Diploma report Implementation and verification of a simple ... - LPAS

where Z 0 = Z 0 (z) = ρ 0 (z)c 0 (z) is the impedance of the medium. This equation might provide
an explanation for the difference in size between the two peaks. Calculations of p ′low and p ′up
are made using the values of w ′ , ρ 0 and c 0 in Table 1 and give 196.7960 Pa and 141.6410 Pa
for the lower and upper wave respectively. These values are close to the values of p ′ obtained
in the simulation that figures in Table 1.
As second test case, a top hat perturbation with half width of 800 m is initialised in
the middle of the domain. This shape of perturbation is generally more difficult to simulate
than a Gaussian shape as it features steep slopes. In this case, oscillations generally appear
near the bottom and the top of these steep slopes. The background state and the integration
method are the same than in the previous example.
400
t=0s
300
t=9s
350
250
300
200
250
150
p’[Pa]
200
150
p’[Pa]
100
100
50
50
0
0
0 2500 5000 7500 10000 12500 15000
z[m]
−50
0 2500 5000 7500 10000 12500 15000
z[m]
300
t=18s
250
t=33s
250
200
200
150
150
p’[Pa]
100
p’[Pa]
100
50
50
0
0
−50
0 2500 5000 7500 10000 12500 15000
z[m]
−50
0 2500 5000 7500 10000 12500 15000
z[m]
Figure 8: Evolution of a top hat shaped pressure perturbation with 12.5 m resolution.
Only a single simulation with 12.5 m resolution is presented here as an illustration. The
evolution of the perturbation is shown in Figure 8. As the perturbation is superimposed
onto a stratified background, the top does not appear flat in the first plot in 8. In the plots
for t = 9, 18 and 33 s, the above mentioned oscillations can be observed and are relatively
pronounced. The different amplitudes and propagation speeds of the two waves are analogous
with the previous case. Finally, in the plot for t = 33 s the slope at the top of the perturbation
is inverted due to the reflection at the boundary.
Further tests with various values of filter coefficient should be performed in order to
attempt to reduce the oscillations. Besides this, simulations with higher grid resolutions
should be done as they might provide better results.
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