There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis



Showing posts with label phasor animations. Show all posts
Showing posts with label phasor animations. Show all posts

Friday, February 15, 2008

Reflection of waves as a process to make standing waves


   In memory of author Hans Christian Andersen and sculptor Edward Eriksen and their The Little Mermaid, Copenhagen, Denmark, 1836 and 1913

Waves usually reflect when they hit any sort of obstruction or change in the media. Sometimes the reflection is total, other times it is partial with the non-reflected waves continuing, being absorbed, or partly continuing and partly being absorbed. Above and below we show animations of the reflection process, as demonstrated with waves on a rope above and with water waves below. Mouse over the animations to start them, off to suspend them, or click on them to restart.

No reflection

We will focus mostly on the animation below which is set up to demonstrate a number of different reflecting conditions and principles. This animation initially starts with "no reflection" active. In this setting, we see a pure traveling wave with no reflection. Some of the features of this wave are:

  • There is a pure traveling wave, propagating from left to right, unimpeded.

  • The light gray region around the surface of the wave shows the envelope of the wave. This is the region swept by the wave (i.e. the water surface) as the wave passes. You can see that for this pure traveling wave, the envelope is constant, does not vary with x, and that all points are swept equally as the pure traveling wave passes.

  • The envelope's minimum amplitude equals its maximum amplitude, as indicated by the little "min" and "max" bubbles. This equality is always true for a pure traveling wave with no attenuation.

  • Below the wave, we see the phasors, for various points along the wave. Note that where the wave is maximum, at the wave crests, the phasors are pointing to the right, totally on the real axis, in which direction the real part is maximum. The phasors at the wave troughs are pointing in the opposite direction, to the left, where the real part is negative.

  • The phasors are all of the same length and point at progressively different angles. This is always true for a pure traveling wave (having no attenuation).

  • Along the bottom edge, a little right of center, we see a number for the standing wave ratio. The standing wave ratio is a measure of the amount of standing waves in the system. It is defined as the envelope's maximum amplitude divided by the envelope's minimum amplitude, as indicated by the max and min bubbles. In this case of zero reflection, it equals one since the min equals the max. This value of one is indicated for the standing wave ratio at the bottom edge of the animation. A standing wave ratio of 1 means no standing waves are present, that the waves are purely of the traveling wave type. We will discuss the standing wave ratio more below.

  • The notation in the center of the bottom edge of the animation indicates that the reflection coefficient is zero, meaning that there is no reflection in this case. The angle of the reflection coefficient is indeterminate since there is no reflection. This is indicated by a question mark.


Click here for a copy of the above animation in a separate window that can be sized (even to fill the entire screen) and positioned independently from the text. Your popup blocker may prevent a separate animation from being launched. In some browsers the separate animation will stay on top if you only use your scroll wheel to scroll down this text, and do not click on the text.

Click here for a slightly different version of a separate window. This version allows you to save the animation to a file by clicking on "file" and then "save page as" to get this animation into your computer. Please read the fair use policy.

Total Reflection, no phase shift

If you click "total reflection +", you see a sea cliff appear on the right hand side which will reflect 100% of the waves.

  • Wait for the wave to propagate up to the cliff, reflect, and travel back across the screen. At this time you will see the formation of a pure standing wave formed by the 100% reflection of the incident wave. As we saw in the last two postings, a standing wave is the superposition of two equal, but oppositely propagating traveling waves, which is exactly the situation in this case where we have 100% reflection.... all of the incident wave is reflected giving an oppositely propagating traveling wave in addition to the incident wave.

  • In this case, you see that the light gray envelope of the pure standing wave is not constant as it was with the no reflection case, but has well defined peaks called antinodes or maximums, and minimums called nodes or zeroes.

  • The minimums of the envelope of a pure standing wave are zero, i.e. the wave has zero amplitude at the minimums, which is again different from the no reflections case. At these points, there is no vertical motion of the water surface, even though the wave field is very active on either side othe these points.

  • A maximum occurs at the reflecting surface.

  • The standing wave ratio (the max divided by the min) is now equal to infinity, since the denominator, the minimum, is zero.

  • The reflection coefficient is now 1.0 or 100%, and the phase angle of the reflected wave is 0°. This means that the phase of the wave immediately after reflection equals that of the wave immediately before reflection.

  • The phasor pointers, while still rotating, are pointing all in the same direction or at 180° from the rest. Their lengths vary in proportion to the amplitude of the envelope at a particular point. This is quite different from the case of the pure traveling wave we looked at first, where all phasors had the same length but different directions.

Total reflection, 180° phase shift

If we click on "total reflection −" we have a setup where the wave undergoes a 180° phase shift upon reflection.

