Moving sources and the 2.5D Helmholtz boundary element method
Christian H. KASESS(1) , Holger WAUBKE(2)(1) Acoustics Research Institute, Austrian Academy of Sciences, Austria, christian.kasess@oeaw.ac.at(2) Acoustics Research Institute, Austrian Academy of Sciences, Austria, holger.waubke@oeaw.ac.at

Abstract
For perception tests on environmental noise, e.g. evaluating the effect of a noise barrier, it can become necessary
to virtualize moving sources. In the framework of the 2.5D boundary element method, used for long and large
structures, the full treatment of sources moving along the length of the structure is possible but can becomecomputationally quite involved. Even in the simplest case, i.e. mono-frequent emission and constant speed,
higher speeds require calculations for ever higher 2D frequencies beyond the 3D target frequency in order toperform the necessary spatial inverse Fourier transform to acquire the time signal. For wide-band signals a
2D inverse Fourier transform needs to be performed for each point in time. Adding a further non-uniform
motion results in yet another dimension that needs to be integrated over. For higher sampling frequencies andsignal durations of a few seconds this can become problematic. Interpolation between data points in either
frequency or space can help to decrease the computational burden in particular when employing different typesof demodulation schemes. Initial data for a sampled source signal is presented to illustrate at which points inthe calculation interpolation can be applied without introducing too high a computational error.
Keywords: moving source, Helmholtz equation, boundary element method

1

INTRODUCTION

In environmental noise research the boundary element method (BEM) is an important tool in particular forsituations were simpler approaches such as those used in noise mapping are not sufﬁcient e.g. for the evaluationof complex barrier designs [1] or other complicated cross-sections such as railway platforms with a canopy [2].
In order to reduce the computational burden, many of these applications involve the use of BEM approaches
applying simplifying assumptions such as a constant cross-section as in the 2D BEM and nowadays in particular
the more general 2.5D approaches [3, 4] which allow for more realistic source models. However, often thesource models used are stationary point sources or incoherent line sources which do not take the motion of thesource as in trafﬁc into account. However, for perception tests on environmental noise, e.g. evaluating the effectof a noise barrier, it can become necessary to virtualize such moving sources.
In situations with a constant cross-section where 2.5D approaches are applicable, a mono-frequent source moving at a constant speed along the length of the structure can be treated quite efﬁciently [3]. Even in the 2.5D
case as soon as non-uniform motion (acceleration or decelaration) and wide-band signals are concerned, thefull treatment of moving sources becomes computationally involved. In this general setting, a three-dimensional
integration needs to be performed comprising a forward Fourier transform and two inverse Fourier transforms.
Whereas the temporal transforms can be treated as discrete Fourier transforms (DFT) in a relatively straightforward manner, this requires, ﬁrst, a large number of 2D calculations and, second, the evaluation of many inverse
Fourier transforms to the spatial domain. The aim of this work is to investigate whether, and to what degree,
the number of calculations can be reduced by interpolating between data points. For this, also a demodulationin frequency and/or space is investigated similar to the method for the 2D calculations as suggested in [3] andsuccessfully used in [4].

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2

METHODS

2.1 2.5D Boundary element method
A brief overview on the treatment of a moving source in a 2.5D setting is provided. For more details the reader
is referred to [3]. The starting point is the wave equation for the pressure p in the time domain of a source
moving along the x-axis (i.e. the inﬁnite dimension of the scatterer):∇2 p(x,t) −1 ∂ 2 p(x,t)= −s(t)δ (x − x0 (t)),
c2 ∂t 2

(1)with c being the speed of sound. x = (x, y, z)T and x0 (t) = (x0 (t), y0 , z0 )T denote the point in space where
the wave equation is evaluated and the source point, respectively. s(t) is the source signal which will also bereferred to as the signal. The Fourier transform along t and x leads to:∇2 p2 (y, z, ω, kx ) − kx2 p2 (x, y, ω, kx ) + k2 p2 (x, y, ω, kx ) =
⎛
⎞
= −⎝

∞ ∞

s(t)δ (x − x0 (t))eiωt e−ikx x dxdt ⎠ δ (y − y0 )δ (z − z0 )

