Reciprocity and the Zwikker-Kosten equations for porous sound absorbing materials

Reciprocity De Bruijn-Janssen Page 1 of 19

Reciprocity and the Zwikker-Kosten equations for poroussound absorbing materials

Alex de Bruijn

Independent acoustical consultant, 7384 AP (51) Wilp Gld, The Netherlands1

Jan H. Janssen †TNO retired, Monster, The Netherlands2

Date: 25 March 2016

Abstract

In 1949 the book Sound Absorbing Materials SAM was published by C. Zwikker and C.W. Kosten. In this

book four fundamental coupled differential equations for sound wave propagation in porous materials have been

proposed. This set of equations has been quoted abundantly and praised for their simple structure. However,

one of the present authors -Janssen- discovered around 1958 that these equation did not fulfill the reciprocity

requirements as explained by Rayleigh. This generated a significant confusion, since it would suggest that

perhaps porous materials did not obey the laws of reciprocity.

In an unpublished succinct note from 1967 to Janssen (JA), D.W. Van Wulfften Palthe (PA) solved the problem

of non-reciprocity of the four differential equations for sound waves in porous materials published by Zwikker

& Kosten and confirmed their validity when a small additional term was inserted in the equation of air(gas)

motion. Of great practical use is that, like PA expected, the reciprocity correction causes minor differences in

calculation results, but provides simple tests for preventing errors, and therefore that SAM remains reliable.

PACS nr.: 43.55.Ev, 43.20.Jr.

1. e-mail: [email protected]. deceased 28 October 2015

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1 Introduction

In the book written by C. Zwikker and C.W. Kosten [1], chapter 3 is devoted to the acoustic properties

of porous materials with an elastic frame. The book was the result of an exhausting research by a Dutch

group of physicists at the Technische Hogeschool in Delft during the years 1935 - 1945. They were the

first who formulated a practical model for the acoustical behaviour of porous, elastic media. In 1949

a survey of all research efforts was published. The book was acclaimed an immediate success. After

more than 60 years it is still in use and the number of references is numerous, even in the last decade.

The book was never reprinted and for this reason the Netherlands Acoustical Society decided to issue

a facsimile reprint in 2012. In 2009 this Society commemorated the book in a special meeting.

The four basic equations on page 53 have been quoted abundantly and praised for their simple structure.

However, they have been criticized also as being too simple and without a sound physical basis. Already

in 1958 Janssen [3] observed that the equations lead to non-symmetrical or non-reciprocal relations.

This author wondered what the background of the inconsistency might be.

In 1967 D.W. van Wulfften Palthe co-operator/colleague of prof. Kosten in Delft- wrote an unpublished

note to Janssen [3] in which he derived the four, virtually similar, equations by using an essential, slightly

different, porosity definition, thus restoring reciprocity of the formulas of the book and explained the

error. This note has been written, however, in a short style not completely explaining some of the basic

notions. Unfortunately Janssen had to postpone studying it until after his retirement. It is the purpose

of this memorandum to elucidate the Palthe’s theory.

A major part of it is also devoted to deriving the reciprocity properties of the equations by Palthe.

In addition numerical results are presented for some acoustical properties of porous materials with aid

of both sets of equations. These data have been obtained with the so-called characteristic equation in

which the propagation constants of the material are expressed as a function of frequency. An important

property of a porous elastic material is the complex frequency-dependent compressibility. By comparing

the results for both sets of equations in the complex plane the minor influence of the additions by

Palthe is made visible.

2 The Zwikker-Kosten equations

The book [1] 3 starts with a treatment of the basic ideas for the general equations governing the wave

propagation in a porous material with a rigid frame. Then a detailed account is given about the viscous

and thermal effects in porous media. Especially the sound propagation in cylindrical tubes and pores is

elaborated extensively. In chapter III the theory of sound absorption by homogeneous porous and elastic

layers is presented. This theory can be considered as the core of the book. The original Zwikker-Kosten

equations of Chapter III in the notation of the book [1] are given below:

−∂p1∂x

= ρ1∂v1∂t

+ s(v1 − v2); SAM (3.01)

−∂p2∂x

= ρ2∂v2∂t

+ s(v2 − v1); SAM (3.02)

3. The acronym SAM is used for matters related to the book, while PA is used for equations etc. referring to Palthe

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−∂p1∂t

= K1∂v1∂x− (1− h)

h· ∂p2∂t

; SAM (3.04)

−∂p2∂t

= h ·K2∂v2∂x

+ (1− h)(K2 − Po) ·∂v1∂x

; SAM (3.05)

in which the coupling coefficient:

s = ωρ2 · (k − 1) + h2 SAM (3.03)

Po = static pressure;

h = porosity, ratio of accessible holes volume to the total volume of the medium, SAM (ch.I, 1);

p1 = force acting on the frame per unit area of cross-section of the sample;

p2 = excess force acting on the air per unit area of cross-section of the sample;

v1 = mean velocity of the air at x;

v2 = mean velocity of the solid material at x;

K1 = specific stiffness of the frame and

K2 = normal (incompressibility) modulus of air, if necessary to be taken complex;

k = structure factor or tortuosity;

σ = resistance constant; SAM (ch.I, 1.26, 1.27);

ρ1 = density of the frame material (i.e. the mass of the solid material per unit porous material);

ρ2 = density of the air in the pores and ρ2 = h · ρo or ρ2 = h · ρ; SAM (3.07).

