Cylindrical Tube With Thermoviscous Losses Using Foster Network

R2026b

Acoustic propagation in a cylindrical tube with thermoviscous losses

Since R2026b

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Cylindrical Tube With Thermoviscous Losses Using Foster Network block

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Libraries:
Simscape Acoustics / Applications / Tubes and Horns

Description

Add-On Required: This feature requires the Simscape Acoustics add-on.

This block models the transmission line of a cylindrical tube with thermoviscous losses. By default, the tube is modeled with auxiliary circuit branches using a Foster network expansion [1]. The tube is modeled using N segments, where each segment includes auxiliary circuit branches that approximate the frequency-dependent viscous and thermal boundary effects. For more information, see Foster Network Equivalent Circuit.

Specify Select tube model as "Simple Model" to model the tube using a single transmission line where the tube is not segmented [2]. The simple model is suitable for tubes with small dimensions compared to the wavelength of the propagating wave.

Examples

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Ports

Conserving

P1 — Left terminal of the network circuit

acoustic

Acoustic conserving port associated with the left terminal of network circuit. The pressure at the input of the tube is measured across P1 and P3, and the volume velocity at the tube input is the volume velocity flowing out of P1.

P2 — Right terminal of the network circuit

acoustic

Acoustic conserving port associated with the right terminal of network circuit. The pressure at the output of the tube is measured across P2 and P3, and the volume velocity at the tube output is the volume velocity flowing out of P2.

P3 — Bottom terminal of the network circuit

acoustic

Acoustic conserving port associated with the bottom terminal of network circuit. The pressure at the input of the tube is measured across P1 and P3, and the pressure at the output of the tube is measured across P2 and P3.

Parameters

Tube radius — Radius of the tube

0.0075 m (default) | positive scalar

Radius of the tube in meters.

Tube length — Length of the tube

0.35 m (default) | positive scalar

Length of the tube in meters.

Select tube model — Selection of tube model

"Differential Branches" (default) | "Simple Model"

Selection of tube model. When you select "Differential Branches", the tube is modeled using a Foster network with a parameterizable number of differential tube sections and order of the LR and CR branches in each section. When you select "Simple Model", the tube is modeled using a Cauer ladder network.

Order of thermoviscous approximation — Order of the viscous and thermal loss auxiliary circuit branches

4 (default) | positive integer

Order of the viscous and thermal loss auxiliary circuit branches, specified as a positive integer no larger than 16. The model approximates viscous loss using series impedance branches and thermal loss using shunt impedance branches. This parameter specifies the number of acoustic inertance and resistance pairs in the series impedance branches and the number of acoustic compliance and resistance pairs in the shunt impedance branches. Higher values improve accuracy across a wider frequency range but increase computation time.

Dependencies

To enable this parameter, set the Select tube model parameter to "Differential Branches".

Number of differential tube lengths — Number of differential tube lengths in the Foster network

20 (default) | positive integer

Number of differential tube lengths in the Foster network, specified as a positive integer. Each segment acts as a lumped transmission line element. Use more segments when the tube length is comparable to or larger than the acoustic wavelength of interest.

Dependencies

To enable this parameter, set the Select tube model parameter to "Differential Branches".

More About

Foster Network Equivalent Circuit

When you set Select tube model to "Differential Branches", the Cylindrical Tube With Thermoviscous Losses Using Foster Network block implements a continued-fraction circuit expansion with N segments. Each segment is modeled using a T-network architecture, resulting in a cascaded T-network or Cauer ladder network lumped-element model. Each component of the ladder network is a subnetwork. The figure shows the high-level architecture for an equivalent circuit with Number of differential tube lengths set to 3.

The block models viscous loss with series impedance components. Each series impedance subnetwork is an infinite network comprised of an initial Acoustic Resistance block followed by pairs of Acoustic Inertance and Acoustic Resistance blocks. The number of pairs is determined by the Order of thermoviscous approximation parameter. The figure shows the equivalent circuit of a series impedance component with an order of 4. The equivalent circuit is derived from Figure 4 of the reference that defined the network model [3], with values calculated using an updated derivation [1].

The block models thermal loss with shunt impedance components. Each shunt impedance subnetwork is an infinite network comprised of an initial Acoustic Compliance block and Acoustic Resistance block, followed by pairs of Acoustic Compliance and Acoustic Resistance blocks. The number of pairs is determined by the Order of thermoviscous approximation parameter. The figure shows the equivalent circuit of a shunt impedance component with an order of 4. The equivalent circuit is derived from Figure 4 of the reference that defined the network model [3], with values calculated using an updated derivation [1]

In practice, the Cylindrical Tube With Thermoviscous Losses Using Foster Network collapses consecutive series impedance blocks from the equivalent circuits into a single block. The resulting model is numerically identical to the equivalent circuit but uses fewer components.

References

[1] Thibault, Alexis, and Juliette Chabassier. “Dissipative Time-Domain One-Dimensional Model for Viscothermal Acoustic Propagation in Wind Instruments.” The Journal of the Acoustical Society of America 150, no. 2 (2021): 1165–75. https://doi.org/10.1121/10.0005537.

[2] Thompson, Stephen C., Thomas B. Gabrielson, and Daniel M. Warren. “Analog Model for Thermoviscous Propagation in a Cylindrical Tube.” The Journal of the Acoustical Society of America 135, no. 2 (2014): 585–90. https://doi.org/10.1121/1.4861237.

[3] Bilbao, Stefan, and Reginald Harrison. “Passive Time-Domain Numerical Models of Viscothermal Wave Propagation in Acoustic Tubes of Variable Cross Section.” The Journal of the Acoustical Society of America 140, no. 1 (2016): 728–40. https://doi.org/10.1121/1.4959025.

Version History

Introduced in R2026b