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Oscillations and Waves

Page history last edited by Joe Redish 7 years, 2 months ago

Course content

 

So far in this class we've studied how objects move in response to the influences (forces) that other objects exert on them. We've developed Newton's laws, the concepts of momentum and energy, conservation laws, and have even looked at what happens when lots of particles (atoms or molecules) interact with each other in a chaotic or random way. In this section, we will consider a particular kind of coherent motion -- oscillation. In oscillation, a motion repeats itself, sometimes for a very long time. (Forever, in our hyper-idealized ignore-friction ignore-thermal-effects world!) A repeating motion is called periodic -- meaning that there is a time period over which it repeats. Periodic motions have tremendous importance in biology, ranging from the beating of an animal's heart to the electrical oscillations in its brain. Many organisms respond to daily changes and have built-in oscillators that lead to circadian ("about a day") rhythms. Oscillations are also knows to occur in ecosystems, for example in the interaction of predators and prey. Simple mechanical oscillations serve a models, analogies, and starting points for writing equations that can be complexified to describe more realistic biological systems.

 

An interesting and important extension of the oscillation of a single object or system is the connected and coherent oscillation of many parts of a system together. In these cases, we get something much more than just a bunch of oscillators. A whole new way of looking at motion emerges -- waves. The study of waves is a major topic in physics and is the basis for our understanding of such biologically important phenomena as sound and light. Wave theory is quite a bit more complex than particle theory, since it is equivalent to treating the motion of a large (perhaps infinite) number of particles. The result is that instead of just studying a dependent function of one variable (what is the object's position as a function of time) we are studying functions of two or more variables (what is the value of what we are looking at as a function of both position and of time). We can get away with this and build some coherent understanding because we look at particular coherent motions.

 

What turns out to be surprising -- if beyond the scope of this class -- is that fact that when we get down to a sub-molecular theory of the particles of which ordinary matter is made, that the fundamental theory that replaces Newton's laws turns out to be a wave theory. The smaller you get, the less you can get away with treating even a single particle as "just a particle". It behaves like something new and that has a fundamental wave nature that appears to be built in, not constructed out of the coherent motion of smaller objects. Our picture of electrons in atoms and molecules and of protons and neutrons in nuclei is a blend of a mental picture of particles and waves -- and really neither. But properties like chemical bonding or how a photon carries energy and momentum require a wave description of how they move and interact.

 

We will begin building up these difficult concepts a step at a time, moving through systems that we can fully understand both conceptually and mathematically so as to develop the tools needed for a description of waves in general. Here is our sequence:

 

  • A mass on a spring -- This is the simplest example and the one that makes connections with the basic concepts of forces and energy through Newton's laws. It provides the basic metaphor for oscillation.
  • Transverse pulses on a long stretched elastic string --  This is the simplest system that shows wave phenomena. It corresponds to a set of many masses connected by springs. It provides the leap into two-variable wave mathematics -- describing the displacement of the spring as a function of distance along the spring and time.
  • Sinusoidal pulses on a long stretched elastic string -- This example allows us to introduce the idea of periodic waves and provides a basic for the understanding of many fundamental concepts in wave theory such as wave length and interference (beats).
  • Water and sound waves -- These bring us into the realm of 2 and 3 dimensional waves and allows us to study the phenomenon of interference in space.
  • Light and other electromagnetic waves -- For many phenomena, light turns out to be best described as oscillating electric and magnetic fields traveling (very fast) in 3D. It includes waves beyond the visible including infrared, ultraviolet, radio, and X-rays.

 

The tools generated in these studies will allow the analysis in your biology classes of a wide range of phenomena, both in studying the sensory capabilities of animals -- for example, sound detection in hearing animals and sonar in bats and dolphins -- and in understanding the wide variety of modern probes that use waves to probe biological systems -- ultrasound, MRI, x-rays, and many more.

 

So what's a "wave"? Talking terminology

An issue that sometimes causes confusion in the discussion of wave theory is the basic question of how we use our terms.  What's a "wave"? Sometimes, in common speech we mean a single motion or change of something as a function of time; the wave of a hand, an ocean wave approaching the beach. Other times we use it to indicate something happening repeatedly; a sinusoidal wave continually oscillating many times, like sound or light. Here, we will use the term as follows:

 

A wave is any change in a variable that depends on both position and time where the space and time changes are correlated.

 

We are this general because some of the waves we will be talking about -- in particular electromagnetic waves -- are not actually describing the motion of anything; just the change of the values of the electric and magnetic fields as a function of position and time. So this includes a single pulse and multiple oscillations. We will use the specific terms

 

  • a wave pulse is a localized variation in the value of some variable in space that propagates (moves through space) in time;
  • a sinusoidal wave is an extended variation in the value of some variable in space that propagates in time that typically looks like a sine or cosine curve at any instant.

 

So, "wave" is the general term including both a "wave pulse" and a "sinusoidal wave". We'll use the latter terms when we want to specify which we mean. Note it is the variation in the variable (displacement, E field, etc.) that moves when we talk about a wave -- not  objects.

 

[Technical comment: It seems funny to pick a particular shape like sines and cosines as a special case. But these turn out to be particularly important -- and common. They allow us to define the concepts of wavelength and frequency. Any pulse or shape can be made up of combinations of them. Analyzing the different sinusoidal components of a signal is called spectral analysis and is what lies behind a large variety of powerful tools.]

 

Joe Redish 3/11/12

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