These equations seem simple, but their numerical evaluation to practical situations is sometimes dif-

ficult. An estimate, moreover, for the structure factor is certainly doubtful, this holds also for the

resistance coefficient . Moreover, reliable measurements or estimates of and are laborious. For years

these equations have been considered to be the unique basis for the study of elastic porous materials.

Many investigators like the well-known Biot in the years fifty have expanded the Zwikker-Kosten equa-

tions with improvements for the coupling coefficient and the interaction of the air with the frame and

moreover with better estimates of the elasticity of the frame.(cf. [4], [5], [7]).

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3 Palthe’s Text

The following two subsections have been taken literally from the original text by Palthe[2]. It presents

Palthe’s ideas about the interaction of the frame and the air in an elegant manner.

3.1 Notation

The symbols used by Zwikker & Kosten and in the present instance are introduced.

Table 1 Symbols, *) indicating that further discussion is to follow.

Zwikker & Kosten Present Remarks, definitions

h, ho h, hs porosity, volumetric fraction of gas, time- and place- de-pendent quantity and static value, respectively.

p1 p1 average pressure in the direction of propagation in theframe, average over frame and pores.

Po ps barometric or static pressure.

p sound pressure, average value in pores only.

p2 force transferred by gas per unit area of material. p2 =

hsp+(1− hs)

ωps∂v1∂x

.*) PA(1)

u volume velocity of the gas per unit area of material.

v2 average particle velocity in the gas, average over poresonly. v2 ∼= u/hs

E, Ev Young’s modulus for the frame under barometric pressureand in vacuum, respectively.

K1 modulus of elasticity for the frame, given by K1 = E +(1− hs)2

hsps. *) PA(2)

ρ1 ρ1 density of the frame in vacuum.

ρo ρs density of gas, static value.

ρ2 mass of gas per unit volume of material; ρ2 ≈ hsρs

K2 K2 bulk modulus of an ideal gas and an isentropic change ofstate , K2 = κps, where κ is the ratio of specific heats (atconstant pressure and volume).

ωµ force transfer impedance, the force exerted by the gas onthe frame per unit volume of pore space for unit relativeparticle velocity of the gas with respect to the frame,

µ = ρs(ks − 1) +σhs

ω. PA(3)

s force transfer impedance, s = ωµhs PA(4)

k ks structure factor, factor giving relative increase of forces ofinertia due to tortuosity of the pores, including boundarylayer effects.

σ σ specific flow resistance of the material for the gas.

S S = µ · hs.

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3.2 The fundamental equations

Fig. 1 Section through the porous material

The equation of motion for the the gas reads:

− ∂p∂x

= ωρsu/hs + ωµ · (u/hs − v1)

= ω(ρs + µ) · u/hs − ωµ · v1. PA(5)

A pressure gradient is required to accelerate the gas and to overcome the inertial and viscous forces exertedon the gas by the frame. The average particle velocity in the pores equalling u/hs, the forces exerted bythe frame are proportional to u/hs − v1.Eq. PA(5) maybe investigated for two extreme cases:• v1 = u/hs

and• v1 = 0

The equation of motion for the frame reads:

−∂p1∂x

= ωρ1v1 − ωµhs · (u/hs − v1)

= ω(ρ1 + µhs) · v1 − ωµ · u. PA(6)

The acceleration of the frame is due to a pressure gradient in the frame and the forces exerted on the frameby the gas. These latter appear to have decreased by a factor hs when Eqs. PA(5) and PA(6) are compared.This is not so: Eq. PA(5) is founded on the unit value of pore space, Eq. PA(6) on that of material.

The equation of continuity for the gas reads:

ωp · hs

K2= −∂u

∂x− (1− hs) · ∂v1

∂x. PA(7)

The left-hand side of Eq. PA(7) essentially represents the change in density of the gas, taking into accountthat a fraction hs of the total space is available to the gas. The right-hand side represents the relative rateof change of the gas volume, due to the gradient in gas particle velocity and the deformation of the frame.The material of the latter is assumed to be incompressible *).

