Sampled Traveling Waves

We may apply sampling theory to solve this problem practically. According to the Nyquist-Shannon sampling theorem, as long the function $y(t,x_0)$ does not contain any energy above the frequency $f_S/2$, we may represent it exactly using

$$y(nT,x_0) = y(t,x_0)$$ (4)

where $n$ is an integer and $T=1/f_S$ is the sampling interval, which is measured in seconds. In a practical sense, this means that any finite-length signal we can measure can be stored using a finite number of points. This is illustrated in Figure 2 where each circle represents a particular value of $n$. Note that because the waves do not change too fast (i.e. they do not contain any energy above $\frac{f_S}{2}$Hz), the sampled or digitized vibrating string representation in Figure 2 behaves analogously to the vibrating string in Figure 1 since they are both initialized with the same triangular pluck.

Figure 2: Sampled traveling waves in a vibrating string

One of the most common sample rates used in audio, which is the sampling rate of compact discs (CDs), is $f_S=44.1$kHz. According to the Nyquist-Shannon sampling theorem, what is the maximum frequency that audio signals on CDs may represent? Consider how this compares to the upper frequency limit of human hearing, which is about 20kHz.

We have effectively also sampled the wave function with respect to $x$. We let $X$ be the spatial sampling interval, which is the distance that a traveling wave in the waveguide travels during one temporal sampling interval $T$. Since $c$ measured in m/s is the wave speed in the waveguide, $X=cT$.