The Physics Philes, lesson 94: Sinusoidal Infection
Last week we started talking about mechanical waves. Now that we have some of the essential knowledge under our belts, let’s dig in a little deeper. It’s time to start talking about periodic waves.
A note: Everything you’re about to read goes for both transverse and longitudinal waves. But it’s easier to visualize, I think, by describing it in terms of transverse waves. Just remember that this stuff works for both types of wave.
Think about a rope stretched out in front of you. If you pick up on end of that rope and shake it up and down once, you’ll see a wave flow through the rope. That wiggle that ripples through the rope is called a wave pulse. Now, imagine that the rope is attached to the wall and you shake the other end up and down at set intervals. The resulting wave will be a periodic wave; motion that repeats itself in a definite cycle. In a periodic wave, each particle undergoes periodic motion.
There is a particular kind of periodic wave that is especially important to talk about. These waves are called sinusoidal waves. Basically, these are waves that move in simple harmonic motion. (Remember simple harmonic motion?) When a wave moves in SHM, there is a symmetrical sequence of crests and troughs. (Crests and troughs are the peaks and valleys of the wave.) Sinusoidal waves are important because they are comparatively easy to analyze and any periodic wave can be represented as a combination of sinusoidal waves.
OK, so we have this wave moving through a medium in SHM. As the wave moves along, any point along the wave is moving in SHM, as well, and does so at the same frequency. This makes some intuitive sense. It would be kind of weird if one part of the wave oscillated at a different frequency as the rest of the wave. (Remember: We’re talking here about how the wave moves, not a particle. A wave moves at a constant speed along the length of the medium. A particle moves up and down or side to side around it’s equilibrium position.)
Since periodic waves are basically just the same wave pulse traveling through a medium over and over again, you may have guessed that the shape of the wave at any instant is is a repeating pattern. For any particular spot on the wave, there is a corresponding point on the next and previous repetition. The distance from one of these spots to the next is called a wavelength (often denoted with the Greek letter λ.) This pattern will travel at a constant speed of one wavelength per period (or, in math, λ/T). Since the frequency of a wave is equal to 1/T, the speed of propagation is equal to the wavelength times the frequency. However, the wave speed is often determined by the mechanical properties of the medium. So, an increase in frequency causes a decrease in the wavelength. That means that the velocity ends up saying the same and waves of all frequencies propagate at the same wave speed.
As I stated at the beginning of this post, all of this holds for longitudinal waves. But there are a couple of things that I should try to explain. Imagine a tube filled with some kind of fluid with a piston at one end. If you push and pull the piston in simple harmonic motion, there will form areas of pressure and density that are more or less than the equilibrium values. Areas of increased density is called areas of compression and areas of decreased density are called areas of rarefaction. But even though the waves look different, the math still holds.
Keep sinusoidal waves in the front of your mind, because we’re going to be talking more about them for the next couple of weeks. next week we’re going to be a bit heavy on the math, as we will all learn about mathematical descriptions of waves, i.e. wave functions. See you then!
Featured image credit: Schristia