Hooke’s law for the frame reads:

−Ev∂v1∂x

=∂

∂t

[p1 − (1− h)(p+ ps)

]. PA(8)

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The above equation expresses the fact that the deformation of the frame is due to the directional stressrepresented by p1 and an isotropic internal pressure caused by the presence of an external pressure. Theframe material having been assumed incompressible, it follows that:

1− h = (1− hs)(

1− 1

ω

∂v1∂x

), PA(9)

or:

h = hs +1

ω(1− hs)

∂v1∂x

.

Defining:

E = Ev + (1− hs)ps.

and using Eq. PA(8) it now follows that:

−E∂v1∂x

= ω[p1 − (1− hs)p

]. PA(10)

Equations PA(5), PA(6), PA(7) and PA(10) are the basic equations sought.For some applications, it is advantageous to bring Eq. PA(7) into a slightly different form. Using Eq. PA(10)it follows that:

−∂u∂x

= ω[ hs

K2+

(1− hs)2

E

]· p− ω(1− hs)p1/E. PA(11)

3.3 Comparison with the fundamental equations of Zwikker and Kosten

Using Zwikker & Kosten’s [1] (p. 54, Eq. SAM(3.06)) definition, it follows that:

p2 = h ·(p+ ps

)− hs · ps.

Elimination of h with aid of Eq. PA(8) yields Eq.PA(1). u, p, µ and ρs are eliminated from Eqs. PA(5),PA(6), PA(7) and PA(10), v2, p2, s and ρ2 being introduced in their place. The result is:

−∂p1∂x

= (ωρ1 + s) · v1 − s · v2, PA(12)

−∂p2∂x

= (ωρ2 + s) · v2 − s · v1 −1

ω· (1− hs) · ps ·

∂2v1∂x2

, PA(13)

−[E + (1− hs)2

pshs

]· ∂v1∂x

= ωp1 − ω(1− hs) · p2hs, PA(14)

−ωp2 = hsK2 ·∂v2∂x

+ (1− hs)(K2 − ps

)· ∂v1∂x

. PA(15)

These equations are compared to Zwikker & Kosten’s (p.53) and it is seen that their:

• Eq. SAM(3.01) is identical with Eq. PA(12);

• Eq. SAM(3.05) is identical with Eq. PA(15);

• Eq. SAM(3.04) is identical with Eq. PA(14); provided K1 satisfies Eq. PA(2),

• Eq. SAM(3.02) differs from Eq. PA(13), the term − 1

ω(1− hs) · ps ·

∂2v1∂2x

present in the right hand

side of the latter, lacking in the former.

It is this discrepancy that gives rise to the apparent violation of the reciprocity theorem in Zwikker &

Kosten’s case. In practice, this discrepancy will be so small as to be insignificant in most cases. It arises

from the fact that p2, on taking the factor hs into account, is not identical with the sound pressure: the

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variation in porosity must be taken into account as may be seen from Eq.(1). Nevertheless, p2 was used

(by Zwikker & Kosten) as a sound pressure in the equation of motion for the gas.

The quantities Ev, E and K1 all differ slightly in their definition. In most cases the actual differences will

be quite small. It should be noted however that there is no direct relationship between these quantities and

Young’s modulus for the frame material. Consider e.g. fibrous materials. The deformation of the frame will

be presumably be due principally to flexing rather than straining of the fibers. This form of deformation

does not involve a change in volume, thus justify the assumption of incompressible frame in this instance.

But even in the general case, the neglection of the compressibility of solid matter with respect to that of

gas appears to be warranted in practically all cases, with exception of pathological frames e.g. consisting

of this films of solid matter in which gas bubbles are enclosed.

? ? ? ? ?

This terminates the literal text by Palthe in his original menorandum of 1967. The memorandum itself

contains more explanatory pages.

4 Electrical equivalent circuit

That the present basic equations satisfy the reciprocity theorem follows from the fact that they can be

represented by an equivalent circuit. Such a circuit for an infinitesimal slice of material thickness dx is

given in Fig. 2. That Eqs.PA(5), PA(6), PA(10) and PA(11) are indeed satisfied follows on composing

the Kirchhoff equations for the circuit.

The memorandum of Palthe contains an equivalent circuit of his four fundamental differential equations.

He offered neither an explanation nor a description how he obtained this circuit. On the basis of

the existence of the circuit one may conclude that the differential equations satisfy the reciprocity

requirements. We believe that first Palthe designed this circuit and adapted later the circuit elements

such that the reciprocity is obtained. For understanding this circuit one works backwards: find the

equations by applying network theory as the Kirchhoff s laws. This was not so easy, since the circuit

under consideration is rather complicated. See the original figure below:

Fig. 2 Electrical equivalent circuit for slice of porous material. Copy of the original figure in ref. [2]

The original circuit can be transformed into a circuit with a number of nodes and meshes:

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Fig. 3 Modified equivalent circuit of the Palthe equations in order to work with nodes and meshes.

