What happens when you pluck the ocean?

Have you ever wondered why A440 sounds different on a piano compared to a violin? And then different again on a flute? Well, the short answer seems awfully simple and almost stupid: it's because the instruments make different sounds.

Anybody who has ever played a violin or a guitar or any other string instrument has probably noticed that there are two main things that affect what note comes off a string when you pluck it: the string length and the string thickness. When you pluck the string, the sound you get is due to how the string vibrates, and then how that vibration resonates within the instrument. Most of what you hear is the pitch of the note you plucked. But the richness of the tone of the instrument, that stuff that makes the violin sound different from the guitar, is due to all the other sounds that also resonate.

The exact physics of that difference in sound becomes very complicated very quickly. For example, most violins look more or less the same, but can sound drastically different from one another. It is because of subtle differences that every violin sounds different. Science describes these differences using tools called eigenvalues and eigenvectors. If you took first year Algebra in university you were probably exposed to these things, and you probably hated them.

In a nutshell, the reason instruments sound different from one another is because when you play a note, a whole bunch of other sound waves (lets call them modes) are also emitted from the instrument when it resonates. Which other sound waves (modes) are excited, and how loud they are relative to one another, is what makes a piano sound different from a squealing toddler. As luck should have it, these modes must obey the rules of some specific mathematical equation. Solving the equation may be quite simple for something like a drum, but can get very complicated for something like a violin. This is because the equation takes into account the shape of the object and its composition (e.g. if it is made of plastic or wood or crystal). Not just any random mode will satisfy the equation, only very specific ones do. These very specific modes are called eigenvectors and each one has an eigenvalue associated with it.

If you ever saw the movie the Red Violin, you might remember scenes where million-dollar isntruments are being tested with equipment resembling oscilloscopes. The scientists there would basically be examining how the instrument resonated, or in science-babble they would be determining what its eigenvectors were.

Q: So why are you writing all this, who cares, and what does it have to do with the ocean?

Well, often when people ask me what I do, I have a hard time explaining it. On the bus home yesterday I came up with an analogy to plucking strings on an instrument. Lately I've been fitting my ocean data to the eigenvectors of the Saint Lawrence Estuary near Tadoussac. You see, the tides "pluck" the ocean at specific frequencies, but then physics causes all these other waves to resonate as well. The speed and shape of the resonated waves are governed by the eigenvalues and eigenvectors of the water column, constrained by Newton's second law and the assumption that matter is not created or destroyed. By knowing which eigenvectors to look for, I've been trying to find specific waves in my data, separate them from everything else in my data, and then figure out which bloody direction they're going.