When a guitarist plucks a string, you might imagine something very simple happening: the ends of the string are fixed in place, while the middle of the string is free to move, which makes the string vibrate forward and backward. The truth, however, is much more complicated than that. A clue to this might come from the fact that a guitar and a piano sound notably different. However, it isn’t just the shape or design of these instruments alone that make them sound different. The thing we must consider first and foremost is how the strings vibrate.
All About Vibration
A strong influence on the sound of any instrument is not only the up-down motion of the string, which is visible to the naked eye. High-speed photography has revealed that the strings actually have an extremely complex range of motion. The richness of the sound comes from the fact that, in any particular solution, there is an infinite number of fundamental solutions to the equation, which can be combined to form something of a ‘super-solution’, in which all the possible variations of the string happen together. The primary solution is what physicists call partials or what musicians in their musical dialect call overtones or harmonics.
You can get a more intuitive and practical idea of this by finding a friend and handing them one end of a long light rope. Stretch it out, each of you taking an end such that the rope does not touch the ground, but is also not too taut. Move your hands in such a way that the rope starts resembling the motion of a skipping rope. This is one fundamental solution. Now, move your hands twice as fast, and you’ll be able to observe a new pattern, with the rope hardly moving at all at its exact centre, but vibrating enthusiastically between the central points and your hands. The point where it isn’t moving is called a node. You may find that by moving your hands even faster, you can produce other patterns of vibrations as well. Each of these is another fundamental solution to the wave equation.
The wave equation is just simply about musical instruments. It applies equally well to electromagnetic waves and waves in fluids. It works for standing waves, such as those produced by a guitar string, as well as traveling waves like the ripples of a pond. The equation is just as good at working with shock waves caused by volcanoes and earthquakes as it is with X-ray waves. It also plays a vital role in the fundamental particles of our existence, and by extension, plays a crucial role in quantum mechanics. The above wave equation is quite general, as it works on any number of dimensions. To investigate the idea more fully, let’s revisit our previously discussed rope analogy involving you and a friend. Any point across the rope can be marked as x. When the rope is pulled tight, all those points are in a straight line at the same height; let’s call this height u, where the starting height u=0. Finally, we want to watch the wave weaving, which requires time, and we will call this t. Now we will able to construct a function u with x and t as input parameters, and call it u(x, t).
Now, using the function u(x, t), we are able to know everything about the wave; given any position along the rope (x) and a time (t), we will be able to say how high a point can be at any given time. From this, we can reconstruct what the rope looks like at any moment, and repeat this for different moments. Finding the function u(x, t) is what we mean by finding a solution for the wave function. On the left-hand side of the equation is the rate of change of u concerning t. Fixing any point x can tell us how the point is accelerating at the moment. On the right is the Laplacian of u. This shows how, if we freeze time, the height is varying close to each point.
The Laplacian operator (the inverted triangle) is multiplied by c2; c itself is the speed at which the wave moves through the material. Either way, squaring it makes the units agree on both sides of the equation, which is essential. Finding a solution to the wave equation—that is, for the function u(x, t) that make it real—was a necessary problem in the 18th century that motivated some fundamental mathematical research. The different ways in which you can stretch the rope to vibrate are all viable solutions. It turns out that these can be modeled using sine waves—simple functions that model the oscillating movement. An acoustic sine wave sounds pure and simple, much like a flute. The famous mathematician Fourier came up with the idea that if you have solutions to the Wave Equation, you can make lots of new ones by simply adding multiples of what you already have. This produces many more complex solutions, all of which represent ways the rope could vibrate if you could only persuade it to do so. In conclusion, any wave conceivable in nature can be generated by adding up several sine-based waves of the wave equation, allowing us to better study and understand them.