Traveling Waves In A Vibrating String

Vibrating strings in many instruments, such as guitars and pianos, behave physically approximately like finite-length one-dimensional waveguides with inverting reflections, so we will first study the physics of this idealized case. As you will recall from the traveling waves laboratory assignment, the traveling-wave solution for an infinitely-long one-dimensional waveguide is

$$y(t,x) = y_l(t,x) + y_r(t,x)$$ (1)

where $y_r(t,x)$ is the right-going traveling wave, $y_l(t,x)$ is the left-going traveling wave, and their sum is $y(t,x)$. For example, if the wave variable is displacement, then $y_l(t,x)$, $y_r(t,x)$, and $y(t,x)$ are displacements and are thus measured in meters.

A vibrating string that is rigidly-terminated at $x=0$m and $x=L$m, will be subject to boundary conditions that cause traveling waves to be reflected with a sign inversion. This kind of reflection is termed an inverting reflection.

$$y_r(t,0) = -y_l(t,0)$$ (2)

and

$$y_l(t,L) = -y_r(t,L)$$ (3)

Figure 1 shows how the vibrating string behaves when it is initialized at $t=0$ according to a triangular pluck for $L=1$m. The top and middle frames show the right-going and left-going traveling waves, respectively. Notice that for each of these two upper frames, the wave traveling off of the right side is inverted and arrives on the left side of the other according to (2) and (3). The traveling wave-based behavior is quite simple.

The lowest frame shows the vibrating string's net displacement. This is what you actually see when you watch a string vibrate, although it happens so fast in real life that it looks like a blur. Due to (1), the lowest frame of the animation is the sum of the upper two frames. You can see that its behavior appears to be more complex.

Figure 1: Traveling waves in a vibrating string

Note that here the waves are represented as functions of time $t$, where $t$ is a real number. Even over a finite window of time, there are an infinite number of points at which these functions may be evaluated. For example, over a one second window (i.e. for $0 \leq t \leq 1$) at the specific point $x_0$ along the waveguide, the function $y(t,x_0)$ may be evaluated at an infinite number of points. Try to imagine how one would go about storing the function $y$ in a computer. It seems like the computer would require an infinite amount of memory!