  • Such a phase shift is rather hard to achieve for water waves, but it is a common occurence in some wave systems such as in the case of a sound wave traveling down a tube...an open end of the tube will cause a near-perfect refection with a 180° phase shift. It is also the most common reflection for waves on a string or rope, such as in the mermaid animation at the very start of this posting. Hold the mouse button down on that animation to achieve longer wave trains to more clearly see the standing waves.

  • The envelope, or light gray region, is very similar in shape to what it was with "total reflection +" case, only that the maximums and zeroes are shifted over from before. Now, a zero or node occurs at the point of reflection. We get a zero because the reflecting object causes a 180° phase shift in the wave, so that the reflected wave is 180° out of phase with the incident wave right at the point of reflection and the two waves cancel there.

  • The complicated reflecting barrier shown (in the second animation), as is needed to cause such a phase shift in water, will not have a single flat reflecting surface, but we will consider the effective "reflecting surface" to be at the right hand edge of the animation. A minimum, i.e. zero, occurs at this "reflecting surface".

  • As before, the standing wave ratio is infinite because the minimum is zero giving a zero in the denominator.

  • The reflection coefficient listed on the bottom of the animation is 1.0 with a phase angle of 180°.

Variable refection coefficient

If we click "variable reflection" the animation uses a random number generator to set, at random, the reflection coefficient and the phase angle. If you don't like the values selected, click on it again for a new set of values.

  • The reflection coefficient and phase angle selected are shown in the center of the bottom edge of the animation.

  • The envelope will now be somewhere between the three cases above, with minimums and maximums, and where the minimums are not zero.

  • The standing wave ratio (the maximum divided by the minimum) will be some number larger than 1.0 but smaller than infinity. This means we have something between a pure traveling wave and a pure standing wave. We can think of this as a mix of the two: perhaps a standing wave with a extra traveling wave added to it.

  • The phasor pointers vary in length and in the direction they point.

  • These waves are fascinating to watch. The wave field has a traveling wave component that continually travels to the right. At the same time the waves have these envelope minimums to slip through. No matter how tight the minimums are, the waves somehow manage to configure themselves to magically slide through the constrictions.

  • We can click "show components" to see a mathematical dissection of the wave into two pure traveling waves. The red part is a traveling wave propagating to the right and the blue part is a traveling wave propagating to the left. The amplitude of the blue wave will be less than that of the red wave, since only a fraction of the red wave is reflected (the ratio of the two equaling the reflection coefficient.) You can watch how the two wave components slide in opposite directions, making the maximums in the black total wave when their peaks pass each other.

  • I have shown as a reflecting barrier, a messy, wave absorbing, seaweed choked cliff with under seas rocks, as might be expected to produce a reflection coefficient with a partial reflection and complicated phase angle.

Math of reflections

We can mathematically express the "incident" wave, as:

equation for the incident wave,

where we have shown both the cosine and the imaginary representations.

The "reflected" wave is just this incident wave times the reflection coefficient and shifted in phase by the phase of the reflection coefficient. We also need to change the sign of the wavenumber κ because the reflected wave is traveling in the reverse direction from the incident wave. In addition, it is important just where the reflection takes place. For the moment, we will assume that it takes place at x = 0, placing the x origin on the far right side of the screen and all the wave activity is in the negative x region. The equation for the reflected wave is:

equation for the reflected wave equation for the reflected wave,

where Γ, capital gamma, is the complex reflection coefficient and is defined as Γ=re where r is the magnitude of the reflection coefficient, is real and is defined as the amplitude of the reflected wave divided by the amplitude of the incident wave. As is typical of the complex form of phasors, both the magnitude and the phase of the reflection can be characterized by one compact complex constant, Γ. The complex reflection coefficient is defined as the complex reflected wave divided by the complex incident wave, both evaluated at the point of reflection (x=0 in our case). The phase shift φ is the phase of the reflected wave just after reflection minus the phase of the incident wave just before reflection (both at x=0 for our case).

We can easily check out if the above equation for the reflected wave is consistent with our description in the text. To do this evaluate the equation for the incident wave at x=0 to get:

equation for the incident wave at x=0.

The above equation for the reflected wave at x=0 gives:

equation for the reflected wave at x=0.

So you see that indeed, the reflected wave equation is just the incident wave multiplied in amplitude by r and shifted in phase by φ.

Another note: there are two standard variations of the lower case phi and internet browsers are very inconsistent on which variation they use. Rest assured that     variation 1 of lower case phi     and     variation 2 of lower case phi      are the same greek letter and stand for the same mathematical quantity. The phi's occurring in equations that are entered as bit maps may be of other variation from those that have been entered as html code, depending on your browser.

