# Why sound waves are drawn with $\sin/\cos$ shape?

I've always curious about one interesting fact. I will explain it with example. Currently, I'm watching an example which shows the following picture and says that box at the left is the speaker and it produces a sound wave.

I wonder why it's being drawn like the above. since it's a longitudinal wave, which actually doesn't have the shape like that. Since particles move in the same direction as wave, their motion is right and left back-and-forth. On the graph though, I get why they end up drawing sin/cos functions, but in the real motion, they don't have the shape as on the above picture. Transverse waves do though which I get why. Thoughts?

• It's much easier to draw a sine wave than a longitudinal wave. May 21 at 15:44
• Sine and cosine waves are used to model sound waves. That doesn't mean they are entirely the same thing.
– Alv
May 21 at 15:50
• If you graph pressure vs distance, you get a sin wave. But the real reason is likely historical. People draw waves that way. When you explain something yourself, you turn to what you have seen before in other explanations. Also it looks like a wave if you think of swells on the ocean. May 21 at 15:58
• Like every diagram in science this is an abstraction. We are representing the change in pressure or a small displacement on one axis and time on the other. Does it represent reality perfectly? No. It's just a tool to make reality easier to describe. May 21 at 16:04
• Thanks everyone. Appreciate your inputs !
– Matt
May 21 at 17:05

The reason is that it's quite difficult to draw a longitudinal wave in just space. For a real longitudinal wave, you would really want animations. You can find plenty, and they are quite useful for visualisation. The sine/cosine wave really just shows you the displacement of particles at each point. So for example, at the peaks of your diagram, you see that the particles have moved forwards from the equilibrium, and at the troughs, they have moved backwards. This also changes, it doesn't just stay that way. The particles that were initially moved forwards compared to their equilibrium will eventually reach the maximum backwards position. The key is that, unlike transverse waves, the displacement is in the direction of motion, so along the x-axis in your diagram.

I hope this made sense, feel free to message me for any clarifications :)

• Thanks. It made sense. Question though - Could we say the same thing for EM wave ? I know they don't need a medium, but I don't think however they move through vacuum would have the shape of sin/cos. and I believe we draw a graph of it as sin due to changing E and B. Do you think this is correct ?
– Matt
May 21 at 17:26
• Yes, I think it's the fields that are changing that are causing these sin/cosine graphs as representations, they are not actually displacing matter. May 22 at 20:15

Don't forget that a picture like that can also be interpreted as a graph, where the vertical axis of the graph can be whatever the person drawing the graph says it is. In this example the vertical axis of the graph could represent any one of the following:

1. longitudinal displacement of the material carrying the sound wave
2. pressure
3. density

Sound waves are't always drawn that way, it's just that they're usually drawn that way, especially as shorthand, or by people who don't quite understand what a compression wave is.

See this video for an example of sound waves shown as compression waves.

I've also seen sound waves draw this way in print, but while I can bring the figure to mind I can't tell you where it was -- I suspect a mid-70's ARRL handbook.

• Could we say the same thing for EM wave ? I know they don't need a medium, but I don't think however they move through vacuum wouldn't have the shape of sin/cos. and I believe we draw a graph of it as sin due to changing E and B. Do you think this is correct ?
– Matt
May 21 at 17:34
• There are more graphic representations of EM waves that don't just show a wiggly line in space -- a good text on electrodynamics should have them. Note that showing a wiggly line in space is a readily understandable and easy-to-draw shorthand, so I'm avoiding saying that one representation is better or worse. May 21 at 17:44