−∞ −∞

⎛
= −⎝

∞

⎞

s(t)eiωt e−ikx x0 (t) dt ⎠ δ (y − y0 )δ (z − z0 )−∞= −S(ω, kx )δ (y − y0 )δ (z − z0 ),(2)with the 3D-wavenumber k = ω/c and the spatial wave number kx . p2 is used to indicate the pressure in 2D (yand z).
When the source is stationary then S(ω, kx ) = ŝ(ω)e−ikx x0 resulting in a straightforward 2.5D problem with asource spectrum weighting and a phase factor depending on the x-offset. The inverse Fourier transform over kx
can be calculated as described e.g. in [4] or [3] leading to p̂(x, ω).
For a moving source, the time signal at the receiver is often more interesting, which requires a second inverse
Fourier transform over ω. Thus the solution of a source moving along direction x reads as follows:1
p(x,t) =
(2π)2
1
=
(2π)2
where q(y, z, ω, kx ) = q(y, z,

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∞ ∞
−∞ −∞

∞ ∞
−∞ −∞

S(ω, kx )q(y, z, ω, kx )e−iωt eikx x dkx dω
⎛
⎝

∞

⎞
iωt 

s(t  )ee

−ikx x0 (t  )dt  ⎠ q(y, z, ω, kx )e−iωt eikx x dkx dω,−∞(ω/c)2 − kx2 ) is the solution of the 2D-problem in the y-z plane.

(3)2.2 Uniform motion
The simplest case is a uniform motion x0 (t) = x0 + vt of a mono-frequent source s(t) = Ae−iω0 t . To begin with,uniform motion leads to⎛⎞∞ ∞ ∞1
⎝ s(t  )eit(ω+kx v) dt  ⎠ q(y, z, ω, kx )e−iωt eikx x dkx dωp(x,t) =
(2π)2
=

1
(2π)2

−∞ −∞
∞ ∞

−∞

ŝ(ω − kx v)q(y, z, ω, kx )e−iωt eikx x dkx dω,

(4)

−∞ −∞

where x0 is set to zero for simplicity. Thus, in the wavenumber domain, the source spectrum is shifted by kx v.
If the signal is a mono-frequent complex exponential, this simpliﬁes further to [3]:p(x,t) =

1
(2π)2

∞ ∞
−∞ −∞

1 −iω0 te=(2π)2Aδ (ω − ω0 − kx v)e−ikx x0 q(y, z, ω, kx )e−iωt eikx x dkx dω∞Ae−ikx (x0 +v−x)
q y, z,((ω0 + kx v)/c)2 − kx2 dkx .

(5)

−∞


The 2D-wavenumbers for which q(.) needs to be evaluated are now given by ((ω0 + kx v)/c)2 − kx2 . Thus,
the integration is not symmetric with respect to kx anymore as in the case of a stationary source. For (ω0 +kx v)/c)2 − kx2 < 0, i.e. kx− < −k(1 + v/c)−1 and kx+ > k(1 − v/c)−1 , the argument of the Hankel functions in
the 2D BEM becomes imaginary and thus the Hankel function decays exponentially. Contrary to the stationary
case, the evaluation of the above expression requires to evaluate 2D-BEM calculations beyond ω0 up to ω0 c(c2 −v2 )−1/2 . E.g. for 4000 Hz, a speed of 180 km/h, and the speed of sound being 343.5 m/s this results of an upper
frequency bound of 4043.06 Hz. While this does not seem a large difference it is important to keep in mind
that in particular the high frequencies require high computational effort and that the original quadrature grid
may not be valid anymore, depending on the quadrature method. Importantly, as soon as the surfaces of thescatterer or the ground are not fully reﬂecting, each signal frequency ω0 requires to calculate a full set of 2D
solutions [5].
2.3 Sampling the source signal
When a sampled signal is used, the Fourier transform of the signal s(t) is replaced by the discrete time Fouriertransform (DTFT) of s(t)eikx x0 (t) . Inserting this into Eq. 3 leads to :1
p(x,t) =
(2π)2

∞ ∞
−∞ −∞

∞

∑ s(nTs )eiωnT e−ikx x0 (nTs ) q(y, z, ω, kx )e−iωt eikx x dkx dω

n=−∞

∞
1
=
s(nTs )∑
(2π)2 n=−∞

∞

−∞

⎛
e−iω(t−nTs ) ⎝∞⎞q(y, z, ω, kx )eikx (x−x0 (nTs )) dkx ⎠ dω,

(6)

−∞

where the expression in the parenthesis is the static 2.5D solution at ω and source-receiver offset x − x0 (nTs )which can be calculated as given e.g. in [4]. Thus, essentially the same set of 2D-calculations can be used
to calculate the different 2.5D solutions. Clearly, for this to be feasible, the signal needs to be truncated in asuitable manner:7477

p(x,t) =N−1

1∑ s(nTs )(2π)2 n=0

∞

−∞

⎛
e−iω(t−nTs ) ⎝∞⎞q(y, z, ω, kx )eikx (x−x0 (nTs )) dkx ⎠ dω,

(7)

