PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Trained Algorithms for Mode Decomposition in Ducts
Stefan Sack1,2; Mats Åbom1
1

KTH Royal Institute of Technology, Sweden

ABSTRACTA new method to split acoustic fields in flow-ducts into upstream and downstream propagating sound waves
is presented. Goal of this new approach to modal decomposition is to create a methodology that does not relyon semi-analytical solutions of the sound propagation and is, hence, applicable to a much broader set of
aeroacoustic problems. An artificial neural network is trained with data gained from a small set of numericreference solutions of the linearized Navier-Stokes equation in the frequency domain in a straight duct. The
network is tested on relevant experimental data. A very good agreement with the analytical solution isdemonstrated in two applications with different flow conditions. The good results obtained for the test casesin this paper make this method a promising addition to the classical methodology;
Keywords: Flow acoustics, two-port, neural network, duct acoustics, modal decomposition

1. INTRODUCTION
Sound created in flow-ducts is a strong contributor to perceived noise disturbance from, forexample, ventilation systems [1] and intake and exhaust systems [2]. In order to develop and improvenoise mitigation strategies, reliable models to describe the sound propagation inside such systems areneeded. Established methodologies, such as the multi-port method [3] and the mode matching method[4], use analytical and semi-analytical expressions the acoustic wave shape and the acoustic wave
propagation [5], often corrected for thermo-viscous dissipation [6]. That limits the methods to hard
walled duct systems with well-defined ﬂow conditions.
In this conference contribution, we present a new approach to analyze in -duct sound without the needfor semi-analytical solutions. Instead, we develop an algorithm that can learn the acoustic propagationof plane waves from a small set of numerical reference solutions of the linearized Navier -Stokes
equation in the frequency domain in a straight duct.
1.1 Low Frequency Propagation of Acoustic Waves in Flow-ducts
In many relevant aero-acoustic applications where in-duct sound is important the acoustic waves
are linear, time-invariant, and stationary. The acoustic field is then a superposition of many tonalcomponents and each tonal component fulﬁlls a convective wave equation in the frequency domain,e.g. the Helmholtz equationሺ‫׏‬ଶ  ൅ ሺɘǡ ሻଶ ሻሺɘǡ ሻ ൌ Ͳ.

(1)Here, p is the acoustic pressure, k is the acoustic wave number, ‫ ׏‬is the Nabla-Operator, ω is theangular frequency, and X = (x,y,z) is a coordinate vector. For a soundwave propagating in a straighthard-walled duct with stationary, uniform background ﬂow, Equation 1 may be solved analytically. At
frequencies lower than the cut-on frequency where only planar acoustic modes can propagate, thesolution can be expressed asሺɘǡ ሻ ൌ  ା ሺɘሻ୧୩శ ୸ ൅ ି ሺɘሻି୧୩ష ୸ Ǥ

(2)Equation 2 describes a superposition of two opposed plane waves with constant pressure in across-section. The two waves propagate in upstream (-) and downstream (+) direction. They have acomplex amplitude ା and ି and the complex wave number  ା and  ି as a function of frequency,
fluid-flow conditions, and duct geometry.
2 Corresponding author: ssack@kth.se5292

1.2 Mode Decomposition
If a sound ﬁeld is sampled along the duct, e.g. with microphones mounted along the duct walls, acommon task in duct acoustics is to split this sound ﬁeld into its upstream and downstream propagatingwaves. With the analytical description of the wave, the complex wave amplitudes can be solved fromat least two samples of the acoustic ﬁeld (p1 and p2 at z1 and z2). Equation 2 is therefore written invector notation୧୩శ ୸భ
ଵቂ ቃ ൌ ൤୧୩ ୸ଶ శమି୧୩ష ୸భ ൨ ቂା ቃ ൌ ‫ ۻ‬.
േି୧୩ష ୸మ ି(3)

