# Superposition of waves in 1D

#### Prerequisites

We have discussed carefully how a pulse propagates on an elastic string and have constructed arguments as to how fast the signal will propagate. Our next step is to consider what happens when two different signals meet each other. This phenomenon — the superposition or adding together of different signals — is the basis of a lot of important and interesting examples, ranging in one dimension (e.g. for waves on a string) from beats (see "Without missing a beat") to standing waves, and in two and three dimensions representing the complex and interesting case of interference, which produces extremely valuable tools that can measure very small distance shifts.

The easiest case to consider is that two separate pulses approach each other from opposite sides of the elastic spring as shown in the figure below.

Let's look at this in the bead & massless spring model. Consider the bead right at the origin.  If only the pulse from the left were approaching, when the pulse first reached the bead at the origin, we would get something like shown in figure A below with a bead on the left pulling up. If only the pulse from the right were approaching, when the pulse first reached the bead at the origin, we would get something like shown in figure B with a bead on the right pulling up. In the situation shown above, where both pulses are coming, we get something like figure C.

In this case, it's easy to see that the effect on the bead in the middle will be the sum of the two forces that are being exerted by the two pulses.

A second case we can look at easily is when one of the pulses is the negative of the other. The situation will then look like the figure below.

In this case, it is easy to see that when the pulses reach the bead at the origin, the beads to either side will be pulling in equal and opposite directions — and they will cancel.

The result will be no net force on the bead — and therefore no acceleration.

In both these cases we can see that the result is that the "signal" that each pulse is giving the bead at a particular point adds together — with the sign mattering. We can express this as a simple rule in words:

When one or more disturbances on an elastic string overlap, the result is that each point displaces by the algebraic sum of the displacements it would have from the individual pulses.

We can express this mathematically very simply:

If two waves $f(x,t)$ and $g(x,t)$ are both present on a string, in any region where both functions say the string should be displaced (they are non zero), the total result at each point $x$on the string and time $t$ is

$$y(x,t) = f(x,t) + g(x,t)$$

that is, the two displacements simply add algebraically.

This result is referred to as the principle of superposition. For displacements on the elastic string, these results rely on the small angle approximation, so for larger displacements, these rules might be violated (usually not by much). But for other situations, such as sound and light, the principle of superposition is extremely good.

## Why it's important

While the superposition of pulses along a long spring might seem like a rather arcane and peripheral topic, the adding together of different oscillatory signals turns out to be of immense significance when dealing with the oscillations of matter or electric fields that correspond to sound and light.  In our follow-ons we will see that adding together travelling waves can produce extremely interesting patterns, and ones that carry significant information. One of the most important is interference, where two waves of the same frequence traveling different paths are brought back together. The pattern produced has the effect of magnifying very tiny effects dramatically. For example, in the laboratory, using only a meter stick and one equation, you can measure the wavelength of light, even though it is much too small to measure directly — fractions of a micrometer. In 2016, this phenomenon was used to make the first measurement of gravitational waves. By comparing how light combined after oscillation down 4 km of travel, scientists were able to detect changes in the length of the 4 km arms to 1 part in 1021. Other important patterns that arise from combining oscillations include beats and standing waves. All three of these phenomena have powerful implications in developing tools for studying microscoping (biological) systems.

Joe Redish 3/30/12

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