Traveling Waves In A Vibrating String

Imagine dropping a pebble in a bucket of still water. After the pebble first touches the water, waves propagate away from the disturbance. Although the waves spread out, their general form remains the same until they reach a change in the medium. In this example, when they reach the bucket wall, they reflect back away from the wall.

Wave propagation is simple in any uniform medium. By considering a few types of reflections, we will be able to understand a large number of wave phenomena in musical acoustics. We will use a one-dimensional waveguide, where wave motion is confined to a line.

Imagine a perfectly uniform, infinitely-long, one-dimensional waveguide. This could for example be an infinitely-long vibrating string. A wave travels either to the right or to the left in such a medium since the wave will never encounter anything that it can reflect off of. Consider a wave traveling to the right. At any time $t$ and at any position $x$ along its length, the displacement of the string due to the wave has a particular value, say $y_r(t,x)$. This means that the right-going traveling wave is described by the function $y_r$. Similarly, $y_l(t,x)$ may describe the left-going traveling wave. Only their sum

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

is observable where $y(t,x)$ is the actual, physical string displacement at time $t$ and position $x$.

Figure 1 shows how an infinitely-long vibrating string behaves when it is initialized at $t=0$ according to a triangular pluck. The top and middle frames show the right-going and left-going traveling waves, respectively. The lowest frame shows the vibrating string's net displacement. Due to (1), the lowest frame of the animation is the sum of the upper two frames.

Figure 1: Traveling waves in an infinite-length waveguide

In this case, the wave variable is displacement, so the appropriate unit for $y(t,x)$, $y_r(t,x)$, and $y_l(t,x)$ is the meter. Other wave variables such as velocity, acceleration, force, and slope waves may describe wave motion in an analogous manner.

Vibrating columns of air also behave approximately like one-dimensional waveguides. In this case, the wave variables may be pressure, velocity, etc [1].