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# 1. Superposition Overview and Basics

### Overview

So far in the course we have discussed how a source could create a wave pulse, a repeating wave or a harmonic wave. By knowing the motion of the source, we have seen that the disturbance keeps its shape and propagates with a speed $$v_{wave}$$. These discussions all assumed that the medium was unperturbed before the wave propagated – what if there was another wave already in the medium? What happens when the two waves collide?

An example of how this could occur is if you and a friend both hold a rope. If you wiggle your end, the wave you make will propagate toward your friend. If your friend propagates her end, her wave will propagate toward you. What happens when your waves collide?  Another example is dropping two stones in a river. Eventually the ripples will overlap; how can we calculate the displacement from equilibrium?

To solve this problem, consider each displacement from equilibrium separately. For each piece of rope add (as a vector) the displacement from each wave. This “combined displacement” will be the total displacement for that piece of rope. This procedure is valid provided the amplitude of the wave is small.

For example, if your friend’s wave would have caused a particular piece of the rope to rise 2 cm, and your wave caused the same piece of rope to rise 1 cm, the actual amount that piece of rope will rise is 3 cm. The idea of adding the individual effects of waves to get the total effect is called superposition.

### Definition of Superposition

Superposition is the concept of adding the effects of two (or more) waves together at the same location at the same time. This gives us the total effect from the two waves. For material waves, "effect" means "displacement," although the principle of superposition works for non-material waves (such as electromagnetic waves, the pressure interpretation of sound waves, and matter waves). For the time being, let us concentrate on material waves. We express superposition mathematically as follows:

$\Delta y_{total}(x,t) = \Delta y_1 (x,t) + \Delta y_2 (x,t)$

where $$\Delta y_1$$ and $$\Delta y_2$$ are the displacement from equilibrium for wave 1 and wave 2 only. The actual displacement of the medium is described by $$\Delta y_{total}$$, as in the picture below.

###### Conventions on Space and Time

Some of our conventions are useful, but a little confusing at first. We have emphasized already that superposition is combining two or more waves acting at the same location at the same time. But so far we have dealt only with single source systems, and we have always chosen the origin of our coordinates to be the location of the source. What do we do if we seek to combine the effects of the two sources at an arbitrary location?

To keep as close as possible to the work we have already done on waves, we adopt the following conventions:

• We use a universal clock $$t$$. As we are combining the effect of the two waves at the same time, we should use the same value $$t$$ in $$\Delta y_1$$ and $$\Delta y_2$$.
• We use a different origin for each source. Even though we are combining the waves at the same location, we have two distances $$x_1$$ and $$x_2$$. Here $$x_1$$ is the distance between source 1 and where we wish to combine the waves; an analogous definition holds for $$x_2$$. We use $$x_1$$ for calculating $$y_1$$ and $$x_2$$ for calculating $$y_2$$ even though we are interested in the same point.

When we are using a sinusoidal wave we also need a convention for $$\phi_1$$ and $$\phi_2$$, the phase constants. The convention we use here is that $$\phi_1$$ determines $$y_1$$ at time $$t=0$$ and $$\phi_2$$ determines $$y_2$$.

The reason that we use these particular conventions, rather than just picking one origin, is that it allows us to keep the formulas

$\Delta y_1 = A_1 \sin \left( \dfrac{2 \pi t}{T_1} \pm_1 \dfrac{2 \pi x_1}{\lambda_1} + \phi_1 \right) \textrm{ and } \Delta y_2 = A_2 \sin \left( \dfrac{2 \pi t}{T_2} \pm_2 \dfrac{2 \pi x_2}{\lambda_2} + \phi_2 \right)$

(Here $$±_1$$ and $$±_2$$ refer to the direction of propagation of wave $$1$$ and $$2$$ respectively, and are independent.)

There is one more convention that is worth noting: we treat $$x_1$$ and $$x_2$$ as positive distances from the source. For a wave that travels outward (this is almost always the case) we would use the − sign. This is because the peak of a wave (for example) gets further away from the source as time increases. We would only use the + sign when waves were traveling inward.

### Constructive and Destructive Interference

While the idea of superposition is fairly straightforward, there is a lot of associated vocabulary that comes with it. Intuitively we can see that if two waves displace in the same direction at the same time, then the resulting wave from will be larger than either of the two initial waves. This is called constructive interference. On the other hand, if one wave displaces up and the other displaces down, for example, the resulting wave will be smaller than the initial waves. This is known as destructive interference. If the waves are the same amplitude, then these waves will cancel each other out completely! Partial interference is any kind of interference that isn't completely constructive or completely destructive. The term "partial interference" is not very descriptive – we can have partial interference that is either almost constructive or almost destructive.

### Limits of Superposition

Because simply adding the waves together is the most obvious thing to do, it is worth pausing and considering if it is the only way we could have combined  the effects of two waves. Actually, we could have combined the waves in much more complicated ways. For very large water waves or sound waves we cannot simply use the principle of superposition presented here. shock waves, such as the ones produced by explosions or sonic booms, are examples of waves for which the principle of superposition simply does not work.

It should be appreciated that the principle of superposition is an experimentally verified result; it's not one that can be derived from purely logical thought.  We are lucky that for “small” waves the principle of superposition is adequate.