter, and a graph of this parameter versus time. The complex plane has been rotated by 90 degrees to facilitate plotting the real component of the complex vector. The center object is a bar graph of the parameter modeled. In this case, we labeled the graph with "voltage" to indicate we are modeling an oscillating voltage, perhaps the voltage in a person's house, or in some electronic device (very much slowed down from most real electric oscillations). Notice that the voltage oscillates with time.

Mouse over the animation to activate it and mouse off to suspend its operation. Click it to reset the animation to zero.

This animation is set up with the rotor angle marked in degrees. Engineers often use degrees as an alternative to radians for simpler problems, remembering to switch back to radians for more difficult calculations. The usual notation is as:

.

For example, an oscillating voltage with an amplitude of 30 volts and an phase angle of 60 degrees might be written as:

.

This animation is extremely similar to an earlier one using phasors. You might be wondering, what is new? Why repeat all this stuff? I'm here to tell you that there is a very important new addition. In fact there are two new things here:

• We are using complex numbers and the complex plane now, and
• We have nice complex notation for the oscillation to allow us to solve equations involving oscillations and their parameters.

Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

With deference to Leonardo da Vinci, pioneering artist, scientist, and inventor.

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Multiplication of an oscillation by a complex constant

There are many physical phenomena and electrical circuit elements that change the amplitude and phase of an oscillation or wave by a fixed amount. Without using complex notation, there is no convenient way to mathematically express this. However with the use of the complex notation, this operation is merely multiplication by a complex constant. To further understand this, consider the following multiplication of an oscillation by the complex constant B:

    .

Here we see that the resulting product has an amplitude equal to the product of amplitudes, and the phase of the constant adds to the phase of the oscillation by the constant amount ϕ. We can compare this with a similar observation above which concerned multiplication of two complex constants. Here one of the factors has the e^iωt term which makes the resulting phase of the oscillation continue to increase with time. After the multiplication, the phase of the rotating vector is shifted by the fixed angle ϕ. In the animation below, we see this shift and the effect of the oscillations, which of course are just the real part of the rotating complex vectors.

An animated diagram showing the multiplication of a complex oscillation by the complex constant B. The value (amplitude and phase) of B, the multiplying complex constant is given in black at the top left of the animation. Note that B is written in the format discussed at the end of the caption in the animation just above. The yellow vector represent the initial oscillation, and the orange vector represents the product of the yellow vector (or oscillation) multiplied by B.

Mouse over the animation to activate it. Clicking it will reset the animation to zero. Each time the animation is reset, a new value for B is generated.

Stop the motion (by mousing off it) and study the phase relationship of the parts. The orange product or output vector leads the yellow input vector by the angle shown at the top of the animation. The output vector has had its phase increased by the phase of B. Thus if B has a positive phase then on the graph, the orange curve will reach the crests before the yellow curve of the input vector.

The vertical axis on the graph is labeled "temperature" and represents oscillation in the temperatures of two sensors. Perhaps the yellow sensor is in the water and the orange sensor is in the air above the water which is heated by the water. The period of oscillation would be one day, meaning that the animation is very much speeded up. While the animation is obviously a simplification, it is meant to illustrate a type of linked oscillations that can be mathematically modeled by the complex notation discussed here. In the center of the animation are bar graphs of the oscillating temperatures.

To sum up, this animation illustrates the time behavior of an oscillation modeled in the complex plane AND a second oscillation which results from multiplying the first oscillation by a complex constant. In general this results in an amplitude change and phase change.

Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

In praise of summer beaches

A sampling of applications of multiplying an oscillation or wave by a complex constant

In each, the final signal, oscillation, or wave can be calculated by multiplying the initial signal by the appropriate complex constant.

• Microphone signal passing through a stereo amplifier.
• Telephone signal passing through 3 miles of wire.
• Fiber optic signal passing through in-line amplifier.
• Sound passing through the bone structure of the inner ear.
• AC power passing through transformers in residential transformers.
• Sound impinging on a microphone and converted to an electrical signal.
• Radar wave impinging on an airplane and returning towards ground antenna.
• Ultrasound signal impinging on fetus and returning towards detector.
• Cell phone wave from tower traveling a half mile to user's phone.
• Starlight passing through interstellar dust.
• Underwater song of a whale passing through 6 miles of ocean water.
• Ultrasonic clicks from a bat impinging on a moth and reflecting towards the bat.
• Oscillations in exterior temperature working their way into the interior of an unheated structure.
• Oscillation of the suspension system of a truck causing oscillations in the structure of a bridge supporting the truck.
• Oscillations in ocean levels caused by tides working their way 60 miles up a river estuary.
• Oscillations in AC power voltage driving AC current in a reactive load, such as motors or fluorescent lights.

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NEXT: More on complex phasors. LAST: More on complex numbers.

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©P. Ceperley, 2007.

Links for complex phasors

• Wikipedia
• An illustrated explanation.
• From Answer.com
• Circuit theory basics.

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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NEXT: Types of Waves I PREVIOUS TOPIC: True Waves

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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