−∞

Here, it is important to note that the number of 2.5D integrals is rather high depending on the duration of
the signal and the source signal sampling frequency. Since the 2D-BEM calculations itself as well as theinverse Fourier transform for kx are approximations, the question arises, whether a certain degree of interpolation
sufﬁces for this purpose.
As e.g. for the use in a perception experiment the signal of interest needs also to be sampled, the inverse
Fourier transform can also be formulated in terms of a inverse DFT (with a potentially different sampling time
Tr ):
⎛
⎞
∞
1
1 Ts L−1 N−1p(x, m) =∑ ∑ s(nTs )e−iωl ((T0 +mTr )−nTs ) ⎝ 2π q(y, z, ωl , kx )eikx (x−x0 (nTs )) dkx ⎠,2π MTr l=0
n=0

(8)

−∞

Essentially, this is a formulation to evaluate the two integrals using a rectangular rule where Ts and (MTr )−1
are the step sizes of the forward and inverse Fourier transform, respectively. If Ts = Tr this factor is simply thetypical normalizing constant M −1 for an inverse DFT.
2.3.1 Interpolation
Each term in the double sum requires the standard 2.5D calculation. Considering signals of durations of a few
seconds as well as typical sampling frequencies in audio applications a very large number of 2.5D calculations
is necessary. For example, for a real signal of 1 s at 48 kHz, 24000 times 48000 inverse Fourier transforms(integral in parenthesis in Eq. 8) need to be evaluated. There is, however, a fundamental difference between
the temporal and the spectral sum. For the time index n each value simply requires the calculation of different
longitudinal offsets, i.e. positions along the x-axis using the set of 2D-BEM calculations for the given frequency
ωl . In contrast, each frequency value requires the calculation of a new set of 2D-BEM problems, which ingeneral takes much longer. Thus, in particular interpolation over 3D-frequencies ωl is important. As in the
2.5D case, a demodulation approach which allows to reduce the number of necessary quadrature points also
works for this setting. An extreme example is the moving point source in free space eikl rn (4πrn )−1 where rn
is the distance between source and receiver at time n. When demodulating this with e−ikl rn the rn -dependence
is relatively smooth as long as the minimal source-receiver distance becomes not too small and the frequency
dependence is a constant. Interpolating the real and imaginary part over rn and/or k to a high accuracy hence
becomes quite simple. Having a reﬂecting half space the problem becomes more interesting, as the 3D-solution
is given as
√2
√2
2
2
2
2
∞
eikl xn +(y−y0 ) +(z−z0 )eikl xn +(y−y0 ) +(z+z0 )1
ikx xnq(y, z, ωl , kx )e
dkx = + 
(9)2π
4π xn2 + (y − y0 )2 + (z − z0 )2 4π xn2 + (y − y0 )2 + (z + z0 )2−∞

From this, it is clear that there is no optimal demodulation due to two superimposed and different oscillations.
2.4 Evaluation
Although the methods described earlier allow for a non-uniform motion as well as wide-band signals, uniform
motion and a mono-frequent stimulus was used for an initial evaluation. As an example to illustrate the potential
savings on computation time, a moving point source above a reﬂecting half space will be used with 4 m source
separation and source and receiver located 0.5 m above the half plane. The smallest distance between source
and receiver is at t = 0.

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Spatial sampling: 0.05 m

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frequency spacing in Hz

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Spatial sampling: 0.25 m

100

frequency spacing in Hz

Figure 1. Effect of the demodulation. The ﬁgure illustrates the median and the inter-quartile-range (IQR) of
the relative error in dB over time when compared to the reference solution. Different frequency spacings anddemodulation settings for the interpolation are shown. The upper panel shows the results for a spatial sampling
distance of 0.05 m, the lower panel shows the results for 0.25 m. The horizontal jitter for the same frequency
spacing is only for visual clarity.