The matrix M is called the modal matrix. More detailed information about the modal decomposition
and its application can be found in Ref [7–9]. In the classical mode decomposition, M is inverted andmultiplied with the sample’s pressure to solve for the complex mode amplitudes ା and ି . The
system can become ill-conditioned and therefore ambiguous in its solution, for example at tones ofcertain wavelengths [10] or close to the cut-on frequencies of higher order modes [7]. Consequently, M
is commonly over-determined by sampling the sound ﬁeld at more than 2 positions. Challenges arestill associated with the analytical solutions of the wave shape and propagation for example at highfrequencies [11] and background flows with greater Mach-numbers (M>=0.3) [12]. In fact, thebackground flow can deform the acoustic modes and influence the propagation speed, which adds
uncertainties in the form of bias errors if the classical mode decomposition (Equation 3) is used.
1.3 Learning Algorithm
Regression problems such as the mode decomposition in Equation 3 can be solved with artificial
neural networks (ANN) [13]. In theory, a network with a single layer between input and output canlearn any desired deterministic regression function, provided that a sufficient number of neurons anda representative set of training data is available (“universal approximation theorem” [14]). The goalof this paper is to show that an ANN can learn the modal decomposition in Equation 3. The
algorithm learns the acoustic mode shapes and propagation from numerical simulations without theneed for explicit knowledge of the governing equations. This allows mode decomposition in complex
flows and arbitrary duct geometries. The numerical scheme can include turbulent flow and boundary
layer effects and hence cover for flow-related acoustic dissipation.
The ANN is implemented as a Feed Forward Back Propagation topology with a single fully
connected hidden layer between input and output. We split the complex numbers of the sampledpressure into a real and an imaginary part and add those as separate input for the model. We use thesame pressure probe number and positions as in our experimental setups. The input is therefore arow vector of 7 elements, namely 3 elements for the real part of sampled pressure transfer functionsat 3 microphone positions, 3 elements for the imaginary part of the sampled pressure transfer
function at 3 microphone positions, and the frequency. The output is a row vector with four elementsin the form of the real and imaginary part of ା and ି .
To create the topology of the neural network, we have tested different numbers of neurons in thehidden layer, namely 10, 30, 50, 100, 200, 500, 1000, 2000, 4000, and 8000 and the solution started toconverge at around 1000 neurons. For the results presented in the paper, we use 2000 neurons.
Equation 3 is linear only if a single frequency is considered. For a frequency-dependent modal
decomposition, the regression is non-linear. We address this by adding non-linear elements in the formof normalized exponential (“softmax”) activation functions to the weights of the neurons in the hidden
layers. The input has no activation functions, and the output has linear activation functions.
1.4 Training Data
The data needed to train the neural network was generated from a small set of numeric solutions.
We solved the linearized Navier-Stokes equation (LNSE) in the frequency domain for empty duct
sections with reflection-free determinations. For each training frequency, we simulated two soundfields, one for plane wave excitation on the upstream and one for plane wave excitation on thedownstream side. The training data could then be generated by superposing phase-shifted and scaled
versions of the two reflection-free reference solutions. This approach allowed us to create theoreticallyinfinitely large training and validation data sets. We trained the ANN on data in 100 Hz steps between
300 and 2600 Hz with 200 000 training samples per frequency. The method is applicable for linear and5293

time-invariant wave propagation. This is true in many applications with sound pressure being lower
than 155 dB (1% of atmospheric pressure) [15].
The numeric solutions were generated in two steps. First, the flow field was computed solving
Reynolds Averaged Navier-Stokes (RANS) equations. In a second step, the acoustic field wascomputed linearized around the stationary mean background flow. We retained all terms containing
viscosity and thermal conductivity, so that the corresponding acoustic dissipation was included in the
training data. The computational domain was inherited from Ref [16] (Figure 1), where good
performance of this method was shown.

Figure 1: Computational domain for empty duct section with reflection-free perfectly matched layers (PML), sources, and six
microphone probes.

1.5 Algorithm Testing
The algorithm was tested on experimental data measured as part of the IFAR Acoustic Liner
Challenge at the 2018 AIAA Conference. The IFAR was a NASA initiative where 3D-printed acoustic
liner configurations were measured on multiple test rigs with the aim to collect and compare the results.
Data was shared with each participant in order to assess how fabrication, data acquisition and analysis
approach influence the results.
Measurements of the acoustic properties of liners are common tasks in aeroacoustics. We therefore
consider the data as a relevant test case for our ANN. To evaluate the results, we plot the scatteringmatrix, which is an anechoic representation of the passive acoustic properties of the liner. To this aim,
we decompose the sound field upstream and downstream of the liner and compute the soundtransmission and reflection. The scattering matrix is defined asɏୢ୳ାቀ ቁ ൌ ቂɒୢା୳ୢɒୢ୳ ୳ିɏ୳ ቃ ቀୢି ቁ,(4)

