Numb3rs 521: Disturbed

This episode contains references to SETI, inferring patterns from incomplete data, and a brief reference to the speed of light, which we'll use as the starting point for a discussion of Hyugen's principle.

SETI and Statistics

In one of the first scenes of this episode, Charlie talks about the SETI program (which stands for Search for ExtraTerrestial Intelligence) to describe an analogy for some work he had done. First we'll talk about the SETI program some, and then describe how ideas used in it might (or might not) be able to be useful for finding criminals.

As you might guess, the goal of SETI is to try to find intelligent life on a planet other than Earth. This goal as stated is definitely too broad to be useful, so some more assumptions and simplifications have to be made. The first (and most important) simplification is that SETI programs only look for sophisticated life that is able to transmit some kind of powerful radiation over long distances. This is because at present we don't have any other way of proving that there is life on some particular planet. This isn't too surprising because convincing observational evidence of the existence of other planets has only been found within the last decade.

A further assumption that SETI programs make is that other civilizations will try to send messages across the galaxy indicating their existence. Initially, most SETI programs will further assume that another civilization will send out strong radio waves in some particular bandwidth. This assumption is mainly based on practical grounds, because it's technologically very difficult to scan many frequencies of radio waves at once. Vastly improved computer technology allows modern SETI programs to search over much wider bandwidths, but there's still a lot of unscanned bandwidth. Even if there are signals, there's a good chance that we would miss them.

Astrology or Astronomy
In one scene the nerdy conspiracy theorist says "Thanks, astrology dude" to Larry Fleindhardt, and he looks visibly peeved and replies "It's astronomy." Astronomy is the scientific study of questions about stars, such as how far apart they are, their chemical composition, the process of their formation, etc. Astrology is the use of the positions of the stars to try to predict human events, and is completely unscientific. Larry gets upset because people get confused between the two areas fairly often, and astronomers (and other scientists) look down on astrology because it has no scientific basis. Calling an astronomer an astrologist is worse, in the astronomer's eyes, than describing Tiger Woods as a good putt-putt golfer.

A third assumption that is made is that other civilizations will want to be discovered, and that in order to be discovered, they will broadcast signals that are recognizable to us. It's not clear that another civilization would want to be discovered, but if they do, then it is clear that they will send out some kind of signals that they think are easy to notice. However, this doesn't necessarily mean that these signals will be easy for us to notice. For example, the other civilization might send out very strong pulses of radio waves at regular intervals, and if these were sent in our direction on a bandwidth that the SETI project is scanning, then we would probably notice them. However, even in this scenario the SETI project might still be unsuccessful. It could be the case that the life span of individuals in the other civilization is extremely long, perhaps 10,000 years, and in this case they might send out a strong pulse every 100 years, which is longer than the SETI program has been in existence. Alternatively, they might have a much better understanding of the laws of physics, and they might send out some other type of signal that they could distinguish from the background radiation in the universe, but that we would be unable to detect because we don't understand background radiation well enough.

  1. If you were given the task of trying to alert alien civilizations about our prescence, what would you do?
  2. Can you think of other ways that alien civilizations might try to send us signals that we wouldn't receive?
  3. Can you think of reasons why we (or alien civilizations) might not want to be discovered by another civilization? (Think about what has happened on this planet whenever a technologically advanced culture encounters a less technologically advanced culture for the first time.)

The relevance of SETI to this particular Numb3rs show comes from the common goal of trying to find a pattern from noisy data. This theme has occured in several Numb3rs episodes, and in this particular episode Charlie suspects that there is an uncaught seriel murderer in Los Angeles and is trying to figure out which unsolved murders he has committed. A difficulty in this particular case is that Charlie suspects that the killer is deliberately trying to avoid patterns so that he would be much harder to catch. Charlie uses this assumption and some unexplained math to find patterns, which lead them to the killer. One problem with this scenario is that if the killer is deliberately trying to avoid patterns, it will be almost impossible for Charlie to find any patterns or to find the killer. In the show Charlie is able to do some fancy (and completely unexplained) analysis to obtain a "spatio-temporal visualization model" that would predict where the killer will strike next. In this case it seems that the show's writers are just using the math as magic.

Huygen's Principle

At one point in the episode Larry says that looking at stars that are very far away is like looking very far in the past, and then he compares this to looking at the past of the seriel killer they're trying to find. This isn't necessarily the best analogy, but it does lead to an interesting discussion about the way that waves travel through material. There is a very important physical principle that is hidden in Larry's statement, which is the fact that nothing in the universe can travel faster than the speed of light. Light moves so fast that without fancy equipment, we can't tell that it actually has a finite speed, but numerous experiments have confirmed the fact that it actually does, and have also measured its speed with extraordinary accuracy.

The wave equation is what's called a differential equation. If you know calculus, this means that we want to find a function u(x,t) which satisfies an equation involving u(x,t) and derivatives of u(x,t). For example, if we wanted to model the behavior of waves in a string that is stretched tight (like a piano string), we would use the 1-dimensional wave equation, which is \frac{d^2u}{dx^2} = \frac{du}{dt} . Usually when we solve the 1-dimensional wave equation we know what the position of the string is and want to figure out what position it will be in at some later time. Mathematically, we can represent this situation by modelling the initial position of the string as a function f(x) and then trying to find a solution to the wave equation that also satisfies u(x,0) = f(x) . The function f(x) is called the initial condition.

The way that light travels is controlled by a differential equation called the wave equation. (To be completely precise, perhaps we should say that the wave equation gives extraodinarily accurate predictions about the behavior of light.) It turns out that the same equation is a good model for the behavior of other things, like the vibrations in a long, tightly stretched string, or the behavior of waves in a deep pool of water. We mentioned above that everything in the universe travels slower than the speed of light, but it turns out that this isn't just a property of light, it's a property of anything whose behavior is modelled by the wave equation. This statement is usually known as the weak Huygen's principle. More formally, this says that if we want to solve the wave equation with some initial condition f(x), then the solution at a fixed point in time and space, say u(x_0,t_0), will only depend on the value of the initial conditions at points x satisfying \lvert x - x_0\rvert \leq t_0. Intuitively it makes sense that the wave equation satisfies Huygen's principle because in each of the physical situations that the wave equation models, the waves have a maximum velocity.

You might ask "why did you call this the weak Huygen's principle instead of Huygen's principle?" It turns out that there's a strong Huygen's principle, which is related to the weak one but, as you might guess, is much stronger. Stated informally, the weak principle says that waves have a maximum velocity, but the strong princple says that waves have a maximum velocity and also don't scatter at all. More formally, it says that the solution at a fixed point in time and space, say u(x_0,t_0), will only depend on the values of the initial condition at points x that satisfy \lvert x - x_0\rvert = t_0.

Short Question
What's the difference between the two formulas for the weak and strong Huygen's principle? Can you figure out what this means geometrically?

Surprisingly, the wave equation satisfies the weak Huygen's principle in any dimension, but it only satisfies the strong Huygen's principle in odd dimensions. This might sound very abstract, but there's actually a real life example which demonstrates this that everyone has seen before.

Two Simple Experiments
  1. Take a pebble and throw it into a still pond (or other relatively large pool of water). Look carefully at the point where the pebble first hits the water. After the pebble hits, does the water where the pebble hit stay still, or continue to move up and down? What about another point in the pond (that's pretty close to where the pebble hit)?
  2. Snap your fingers, clap your hands, or make some other sharp noise. Do you hear the clap just once, or more than once?
  3. How do these two experiments demonstrate the statement that the wave equation only satisfies the strong Huygen's principle in odd spatial dimensions?