The total wave field

  • The total wave field consists of the sum of the two waves discussed above. Thus we sum the incident wave and the reflected wave:

    cosine equation for the incident plus the reflected wave,

    as written in the cosine notation.

  • The same wave field written in the complex notation is:

    complex equation for the incident plus the reflected wave

    complex equation for the incident plus the reflected wave.     (1)

    This last equation sees extensive use in the design of electronic communication devices that use waves to transmit information.

Special cases, total reflection

  • In the special case of positive total reflection, where φ=0 and r=1, then Γ=1 and the last equation becomes

    complex equation for postive total reflection.

    This is the same as derived in the previous posting for a standing wave.

  • In the case of negative total reflection, r=1 as before, but φ=180°. This makes Γ = re = -1, making our equation become:

    complex equation for negative total reflection.

    This is similar to the above equation for the φ=0 case, however it is shifted both in position (by π/2κ) and in phase (by i = eiπ/2 or 90°). We see this shift in the animation if we compare the standing wave pattern of the "total reflection+" case with that of the "total reflection −" case.

In the above we have used the following relationships, which are easily derivable from Euler's formula (just substitute Euler's formula in for the e and e−iα to verify these):

equation for the sine in terms of exponentials  (1a)

equation for the cosine in terms of exponentials.

The standing wave ratio

The standing wave ratio is defined as the maximum amplitude in the wave field divided by the minimum amplitude:

definition of the standing wave ratio.     (2) graph of the standing wave ratio versus the reflection coefficient

This can be expressed in terms of the reflection coefficient by realizing that the maximum of the amplitude occurs where the incident and reflected waves are in phase. Look at the phasor dials in the animation above to see this (with "show components" and "total reflection" active). Similarly, the minimum occurs where the two are out of phase. When in phase they add up to an amplitude of A+rA and when out of phase they add to ArA. Thus we can write the reflection coefficient as:

equation of s in terms of r.

This equation is graphed to the right. We can also use this to solve for the reflection coefficient r in terms of the standing wave ratio s to give:

equation of r in terms of s.     (3)

Position of the envelope maxima

We can also use these concepts to calculate the position at which the maximums and minimums will occur. If we are a distance  l  from the reflection, then the phase shift for the incident wave to travel up to the reflection is κl . There it undergoes a phase shift of φ upon reflection and then an additional phase shift of κl returning to our spot as the reflected wave. Thus the phase difference between the incident and reflected waves at this spot is 2κl+φ. To make a maximum, this phase difference must be an integer multiple of 2π radians. Putting this into an equation gives:

relationship for maximum to occur,    (4)

where n is an integer and represents the number of the particular maximum we are considering. The first maximum from the reflecting object (just to the left of the object) would have an n = 1, the second one from the object has n = 2, and so on. The wavenumber κ can be calculated from the wavelength λ (the distance between wave crests of the incident wave) as:

κ = 2π/λ.     (5)

We can solve Equation (4) for the necessary distance l for an envelope maximum to occur:

distance for maximum to occur.

where φ is the phase of the reflection coefficient. Alternately, we can solve for the phase of the reflection coefficient in terms of  l:

angle of reflection coefficient.    (6)

The most common utility of the standing wave ratio is that it allows determination of the reflection coefficient, angle and magnitude, from simple measurements on the wave field. Observing the waves, one can obtain the wavelength λ and from that the wavenumber from the Equation (5) above. One can also measure the wave amplitudes at the maximum and minimum and calculate their ratio, or s (Equation (2) above). Using the Equation (3) above this gives the magnitude of the reflection coefficient r. Measuring the distance l from the reflecting obstacle to the first maximum and using the Equation (6) above with n=1, gives the angle of the reflection coefficient.

Location of the envelope minima

If one wishes instead to measure the distance to the minimum, then we need to alter the relationship Equation (4). In this case, we need the phase length to be some odd integer of π radians or 180° so that the incident and reflected waves will be 180° out of phase and try to cancel. So setting the phase shift due to the distance and reflection ((2κl+φ) to the odd integer of π gives:

relationship for minimum to occur,

where m is an odd integer (n=1, 3, 5, or 7, etc.). This can be solved to yield:

distance for minimum to occur

and

angle of reflection coefficient.