where p u- and pd- are amplitudes of plane pressure waves incident of the lined section, p u+ andp d+ are amplitudes of palne pressure waves outgoing from the sample, and Ud and Uu are reflection
coefficients with u and d denoting upstream and downstream, respectively. W du and Wud aretransmission coefficients from the downstream to the upstream and the upstream to thedownstream side. The procedures to extract the scattering with the multi-port method and to performthe necessary measurements are explained in Ref [7].
The experiments for this paper were carried out at the KTH modular advanced flow-rig facilities,
using 6 microphones and 4 loudspeakers. Sound fields were excited upstream and downstream of theliner using the four sources simultaneously, each playing a different sinusoidal tone between 300 and
2600 Hz in 20 Hz steps and we measured the frequency response functions between the microphones
and the loudspeaker reference signals. Note, that we have chosen a finer frequency resolution for theexperiment than for the data included in the training data set. We can therefore evaluate the ability of
the ANN to learn the actual regression functions by testing frequency points that were not included in
the training data set.
In this paper, we compare scattering data gained from the experiments, using i) the two-port methodwith mode decomposition based on analytical solutions of the wave equation and ii) the learningalgorithm trained on numerical data. We test two different flow conditions. Firstly, a case without flow
(M=0) and secondly a case with moderate flow (centerline Mach number M=0.3).

2. Results and Discussion
The scattering matrix of the tested liner is presented in Figure 2 and 3 for the no flow (M=0) case,and in Figure 4 and 5 for the moderate (M=0.3) flow. The performance of the neural network comparedto the analytical modal decomposition is presented.
For the no flow (M=0) case, the wave propagation inside the straight duct is governed accurately by
the Helmholtz equation and the analytical solution can therefore be used for quality assessment. We
found, that the agreement of the analytical solution and the neural network is distinguished in the5294

mid-frequency range between 750 Hz and 2100 Hz for both, the amplitude and the phase (Figure 2 and
3). For low and high frequencies, the accuracy of the neural network decreases for the amplitude, but
remains of good quality for the phase. This indicates a general good decomposition of incident andoutgoing waves on each side over the entire frequency range, but a bias in the amplitude of both p+ andp- waves on each side towards the limits of the trained frequencies.
We could find similar agreement for the moderate flow case (M=0.3) depicted in Figure 4 and 5. All
important features of the scattering matrix are captured with good accuracy. The neural network
furthermore approximated the scattering matrix smoother than the analytical solution which becomesapparent for the noisier upstream reflection coefficients.

Figure 2: Performance of neural network compared to analytical solution for M= 0. Amplitude of the scattering matrix of thetested liner.

Figure 3: Performance of neural network compared to analytical solution for M= 0. Phase of the scattering matrix of thetested liner.

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Figure 4: Performance of neural network compared to analytical solution for M= 0.3. Amplitude of the scattering matrix ofthe tested liner.

Figure 5: Performance of neural network compared to analytical solution for M= 0.3. Phase of the scattering matrix of thetested liner.

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3. Conclusion
We present a new method to decompose acoustic fields in ducts into upstream and downstream
propagating plane waves. The novelty of the presented methodology is the algorithm’s ability to learn
the decomposition without explicit knowledge of the governing equations. This makes the method
applicable to experiments where the analytical solution of the wave propagation is unavailable. Thisapplies for example to duct systems with complicated cross-sections, soft walls, and high flow
velocities.
We developed a learning algorithm that can be trained with numerical data. The algorithm contains
a single layer feed forward - back propagation artificial neural network. The data needed to train the
network is created from two reference solutions of the Linearized Navier-Stokes equation in a straight
duct, solved for a number of training frequencies. Such computations are generally computational
inexpensive although they contain the significant sound dissipation mechanisms.
We tested the trained network on experimental data measured for an acoustic liner, which
represents a relevant application. By comparing the liner’s scattering matrix extracted with the neuralnetwork and the classical two-port method for a case without flow, we found that the learningalgorithm performs well in the mid-frequency range of the training area, whereas the accuracy
decreases slightly towards the higher and lower training frequencies. In the mid-frequency range, thealgorithm reached very good performance for most of the frequencies even though the network was
only trained on 20 % of the tested frequencies. This indicates a good ability of the presented network
to learn the non-linear regression functions without overfitting the trained points. A more complex
neural network with several layers might be able to increase the accuracy towards the borders of the
trained frequencies.
To improve the method, the network could be trained to also predict the wave numbers of theextracted duct modes. This would allow to move the modal data from the reference cross section toarbitrary cross sections, as commonly done in the two-port method.

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