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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''
'' means ``identically equal to,'' i.e.
, equal for all
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
A digital simulation diagram
rigidly terminated ideal string is shown in Fig.6.3
A ``virtual pickup
'' is shown at the arbitrary location
The rigidly terminated
ideal string, with a displacement output indicated at position
. Rigid terminations reflect traveling displacement, velocity,
or acceleration waves with a sign inversion. Slope or force waves
reflect with no sign inversion.
Previous: Ideal Acoustic TubeNext: Velocity Waves at a Rigid Termination
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA)
, teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/