Using graphs to represent waves

It is useful to represent a wave with a graph. In a rope wave, this can mean plotting the displacement of each point on the rope against position along the rope. In a sound wave, it could be plotting the air pressure against distance along one direction of the sound.

One advantage of using such a graph is that we can tell at one look exactly what the displacement at any point on the rope is. For sound wave, it is especially useful. Even if we can see the molecules moving, a longitudinal wave does not look like the familiar waves on a rope or on the water surface. One look at a graph of molecule displacements against distance, and it is immediately obvious that sound is a wave.

This is what a graph of a rope wave may look like. Position along the rope is given by x, and displacement at each point by y. The solid curve is just like a snapshot of the rope at a particular instance in time, say t = 0 s. The dashed curve is the rope after a short time. An important feature of a wave is that there is a pattern that keeps repeating itself. The distance of such a pattern is called the wavelength, λ.

There is another way we can plot a graph about the rope - we can plot one against time. We need to look at one point. Lets look at the point at x = 0. After a short time, the point at 0 moves down. So a graph of its displacement against time would look like the solid curve below. Compare this with the solid curve above. It is upside down! We must be careful not to think of them as referring to the same thing. The graph above is for the whole rope at one time. The one below is for one point on the rope at all time. In this graph against time, it also looks like a wave. This time, the repeating pattern is the period, T. The dotted curve below is the displacement-time graph for A, a point that is next to x = 0 above.

Now lets look at sound wave. In the figure below, the little grey balls are the air molecules. The little ticks on the x axis are their original positions when it is silent. Then someone shouts, and the molecules get displaced. They knock into each other and this oscillation is made to travel in the same direction as the displacements. The little arrows show their displacements at a particular time. It looks nothing like the wave we have seen on the rope above.

Let us plot a graph of these displacements against the position x. Remember that displacements have directions. I shall choose right to be positive, and left to be negative. The graph is then shown by the vertical arrows in the figure below. I have drawn a dashed curve through the tips of the arrow. Does it look like a wave now?

There is something interesting here. Do you see that where the molecules come together, the displacement is zero? Where the molecules come together, the pressure would also be high. So this is the compressed region. You can also see that where the molecules move apart, the displacement is also zero. This is where pressure is low, and is called the rarefied region. So maximum pressure does not happen at the same place as maximum displacement. We say that they are out of phase. In this case, they are out of phase by a quarter of a wavelength, or by 90°.




Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 1 June 2011.