The simulation of displacement wave motion in a string rigidly terminated at both its ends (and without losses) is shown in the figure below.
Figure 2:
Digital waveguide simulation of ideal lossless wave propagation on a string fixed at both ends (with pluck initialization).
An ideal plucked string is defined as having an initial displacement and zero initial velocity. In the model, the delay lines should be initialized with displacement data corresponding to some arbitrary initial string shape (as illustrated in the figure above by the dashed lines).
Because the physical displacement of a string is given by the superposition of left- and right-going traveling waves, the initial amplitude of each delay-line section should be half the amplitude of the initial, physical string displacement.
The initial displacement shape must be bandlimited to half the discrete-time sample rate. Because sharp corners imply an infinite bandwidth, plucking points should be rounded to some extent. In fact, this is physical given the stiffness of real strings and the finite size of plectra.
It is possible to use digital waveguide models to simulate other wave variables as well, such as velocity or acceleration waves. Note that an ideal pluck shape corresponds to positive and negative acceleration impulses.
The simulation of velocity wave motion in a string rigidly terminated at both its ends (and without losses) is shown in the figure below (for the pluck displacement shown above).
Figure 3:
Digital waveguide velocity-wave simulation of ideal lossless wave propagation on a string fixed at both ends (with pluck initialization).
For the moment, we ignore the fact that the string vibrations are influenced and filtered by the body resonances before being transmitted into the air.
