The derivation of the specific admittance of the wall for modeling the thermoviscous losses can be found in Pierce's book, Acoustics: An Introduction to Its Physical Principles and Applications (2019, 3rd edition). This article is to note down the derivation with the key steps and equations copied from the book.

Derivation

The derivation is explained in Chapter 10.4, Acoustic Boundary-Layer Theory, while the entire Chapter 10 is about Effects of Viscosity and Other Dissipative Processes.

  • At frequencies of normal interest, any disturbance governed by the linear equations derived in the previous setion can be considered as a superposition of vorticity, entropy, and acoustic modal wave fields.

  • The linear equations mentioned here refer to the mass conservation equation, Navier-Stokes equation, and the Kirchhoff Fourrier equation. (Eq. 10.2.2)

  • The derivation starts from the notation \(\psi_n(\boldsymbol{x},t) = \text{Re}\{\hat{\psi}_n e^{-i\omega t} e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\}\), Eq. (10.3.1), where \(\psi_n\) can be the plane-wave disturbance of either of the three modal wave fields mentioned above.

  • There are only a small number of \(k^2\) for a given \(\omega\) for which a nontrivial solution (at least one \(\hat{\psi}_n\) not zero) exists.

  • The resulting relations between \(k^2\) and \(\omega\) are the dispersion relations for the possible modes of propagation.

  • Three different modes correspond to three different dispersion relations, leading to three different PDEs, where the nontrivial solution exists.
    • Vorticity mode: \(k^2 = i\omega\rho/\mu\) (\(\nabla\times\boldsymbol{v}\neq 0\), \(\boldsymbol{k\cdot v}=0\)) \[\nabla^2\boldsymbol{v}_{vor} = \dfrac{\rho}{\mu}\dfrac{\partial \boldsymbol{v}_{vor}}{\partial t}\qquad \text{(Eq. 10.3.11)}\]
    • Acoustic Mode: Eq. (10.3.7a) (\(\boldsymbol{k\times v} = 0\)) \[\nabla^2 p_{ac} - \dfrac{1}{c^2}\dfrac{\partial^2 p_{ac}}{\partial t^2} + \dfrac{2}{c^4}\delta_{cl}\dfrac{\partial^3 p_{ac}}{\partial t^3} = 0, \qquad \text{(Eq. 10.3.13)}\]
    • Entropy Mode: Eq. (10.3.7b), or \(k^2\approx i\omega\rho c_p/\kappa\) (\(\boldsymbol{k\times v} = 0\)) \[\nabla^2s_{ent} = \dfrac{\rho c_p}{\kappa}\dfrac{\partial s_{ent}}{\partial t}\qquad\text{(Eq. 10.3.15)}\]

  • Any superposition of vorticity-, acoustic-, and entropy-mode fields will satisfy the linear equations for a fluid with finite viscosity and thermal conductivity. ... any disturbance satisfying those equations can be represented as such a superposition, ... \[\boldsymbol{v} = \boldsymbol{v}_{vor}+ \boldsymbol{v}_{ac} + \boldsymbol{v}_{ent}\qquad \text{(Eq. 10.4.1)}\]

  • Boundary layer thickness, determined by \(1/|k_l|\), where \(k_l\) is calculated using the dispersion relations.
    • \(l_{vor} = \sqrt{\dfrac{2\mu}{\omega\rho}}\), N.B., there is a typo in the book
    • \(l_{ent} = \sqrt{\dfrac{2\kappa}{\omega\rho c_p}} = \dfrac{l_{vor}}{\sqrt{\text{Pr}}}\)
    • Question: why not use \(2\pi/|k_l|\) to define the boundary layer thickness as \(k_l\) is the wavenumber.
  • The field varies much more rapidly with the \(z\) coordinate than with the \(x\) and \(y\) coordinates (\(\partial/\partial z > \partial/\partial x \sim \partial/\partial y\)), as the sound wavelength is much larger then \(l_{vor}\) and \(l_{ent}\), so the \(\nabla^2\) is approximated by \(\partial^2/\partial z^2\) in Eq. 10.3.11 and Eq. 10.3.15, and leads to
    • \[\dfrac{\partial^2}{\partial z^2}\hat{\psi}(x,y,z) = -\dfrac{2i}{l^2}\hat{\psi}(x,y,z)\qquad\text{(Eq. 10.4.5)}\] with the solution \[\hat{\psi}(x,y,z) = \hat{\psi}(x,y,0)e^{-(1-i)z/l}\qquad\text{(Eq. 10.4.6)}\] (substitute Eq. (10.4.6) into Eq. (10.4.5) will lead to the \(-2i/l^2\) on the LHS) with
    • \(l= l_{vor}\) or \(l_{ent}\), and
    • \(\hat{\psi}=\hat{s}_{ent}\) or \(\hat{\boldsymbol{v}}_{vor}\).
  • The \(\partial/\partial z\) is replaced with \(-(1-i)/l\) based on Eq. 10.4.6

  • Based on the polarization relations $_{vor} =0 $ and \(\boldsymbol{v}_{ent} \approx \left(\dfrac{\beta T \kappa}{\rho c^2_p}\right)_o\nabla s_{ent}\), and the fact that \(\nabla = \nabla_T + \nabla_\boldsymbol{n}\), the Eq. (10.4.7) can be derived

