## Rigid Terminations

A *rigid termination* is the simplest case of a string (or tube)
termination. It imposes the constraint that the string (or air) cannot move
at the termination. (We'll look at the more practical case of a *yielding
termination* in §9.2.1.) If we terminate a length ideal string at
and , we then have the ``boundary conditions''

where ``'' means ``identically equal to,''

*i.e.*, equal for all . Let denote the time in samples to propagate from one end of the string to the other and back, or the total ``string loop'' delay. The loop delay is also equal to twice the number of spatial samples along the string.

Applying the traveling-wave decomposition from Eq.(6.2), we have

Therefore, solving for the reflected waves gives

(7.10) | |||

(7.11) |

A digital simulation diagram for the rigidly terminated ideal string is shown in Fig.6.3. A ``virtual pickup'' is shown at the arbitrary location .

### Velocity Waves at a Rigid Termination

Since the displacement is always zero at a rigid termination, the velocity is also zero there:

Such inverting reflections for velocity waves at a rigid termination are identical for models of vibrating strings and acoustic tubes.

### Force or Pressure Waves at a Rigid Termination

To find out how force or pressure waves recoil from a rigid termination, we may convert velocity waves to force or velocity waves by means of the Ohm's law relations of Eq.(6.6) for strings (or Eq.(6.7) for acoustic tubes), and then use Eq.(6.12), and then Eq.(6.6) again:

Thus, force (and pressure) waves reflect
from a rigid termination with *no sign inversion*:^{7.3}

The reflections from a rigid termination in a digital-waveguide acoustic-tube simulation are exactly analogous:

Waveguide terminations in acoustic stringed and wind instruments are
never perfectly rigid. However, they are typically *passive*,
which means that waves at each frequency see a reflection coefficient
not exceeding 1 in magnitude. Aspects of passive ``yielding''
terminations are discussed in §C.11.

**Next Section:**

Moving Rigid Termination

**Previous Section:**

Ideal Acoustic Tube