The animation below shows two types of waves: transverse waves on a rope and longitudinal compression (i.e. very slow acoustical) waves propagating through a line of squishy balls full of water. The compression waves are made clearer by having the color of each ball change in response to the compression of the ball, to red as a ball is compressed and to blue as it is stretched. To see the animation, mouse over it. To stop it, mouse off it. To restart it, click on it.
As the animation first opens (with only the pulse option clicked on... having a red square), it shows two short bursts of traveling waves being launched from the two ends of each wave system. At first, these bursts separately travel towards each other. At some point, they overlap and cause a complicated interference pattern. Then once they are through each other, they resume their simple sinusoidal shape and travel to the end where they are absorbed. During the interference time, we see a brief glimpse of the standing waves we will be discussing below.
If we click on "phasors" we see a phasor diagram in each of the balls, showing the "state of the wave" at each point along the medium. Just to remind you, a phasor is a vectorial diagram showing the amplitude and phase of a sinusoidal wave at a particular point. The phasor in a particular ball is correct, both for the longitudinal wave going through that ball and for the transverse wave in the rope directly above the ball. The phasor diagrams normally would be drawn so they are stationary, but we took artistic liberties on this, having them move back and forth with each ball. They also compress and stretch with the balls.
We can use the animation to understand the phasors and the waves they represent in three regions:
- In the non-interfering part of the wave, i.e. inside the pure traveling wave part, we see that the amplitudes (magnitude or length of the vectors) are constant, while the phases (the angles of the phasors) are continually changing, i.e. the phasors constantly rotate in a counter-clockwise direction.
- Where there is no wave, the magnitudes are zero. The angles, i.e. phases, are indeterminate since the vectors have zero lengths.
- In the interference zone, the phasors have different magnitudes at the different points and they also still rotate.
If we click "components" we now see extra red and blue ropes once the waves are launched. Note that in the sections with no waves, all three colored ropes are bunched together and hard to distinguish. The red rope shows only the wave launched from the left, while the blue one shows the wave launched from the right. The black rope shows the sum of the two waves. That is, at each point in the horizontal direction, if we sum (for a particular x value) the deflections of the red and blue ropes, we will get the deflection of the black rope. This is the principle of superposition, that in a linear system, such as these, when two waves travel through each other, the response of the medium is just to add the deflections of the two waves at each point. We also see that in the interference region, the phasors now show red and blue vectors in addition to the black one. Note that in each ball, the black vector is the vector sum of the red and blue vectors. Note also that the red phasor is the correct phasor for the red rope's wave, and the blue one is correct for the blue wave. Because the red phasor is in front of the blue one which is in front of the black one, the red tip is made a little smaller than the blue tip which is smaller than the black tip to allow you to see a little of all of them when they are together. At the same time, they are very small and it takes sharp eyes to distinguish them in this case. For example, you will have to look very closely to see the red and blue phasors in the interference region at the very center where all three vectors are aligned.
|Animation showing the process of superposition of two traveling waves to make a standing wave|
If we click the "continuous" button, a continuous sinusoidal wave will be launched from the two ends (this may take a bit of time to get started). At first, we see two waves traveling towards each other and interfering. In time, the interference zone takes over the whole space. When this happens, we have what is called a standing wave on the black rope. A standing wave results from the superposition of two traveling waves. You can probably see the standing wave best by unclicking the "components", i.e. just click on "components" again to turn off its red indicator.
- A standing wave has a different appearance from a traveling wave.
- A traveling wave keeps its shape and appears to slide along (focus on just the red wave to see the "sliding" of a pure traveling wave).
- A standing wave, the black wave, oscillates in place and appears stationary.
- A standing wave has "nodes", places where the wave amplitude is zero, pointed out on the animation above by the stationary black arrows.
- It also has "anti-nodes" half way between the nodes where the wave oscillates at its maximum amplitude.
- This author prefer the terms "zeroes" and "maxima" instead of nodes and anti-nodes.
Looking at the waves on the rope, we can understand the mechanics of the standing wave formation. The red and blue traveling wave, propagate to the right and left, respectively (click again on "components" to see this and focus on the red or blue wave). The zeroes (nodes) are located at points where the red and blue waves are mirror images of each other and cancel, producing zero sums at these locations. We see in the phasors, below the zeroes, that the red and blue vectors (or phasors) are pointing in opposite directions (180° apart) and the black sum vector is zero, and therefore not visible.
The maximums (anti-nodes) occur where the two traveling waves equal each other. When the red wave is maximum, the blue wave is also maximum at these points. It is rather fascinating to center your attention on a maximum and watch the red and blue waves slide in a symmetrical way towards this point from opposite directions. A look at the phasors below these point reveals red and blue vectors pointing in the same direction, which is also the direction of their sum, the black phasor. As pointed out before, when the vectors are aligned it takes a sharp eye to distinguish the individual vectors. At these maximums (anit-nodes), the red and blue waves are "in phase" with each other, have 0° phase difference, and add in a maximal way.
At other points besides the zeroes and maximums, the phase shift is somewhere between 0° and 180°. At these points, you can see all three vectors: the red, blue, and black (sum) ones. Note that the phasors in all the balls rotate counterclockwise. Also note that the black, sum phasors, all are pointing in the same, or in exact opposite, directions at any point in time (mouse off the animation to freeze the motion). On the other hand, the traveling waves (such as the red or blue waves) have phasors that vary in direction along the length of the wave.
Math of standing waves done with cosines
The cosine equations for transverse traveling waves propagating in the positive and negative directions are
The similar equations for the pressure in longitudinal compression (acoustical) waves are
The above is correct for transverse waves. For the compression wave, the equation is practically the same:
By using the cosine equation,
we can change these sum formulas into
The right most part can be rewritten with emphasize on the relevant terms as
Thus we see that the basic x sinusoid, cos κx, is multiplied by the time varying function 2A cos ωt. By the process of adding two shifting, traveling waves together, by interfering them, we no longer have a shifting function f(ωt ± κx), and instead have a stationary cosine whose amplitude cyclically varies with time. This is just what the animation above shows. When we have a fully developed standing wave, the magnitudes or lengths of the phasors vary in proportion to cos κx.
Almost as if by magic, two oppositely moving traveling waves interfere to produce a quite different, standing wave.
Math of standing waves with complex phasors
Complex phasors can be used to simplify the above math. We do that here, repeating the above work using complex notation for the waves.
The sum of the two oppositely moving traveling waves is
In contract to cosines, exponentials are easy to factor using
Thus our complex superposition equation becomes
This can be further reduced with the equation
which is easy to derive from Euler's formula. Using this gives
Just as in the cosine derivation, we end up with the conclusion that the sinusoid, cos κx, is multiplied by the time varying function 2A exp iωt. Of course there is the implied "real part of" associated with complex phasors. If we take the real part of the last equation, we get:
which is the same as the result in the cosine derivation.
If we hadn't explained all the steps, our derivation would have looked like:
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|Good references on WAVES||Good general references on resonators, waves, and fields|