| Summary: | A thesis submitted in ful lment of the requirements
for the degree of Master of Science
in the
School of Computer Science and Applied Mathematics, 2018 === A linear and a non-linear partial di erential equation is used as a phenomenological
model to describe the synthesis of a vibrating string. The models are numerically
simulated using nite di erence method and an exact solution of the linear partial
di erential equation is derived using a functional transformation method. The
functional transformation method is based on the transformation to the frequency
domain in time and in space by applying the Laplace transformation and the Finite
Sine transformation. The string is excited through the initial conditions, initial
displacement and an initial velocity, of the system. Another excitation mode that
is considered is the force density, where the initial conditions are set to zero. The
force density is added as a Source term to the partial di erential equation and
the initial excitation enters the system by the use of this term. To analyse the
model di erent observation positions are considered, the observation points are
the positions of the string where the vibrations are observed. The exact solution
is used to verify the Finite di erence approximation and a spectrogram analysis of
the di erent excitation points is used to asses the methods. The nite di erence
method is then used to analysis the vibrations of the string. The nonlinear model
include the e ects of tension modulation, which is de ned by the Kirchho -Carrier
equation. The results are analysed using a quantitative analysis and a Fourier
analysis. === XL2019
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