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0
−100

relative error in dB−200

71 points, uniform
58 points, non−uniform

0.5 Hz

−0.4

−0.2

0.0

0.2

0.4

−100
−200

relative error in dB

0time in s

71 points, uniform
58 points, non−uniform

100 Hz

−0.4

−0.2

0.0

0.2

0.4

time in s

Figure 2. Temporal/spatial distribution of the error for coarse spatial sampling. The ﬁgure illustrates the relative error in dB as a function of time for a spatial sampling of 1 m and 0.5 Hz (upper panel) as well as
100 Hz (lower panel) spectral sampling distance. The line color denotes uniform (black) and non-uniform (red)sampling.
Different frequency as well as spatial spacings are compared in terms of deviations from the reference solution.
A cubic spline was used for interpolation as implemented in the function spline of the R programming language[6]. The reference solution was calculated using the retarded potential formula (see e.g. [3]). The difference
between the reference and the approach presented here is expressed in terms of the relative error expressed in
dB. A data window was applied to the source signal to avoid potential problems at the endpoints of the signal.
To exclude the data window from the analysis, 0.5 s at the beginning and the end of the receiver signal werediscarded before the error was calculated.

3

RESULTSThe effect of the spatial as well as the spectral demodulation for different spectral sampling intervals is illustrated in the upper panel of Fig. 1 for a source speed of 50 m/s, a frequency of 800 Hz, a sampling frequency of2000 Hz, and a spatial sampling distance of 0.05 m. Note, that a full sampling would necessitate the calculationof 2.5D transfer functions every 0.025 m.
The effect of spectral demodulation is apparent. Due to the full signal length of 2.5 s in the calculation the theoretical spacing is 0.4 Hz. When increasing the spectral spacing the error increases steadily unless demodulation
is applied. Since the spatial sampling is coarser than required in theory, using spatial demodulation also results
in a reduction of the relative error. When increasing the sampling distance (0.25 m, lower panel Fig. 1), spatial
demodulation becomes more important.
Eventually, for yet a coarser sampling the error starts to increase even in the demodulated setting. Fig. 2
illustrates the relative error over time for a spatial sampling at 1.0 m and two different spectral sampling intervals(0.5 Hz upper panel, 100 Hz lower panel, solid black lines).

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60
50
40
30

x−offset in m

20
10
0
0

100

200

300

400

500

number of required data points

Figure 3. Non-uniform sampling. The ﬁgure illustrates the number of points required to sample up to a given
x-offset for the different non-uniform sampling functions as distinguished by line type and color.
From this ﬁgure it can be seen that the frequency sampling has a only a minor effect due to the use of thespectral demodulation. The uniform spatial sampling leads to an increased error when the source is in thevicinity of the receiver, as the change of change in the spectrum as a function of position is higher. Thus, inthis region a denser sampling is required. A non-uniform spatial sampling has been used, where the spacing of
the data points used for interpolation is proportional to the distance in x between the source and the receiver.
Fig. 3 illustrates this sampling for different proportionality factors. It can be seen that the region with smaller
offsets is more densely sampled. Fig. 2 (dashed red lines) shows, that the error can be controlled by using a
non-uniform sampling density, although the sampling towards higher offsets was chosen too coarse in this case.

4

CONCLUSIONSThis preliminary work deals with the potential saving on computation time when dealing with moving sources inacoustic scattering. Results on a uniformly and mono-frequent moving point source above a reﬂecting half-spaceshow that the number of data points for which the BEM needs to be calculated can be reduced signiﬁcantly
when using a demodulation approach before interpolation. For a single frequency and constant motion, thepresented approach may not be optimal. However, the method is applicable to more general situations andinitial data on non-uniform motion also show increased efﬁciency when using spatial and spectral interpolation.
The example illustrated did not involve a scatterer and thus only required the calculation of the Green’s function
instead of real 2D BEM calculations. Preliminary data on a noise barrier example indicate similar tendencies
although higher overall errors were observed. However, a thorough analysis of different settings is necessary.
Furthermore, other interpolation methods than a cubic spline may also be more efﬁcient.

REFERENCES[1] C. H. Kasess, W. Kreuzer, and H. Waubke. Deriving correction functions to model the efﬁciency of noisebarriers with complex shapes using boundary element simulations. Applied Acoustics, 102:88 – 99, 2016.

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[2] C. H. Kasess, H. Waubke, M. Conter, C. Kirisits, R. Wehr, and H. Ziegelwanger. The effect of railway
platforms and platform canopies on sound propagation. Applied Acoustics, 151:137 – 152, 2019.
[3] D. Duhamel. Efﬁcient calculation of the three-dimensional sound pressure ﬁeld around a noise barrier.
Journal of sound and vibration, 197(5):547–571, 1996.
[4] C. Kasess, W. Kreuzer, and H. Waubke. An efﬁcient quadrature for 2.5D boundary element calculations.
Journal of Sound and Vibration, 382:213–226, 2016.
[5] D. Duhamel and P. Sergent. Sound propagation over noise barriers with absorbing ground. Journal of soundand vibration, 218(5):799–823, 1998.
[6] R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2018.

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