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