Overview: Waves in 1D


Our first step in building up an understanding of wave phenomena was to look at the oscillation of a single mass that had a stable point, a restoring force, and a tendency to overshoot as it came back towards the stable point. This led it to oscillate in response to a displacement. Our next step is to look at the coordinated oscillations of many connected masses. Eventually, we will take what we have learned about oscillations of many objects distributed in space and "cut loose" from the masses so we can talk about oscillations of things that aren't masses — like electromagnetic fields and quantum wave functions.

But understanding how this complex coordination builds up in space and time is quite challenging. We'll begin by considering (as we usually do) the simplest possible system that shows the phenomenon — transverse waves on a stretched elastic string (or spring). This has the advantage that we only have one space variable so we can figure out how it coordinates with the time variable. And the motion of the matter is in a different direction (transverse) from the motion of the wave so another confusing issue can be clarified.

Just as for the mass on a spring, the motion of the bits of an elastic spring are controlled by there being a stable position, a restoring force, and a tendency to overshoot. But instead of having one mass whose motion we have to pay attention to, we have an infinite number of them! Getting this straight is tricky, so it is definitely worth the effort to understand this system before moving on to the more interesting 2 and 3 dimensional examples.

Here's an outline of how we will analyze the transverse motion of an elastic string (or spring).

  1. Build a toy model — We'll start by modeling the string as massive beads connected by massless springs. That allows us to focus on the restoring force and the inertia separately. (You can think of these as molecules and molecular forces if you like.)
  2. Consider the propagation of a pulse — We'll work out the math of how to describe this system in a simple example: the propagation of one pulse moving down the string. 
  3. The superposition principle — We develop the rules for what happens when multiple pulses overlap together at the same place and time.
  4. Sinusoidal waves — This will take us to look at periodic (repeated) pulses: motions in which each bit of the string oscillates like our SHM, but all the different oscillations are coordinated in particular ways.
  5. Combined sinusoidal waves — Finally we will consider a variety of interesting phenomena including beats, standing waves, and spectral analysis. Besides being important phenomena on their own, these are good preparation for looking at interference phenomena in 2 and 3 D).

Once we have these concepts and tools down pat, we will break loose from 1D and consider water and sound waves.

Joe Redish 6/11/19


Article 682
Last Modified: June 11, 2019