Sampled Traveling Waves in a String

For discrete-time simulation, we must sample the traveling waves

• Sampling interval $\mathrel{\stackrel{\mathrm{\Delta}}{=}}T$ seconds
• Sampling rate $\mathrel{\stackrel{\mathrm{\Delta}}{=}}f_s$ Hz $= 1/T$
• Spatial sampling interval $\mathrel{\stackrel{\mathrm{\Delta}}{=}}X$ m/s $\mathrel{\stackrel{\mathrm{\Delta}}{=}}cT$
  $\Rightarrow$ systolic grid

For a vibrating string with length $L$ and fundamental frequency $f_0$ ,

$$c = f_0 \cdot 2L \quad\quad \left(\frac{\hbox{meters}}{\hbox{sec}} = \frac{\hbox{periods}}{\hbox{sec}} \cdot \frac{\hbox{meters}}{\hbox{period}} \right)$$

so that

$$X= cT = (f_0 2L)/f_s= L [ f_0 / (f_s/2) ]$$

Thus, the number of spatial samples along the string is

$$\zbox{L/X= (f_s/2) / f_0}$$

or

$$\zbox{\hbox{Number of spatial samples} = \hbox{Number of string harmonics}}$$

Examples:

• Spatial sampling interval for CD-quality digital model of Les Paul electric guitar (strings $\approx$ 26 inches)
  • $X=L f_0/(f_s/2) = L 82.4/22050 \approx 2.5$ mm for low E string
  • $X\approx 10$ mm for high E string (two octaves higher and the same length)
  • Low E string: $(f_s/2)/f_0 = 22050/82.4 = 268$ harmonics (spatial samples)
  • High E string: 67 harmonics (spatial samples)
• Number of harmonics = number of oscillators required in additive synthesis
• Number of harmonics = number of two-pole filters required in subtractive, modal, or source-filter decomposition synthesis
• Digital waveguide model needs only one delay line (length $2L$ )

  Examples (continued):

• Sound propagation in air:
  • Speed of sound $c \approx 331$ meters per second
  • $X=331/44100=7.5$ mm
  • Spatial sampling rate $= \nu_s = 1/X=133$ samples/m
  • Sound speed in air is comparable to that of transverse waves on a guitar string (faster than some strings, slower than others)
  • Sound travels much faster in most solids than in air
  • Longitudinal waves in strings travel faster than transverse waves
    • typically an order of magnitude faster