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Rigid Terminations

Waveguides in real musical instruments do not have infinite length, rather they begin and end at terminations. Since a termination may be interpreted as a change in the medium, it is almost always responsible for a reflection. Let's start by considering the terminations of a typical vibrating string.

The vibrating string on a guitar is terminated at the bridge ($x=0$m) and the nut ($x=L$m). Roughly speaking, the terminations are rigid, meaning they force the local string displacement to be zero. They can be considered to be nodes.

Mathematically, we have the boundary conditions $y(t,0) = 0 = y(t,L)$, and so due to (1)


\begin{displaymath}
y_r(t,0) = -y_l(t,0)
\end{displaymath} (2)

and


\begin{displaymath}
y_l(t,L) = -y_r(t,L)
\end{displaymath} (3)

This means that a rigidly-terminated vibrating string causes displacement waves to change sign when reflected at terminations. This kind of reflection is known as an inverting reflection. Figure 2 shows how the rigidly-terminated vibrating string behaves when it is initialized at $t=0$ according to the same triangular pluck as Figure 1, where $L=1$m. The top and middle frames again 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 side is inverted and arrives on the same side of for the wave traveling in the opposite direction according to (2) and (3). The traveling wave behavior is quite simple.

The lowest frame shows the vibrating string's net displacement. This is the quantity that you can actually see when you watch a string vibrate, although it happens so fast in real life that it looks like a blur. Here we slow down the behavior so that you can see and understand how it happens.

Figure 2: Waves in a rigidly-terminated vibrating string
Image continuous-terminated

The triangular pluck, or initial condition, is idealized for simplicity. In real life, it would be hard to start a string vibrating with this condition. For example, you would need a pick for each of the three kinks. Most guitarists use at most one pick per string-certainly not three picks for one string!


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Download travelingwaves.pdf

``Traveling Waves In A Vibrating String'', by Edgar J. Berdahl, and Julius O. Smith III,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2007-06-10 under the Creative Commons License (Attribution 2.5), by Edgar J. Berdahl, and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA