Periodic Signal Generation using an Approximation to the Analytical Wave Equation Solution

Resumo

A major issue in analysing wave propagation problems is the geometric and time domain discretizations,
which if it is not properly performed, may lead to poor results or computational issues such as memory overflow.
Ideally, an analytical solution for the Partial Differential Equation is desired since it may provide results for any

given pair (~x, t) independently of mesh size, time step and without any previous iteration. In this paper, an approx-
imate analytical solution of the wave equation is used in order to simulate the behavior of any periodic signal in

one dimensional domains. The periodic signal is approximated using Fourier series with the sine and cosine terms
evaluated analytically and their respective coefficients evaluated numerically using Gauss-Legendre integration
rule. A Python 3.7 Application Programming Interface was developed using an object oriented approach, allowing
user defined periodic input function, spatial domain size, time range, number of Fourier terms used and static and
dynamic solution plotting. Dirichlet Boundary Conditions defined as time periodic functions are considered and
three different functions, rectangular pulse, sawtooth wave and Gaussian pulse, are evaluated and their respective
results compared with the corresponding Finite Difference Method solution presenting a mean-squared-errors of
order 10−3
.