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Atmospheric Sound Propagation Over Large-Scale Irregular Terrain

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Abstract

A benchmark problem on atmospheric sound propagation over irregular terrain has been solved using a stable fourth-order accurate finite difference approximation of a high-fidelity acoustic model. A comparison with the parabolic equation method and ray tracing methods is made. The results show that ray tracing methods can potentially be unreliable in the presence of irregular terrain.

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Authors and Affiliations

  1. Department of Information Technology, Uppsala University, Lägerhyddsvägen 2, 752 37 , Uppsala, Sweden

    Martin Almquist & Ken Mattsson

  2. KTH Royal Institute of Technology, Stockholm, Sweden

    Ilkka Karasalo

Authors
  1. Martin Almquist
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  2. Ilkka Karasalo
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  3. Ken Mattsson
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Correspondence to Martin Almquist.

Appendix: Finite Difference Operators

Appendix: Finite Difference Operators

For completeness we present the fourth order SBP operators. Here \(h\) denotes the grid-spacing. The interior stencils (in \(D_1\) and \(M^{(b)}\)) are the standard central 4th order accurate finite difference stencils. At the boundaries we use one-sided stencils that are formally second order accurate. The discrete norm \(H\) is defined:

The first derivative SBP operator is given by,

The third-order accurate boundary derivative operator \(S_0\) is given by,

$$\begin{aligned} S_0=\frac{1}{h}\begin{bmatrix} {\frac{11}{6}}&-3&\frac{3}{2}&-\frac{1}{3}&0&0&\ldots \\ \end{bmatrix} \end{aligned}$$

The interior stencil of \(-h\,M^{(b)}\) at row \(i\) is given by \((i=7\ldots N-6)\):

$$\begin{aligned} m_{i,\,i-2}&= \frac{1}{6}\, b_{i-1}-\frac{1}{8}\, b_{i-2}-\frac{1}{8}\, b_{i}\\ m_{i,\,i-1}&= \frac{1}{6}\, b_{i-2}+\frac{1}{6}\, b_{i+1}+\frac{1}{2}\, b_{i-1}+\frac{1}{2}\, b_{i}\\ m_{i,\,i}&= -\frac{1}{24}\, b_{i-2}-\frac{5}{6}\, b_{i-1}-\frac{5}{6}\, b_{i+1}-\frac{1}{24}\, b_{i+2}-\frac{3}{4}\, b_{i}\\ m_{i,\,i+1}&= \frac{1}{6}\, b_{i-1}+\frac{1}{6}\, b_{i+2}+\frac{1}{2}\, b_{i}+\frac{1}{2}\, b_{i+1}\\ m_{i,\,i+2}&= \frac{1}{6}\, b_{i+1}-\frac{1}{8}\, b_{i}-\frac{1}{8}\, b_{i+2}. \end{aligned}$$

The left boundary closure of \(-h\,M^{(b)}\) (given by a \(6\times 6\) matrix) is given by