This brings us to an interesting point. The phase angle of the reflection coefficient depends critically on the location we select as the reflecting point. In some situations, this reflecting point is clear, as it was with our "total reflection+" case where there was a sheer cliff. However, often there is not an exact point that reflects the waves, such as in the "total reflection-" or "variable refl" cases. Then, in the interest of maintaining simplicity, one chooses an "effective" reflection point. With regard to the above animation, we chose the right edge of the animation, for simplicity, and calculated a set of reflection angles based on this. If we had chosen another effective reflecting point, then we would have calculated another set of reflection coefficient angles for the same total wave fields, in order to account to the different phase lengths involved with a different idealized reflection point.

Equation for the envelope

To get an equation for the envelope or magnitude of the total wave field as a function of x, we revisit Equation (1) above. One approach is to use a carefully drawn phasor diagram of Eq. (1) and the law of cosines to arrive at an expression for the magnitude of the wave field. We shall do that first. Another approach is to simply separate out the real and imaginary parts of (1), square each, take the square root, and simplify a bunch to find the magnitude. We do that secondly, below.

We do the vector sum of phasors as follows:
  • To the right, we have drawn a phasor diagram (in the complex plane) showing the vector addition of the left side of Eq. (1) with the A eiωt factored out, i.e.     

  • There we see red vector representing eiκx  is of length 1 and is drawn at an angle of κx with respect to the real axis. vector diagram

  • The blue vector represents  rei(φ-κx)  and is of length r and angle (φ−κx).

  • The angle B between the red and blue vectors is given by B = (φ−κx)−κx = φ−2κx.

  • The angle C equals angle B (and therefore also equals φ−2κx ), because their two sides are parallel.

  • The supplement to C, angle D, is given by D = πC = π − (φ−2κx), where π radians equals 180°.

  • By the law of cosines, the magnitude M is given by:
         law of cosines.

  • We can simplify the cosine:
         simplifying the cosine simplifying the cosine.

  • This allows the magnitude to be written as:
         magnitude of the vector sum.

  • Putting in the amplitude A that we left out, gives us the amplitude (half the thickness of the envelope) as a function of position:
         magnitude of the envelope.

    Note that A is the amplitude of the incident wave by itself.

An alternative way to derive the above expression that doesn't involve drawings of vectors is as follows:
  • We start with the same complex phasor expression of incident and reflected wave, Equation (1), but we factor out half of the reflection angle as:

    .
    .

  • The first factor is simply a magnitude times the complex temporal rotor times a constant phase shift. We ignore this for now and focus on the rest, which we will call M1.   M1 is similar to the expression for the cosine (Equation (1b) above), except for that factor of r. To get around the r, we peel off an r amount of the first term and combine it with the second term to make a cosine. Also, we still have the rest of the first term, which we put first:


    .

  • The cosine term is totally real, so only has a real term, and the other term has both a real and an imaginary term, via Euler's formula. To take the magnitude of M1, we use the square root of the sum of the squares of the real and the imaginary parts. So under the square root, we need to add the two real terms and square their sum, and to that add the square of the imaginary term.
    .

  • We expand and combine terms and finally use the identity sin2x+cos2x=1:
    .
    .
    .

  • Using the double angle formula for cosine, we get:
    .

  • Symmetry of the cosine, i.e. cos(−x)=cos(x) allows us to write it as:
    .

  • This is the same expression as we got above.

Practical applications of reflection concepts

The concepts discussed above are of great use in modern communication systems, where electromagnetic waves carry information. These waves may be in air, in space, in a cable, or in an optical fiber. In most cases engineers seek to minimize reflection, so that the waves are almost completely absorbed by a receiver, where they are most needed and will not reflect and cause spurious ghosts or extra noise in the system. Engineers use these equations to design receivers and other components to minimize reflections and optimize other performance features. The commonly used Smith Chart is a graphical method to approach reflection problems and is based on the above equations. A biography of the inventor, Mr. Phillip Smith, is here and a very short history is here.

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, 2008.


NEXT: Water waves PREVIOUS TOPIC: Complex phasor representation of a standing wave
Good references on WAVES Good general references on resonators, waves, and fields
Scroll down farther for a list of the various related topics covered in postings on this blog.

Sunday, October 28, 2007

Phasors

illustration, space woman shooting phasor

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.

digram of phasor showing the angle and projection of the rotating vector on the x axis

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.

a clock serves a similar function as a phasor

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:

equation, amplitude of sum of two waveforms and

equation, tangent of phase angle of the sum of two waveforms

diagram of the phasor for a sum waveform

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:

equation, x component of phasor for first waveform

and

equation, y component of phasor for first waveform

making our equations become:

equation, amplitude of vector sum

equation, tangent of angle of vector sum .

diagram showing the summing of two vectors

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):

equation, magnitude of sum vector

equation, tangent of angle of vector sum

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.

NEXT: More on Complex Numbers.

© 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.