  • Assume the surface is oscillating as a rigid body such that every material point on the surface has a verlocity with complext amplitude \(\hat{v}_{wall}\), \[\hat{\boldsymbol{v}}_{wall} = \hat{\boldsymbol{v}}_{vor} + \hat{\boldsymbol{v}}_{ac} + \hat{\boldsymbol{v}}_{ent} \qquad\text{(Eq. 10.4.8)}\]
  • \(\nabla_T\cdot(\text{10.4.8})\) + Eq. (10.4.7a) leads to Eq. (10.4.10) which express \(\hat{\boldsymbol{v}}_{vor}\) in terms of \(\hat{\boldsymbol{v}}_{ac,T}\) \[\hat{\boldsymbol{v}}_{vor} = -\dfrac{1+i}{2}l_{vor}\nabla_T\cdot\hat{\boldsymbol{v}}_{ac,T}\qquad\text{Eq.a}\]
  • Take the normal component of Eq. 8 \(\rightarrow(\text{Eq. 10.4.8}\cdot\boldsymbol{n})\),
    • Applying Eq. a, Eq. 10.4.7c, Eq. 10.3.14, leads to the boundary condition **Eq. 10.4.12, expressing \(\hat{\boldsymbol{v}}_{wall}\cdot\boldsymbol{n}\) only in terms of acoustic variables \(\hat{\boldsymbol{v}}_{ac}\) and \(\hat{p}_{ac}\)
    • It allows an examination of the effects of viscosity and thermal condition on the reflection of plane waves.
  • Rewrite the Euler's equation \[\nabla_T\cdot\boldsymbol{v}_{ac,T} = -\dfrac{\sin^2\theta_i}{\rho c^2}\dfrac{\partial p}{\partial t}\qquad\text{(Eq.10.4.17)}\]

  • Inserting Eq. 10.4.17 into Eq. 10.4.12 leads to the specific admittance of the surface \[\dfrac{1}{Z} = Y = -\dfrac{\hat{\boldsymbol{v}}_{ac}\cdot\boldsymbol{n}_{wall}}{\hat{p}} = \dfrac{1}{2}(1-i)\dfrac{\omega}{\rho c^2}[l_{vor}\sin^2\theta_i + (\gamma-1)l_{ent}]\qquad\text{(10.4.18)}\]

Different expressions

The definitions of the specific admittance of the wall are different between Chaigne & Kergomard (2016) and Pierce (2019)

  • Chaigne & Kergomard (2016): \[Y_p = \dfrac{1}{\rho c}\sqrt{\dfrac{j\omega}{c}}[\sin^2\theta\sqrt{l_v} + \sqrt{l_t}] \qquad (5.141)\] where the characteristic lengths \(l_v = \dfrac{\mu}{\rho c}\) and \(l_t = \dfrac{\kappa}{\rho c C_p}\) in Eq. (5.136), are defined differently with \(l_{vor}\) and \(l_{ent}\).

  • Pierce (2019): \[\dfrac{1}{Z} = \dfrac{1}{2}(1-i)\dfrac{\omega}{\rho c^2}[l_{vor}\sin^2\theta_i+(\gamma-1)l_{ent}]\qquad (10.4.18)\]

  • Rewrite Eq. (10.4.18) in terms of variables defined in Chaigne & Kergomard's book \[\dfrac{1}{Z} = \dfrac{1}{\rho c}\dfrac{1-i}{1+i}\sqrt{\dfrac{j\omega}{c}}[\sin^2\theta\sqrt{l_v} + \sqrt{l_t}]\qquad Eq. (b)\]

  • Comparing Eq. (b) and Eq. (5.141), the difference is the term \(\dfrac{1-i}{1+i}\) in Eq. (b).
    • If the \(1-i\) in Eq. (10.4.18) is \(1+i\), then the definitions in two books are identical to each other.

  • After tracing back the derivations in Pierce (2019), I found the origin of the \(1-i\) comes from the definition of the phaser.

  • In Kinsler (1999) Fundamentals of acoustics, Ch. 1.5, it is mentioend there are different conventions to represent the time dependence of scillatory functions
    • the engineering convention: \(\exp(j\omega t)\)
    • the physics convention: \(\exp(-i\omega t)\)
  • In Chaigne & Kergomard's book, it is defined as \(\psi = \hat{\psi}e^{j\omega t}\), while in the Pierce's book, it is defined as \(\psi = \hat{\psi}e^{-i\omega t}\). It is this difference that leads to the differences in the definition of the wall losses
  • Different conventions are also reflected in the definition of the wave number involving the attenuation coefficient as shown in Eq. (10.5.10) in Pierce's book, which can be used to calculate \(\Gamma\) for the transfer matrix of a cylinder. \[k = \omega/c + (1+i)\alpha_{walls} \qquad \text{(10.5.10) in Pierce's book}\]
    • If the engineering convention is applied, Eq. 10.5.10 becomes, \(k=\omega/c + (1-i)\alpha_{walls}\), which is the one used in some papers,
      • Van Walstijn, M. et al. (2005) Wideband measurement of the acoustic impedance of tubular objects
      • Mason, W.P. (1928) The propagation characteristics of sound tubes and acoustic filters

References

  • https://www.comsol.com/blogs/theory-of-thermoviscous-acoustics-thermal-and-viscous-losses/
  • Pierce, A. D., 2019, Acoustics: An Introduction to Its Physical Principles and Applications, Springer International Publishing.
  • Chaigne, A., and Kergomard, J., 2016, Acoustics of Musical Instruments, Springer-Verlag, New York.
  • Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V., 1999, Fundamentals of Acoustics, Wiley.