$$\begin{aligned} m_{1,\, 1}&= {\frac{12}{17}}\, b_1+{\frac{59}{192}}\, b_2+{\frac{27010400129}{345067064608}}\,b_3+{\frac{69462376031}{2070402387648}}\,b_4\\ m_{1,\, 2}&= -{\frac{59}{68}}\, b_1-{\frac{6025413881}{21126554976}}\, b_3-{\frac{537416663}{7042184992}}\, b_4\\ m_{1,\, 3}&= \frac{2}{17}\, b_1-{\frac{59}{192}}\, b_2+{\frac{213318005}{16049630912}}\, b_4+{\frac{2083938599}{8024815456}}\, b_3\\ m_{1,\, 4}&= {\frac{3}{68}}\, b_1-{\frac{1244724001}{21126554976}}\, b_3+{\frac{752806667}{21126554976}}\, b_4\\ m_{1,\, 5}&= {\frac{49579087}{10149031312}}\, b_3-{\frac{49579087}{10149031312}}\, b_4\\ m_{1,\, 6}&= -{\frac{1}{784}}\, b_4+{\frac{1}{784}}\, b_3\\ m_{2,\, 2}&= {\frac{3481}{3264}}\, b_1+{\frac{9258282831623875}{7669235228057664}}\, b_3+{\frac{236024329996203}{1278205871342944}}\, b_4\\ m_{2,\, 3}&= -{\frac{59}{408}}\, b_1-{\frac{29294615794607}{29725717938208}}\, b_3-{\frac{2944673881023}{29725717938208}}\, b_4\\ m_{2,\, 4}&= -{\frac{59}{1088}}\, b_1+{\frac{260297319232891}{2556411742685888}}\, b_3-{\frac{60834186813841}{1278205871342944}}\, b_4\\ m_{2,\, 5}&= -{\frac{1328188692663}{37594290333616}}\, b_3+{\frac{1328188692663}{37594290333616}}\, b_4\\ m_{2,\, 6}&= -{\frac{8673}{2904112}}\, b_3+{\frac{8673}{2904112}}\, b_4\\ m_{3,\, 3}&= {\frac{1}{51}}\, b_1+{\frac{59}{192}}\, b_2+{\frac{13777050223300597}{26218083221499456}}\, b_4+{\frac{564461}{13384296}}\, b_5\\&\quad +{\frac{378288882302546512209}{270764341349677687456}}\, b_3\\ m_{3,\, 4}&= {\frac{1}{136}}\, b_1-{\frac{125059}{743572}}\, b_5-{\frac{4836340090442187227}{5525802884687299744}}\, b_3-{\frac{17220493277981}{89177153814624}}\, b_4\\ m_{3,\, 5}&= -{\frac{10532412077335}{42840005263888}}\, b_4+{\frac{1613976761032884305}{7963657098519931984}}\, b_3+{\frac{564461}{4461432}}\, b_5\\ m_{3,\, 6}&= -{\frac{960119}{1280713392}}\, b_4-{\frac{3391}{6692148}}\, b_5+{\frac{33235054191}{26452850508784}}\, b_3\\ m_{4,\, 4}&= {\frac{3}{1088}}\, b_1+{\frac{507284006600757858213}{475219048083107777984}}\, b_3+{\frac{1869103}{2230716}}\, b_5+\frac{1}{24}\, b_6\\&\quad +{\frac{1950062198436997}{3834617614028832}}\, b_4\\ m_{4,\, 5}&= -{\frac{4959271814984644613}{20965546238960637264}}\, b_3-\frac{1}{6}\, b_6-{\frac{15998714909649}{37594290333616}}\, b_4-{\frac{375177}{743572}}\, b_5\\ m_{4,\, 6}&= -{\frac{368395}{2230716}}\, b_5+{\frac{752806667}{539854092016}}\, b_3+{\frac{1063649}{8712336}}\, b_4+\frac{1}{8}\, b_6\\ m_{5,\, 5} \!&= \! {\frac{8386761355510099813}{128413970713633903242}}\, b_3\!+\!{\frac{2224717261773437}{2763180339520776}}\, b_4\!+\!\frac{5}{6}\, b_6\!+\!\frac{1}{24}\, b_7\!+\!{\frac{280535}{371786}}\, b_5\\ m_{5,\, 6}&= -{\frac{35039615}{213452232}}\, b_4-\frac{1}{6}\, b_7-{\frac{13091810925}{13226425254392}}\, b_3-{\frac{1118749}{2230716}}\, b_5-\frac{1}{2}\, b_6\\ m_{6,\, 6}&= {\frac{3290636}{80044587}}\, b_4+{\frac{5580181}{6692148}}\, b_5+\frac{5}{6}\, b_7+\frac{1}{24}\, b_8+{\frac{660204843}{13226425254392}}\, b_3+\frac{3}{4}\, b_6 \end{aligned}$$

The corresponding right boundary closure is obtained by replacing \(b_{i}\rightarrow b_{N+1-i}\) for \(i=1,\ldots ,8\) followed by a permutation of both rows and columns. Let \(m_{i,j}\) be the entry at row \(i\) and column \(j\) in \(M^{(b)}\). The matrix \(M^{(b)}\) is symmetric, which means that it is completely defined by the entries on and above the main diagonal, i.e., \(m_{j,i}=m_{i,j},\,i=1,\ldots ,N, \quad j=i,\ldots ,N\).

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Almquist, M., Karasalo, I. & Mattsson, K. Atmospheric Sound Propagation Over Large-Scale Irregular Terrain. J Sci Comput 61, 369–397 (2014). https://doi.org/10.1007/s10915-014-9830-4

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