In the figure a number of symbols have been used to make the drawing transparent. The following

symbols are used to denote the various impedances in the Palthe circuit:

• 1 =1

ω

/[ hsK2

+(1− hs)(2− hs)

E

]dx

• 2 =1

ω

/[ (2− hs)E

]dx.

• 3 = − 1

ω

/[ (1− hs)E

]dx.

• 4 = ω[ρs + (1− hs) · µ

]dx/hs

• 5 = ωµdx.

• 6 = ω[ρ1 − (1− hs) · µ

]dx

The first law of Kirchhoff denotes that the sum of the incoming and outgoing currents must be zero.

Nodes are given in the figure by a, b, c and d. Note the outgoing currents are denoted with ”+” and

ingoing currents with ”-”. The equations at the various nodes are:

a: u− J I− J IV + J V + J II− v1 = 0;

b: −(u+ du) + (v1 + dv1) + J IV − J V = 0;

c: J I + du+ J III = 0;

d: −J II− J III− dv1 = 0,

noting that J I = J II− du+ dv1.

The loops represent currents:

• J I = p/ 1 ;

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• J II = p1/ 2 ;

• J III = (p+ p1)/ 3 ;

• J IV = u+ du;

• J V = v1 + dv1.

We start with loop J IV:

(u+ du) · 4 + p+ dp− p+ (J IV − J V) · 5 = 0 (4.1)

This leads to:

−dpdx

= ω[ρshs

+(1− hs)µ

hs

]· u+ ωµ · (J IV − J V). (4.2)

Since (J IV − J V) ≈ u− v1, hence

−dpdx

= ωρshs· u+ ωµ

( uhs− v1

). (4.3)

This is just ”PA (5)”

We consider now

−dp1dx

= ω[ρ1 − (1− hs)µ

]· (v1 + dv1) + ωµ

(J IV − J V

)(4.4)

since

−dp1dx

= ω(ρ1 + hsµ)(v1 + dv1)− ωµ[(v1 + dv1)− J V + J IV ≈ u

].

Hence:

−dp1dx

= ω(ρ1 + hsµ)(v1 + dv1)− ωµ[(v1 + dv1)− J V + J IV ≈ u

]. (4.5)

−dp1dx

= ω(ρ1 + hsµ) · v1 − ωµ · u

We consider now J I and c+ d:

J I = ωp[ hsK2

+(1− hs)(2− hs)

E

]dx =

[J II = ωp1

(2− hs)E

dx]− du+ dv1.

J III =(p+ p1)

3= −

[J II =

(p1)

2

]− dv1 (4.6)

=⇒ ωp1 ·[ 1

3+

1

2=

1

E

]dx = ωp

1− hsE

dx− dv1.

Multiply Eq. (4.6) both the left-hand and the right-hand-sides with (2− hs) and eliminate herewith p1

from Eq. (4.5):

ωp · hsK2

= −dudx− (1− hs) ·

dv1dx

,

being ”PA(7)”, while Eq. (4.6) does agree with ”PA(10)”.

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5 Considerations about reciprocity

5.1 Lagrange symmetry and reciprocity

The major question is whether the equations given by Palthe are a sound physical system. A piece

of sound absorbing material is in itself a mechanical system with its own vibrational behaviour. The

interaction between the air in the pores and the skeleton constitutes a mechanical system which should

satisfy the formulation by Euler-Lagrange. In principle the differential equations obtained by Palthe

should show the property they can be reworked to typical Euler-Lagrange equations.

To elucidate this method a general introduction into mechanical properties of small vibrations is given.

In general the Lagrange function is the difference of the kinetic energy T and the potential energy of

moving system V: L = T−V.

An expression for the kinetic energy for small oscillations can be written as, cf. ref. [10], p.320:

T =1

2

∑i,j

Ti,j · η̇i · η̇j . (5.1)

and similarly for the potential energy:

V =1

2

∑i,j

Vi,j · ηi · ηj . (5.2)

in which ηi is the displacement in a generalized coordinate qi . The constants Ti,j and Vi,j must

be symmetric, since the single terms are not affected by an interchanging of the indices. This a most

important property, since this leads directly to the reciprocity of mechanical systems excited in vibrations

with small amplitudes, thus linear behaviour.

To include friction the Rayleigh dissipation function F can be added to this equation. This can be

defined as follows:

F =1

2

∑i,j

Fi,j · η̇i · η̇j . (5.3)

The Lagrangian equations can be written as:

d

dt

( ∂L∂q̇i

)− ∂L

∂qi+∂F

∂q̇i= Ψ (5.4)

where L denotes the Langrangian function and Ψ is the force which excites the system.

This can be elaborated by taking the η′s as the generalized co-ordinates as follows:

∂

∂t

( ∂T∂ηi

)+∂F

∂η̇i+∂V

∂qi= Ψ (5.5)

We assume that the present system viz, the porous material has a kind of two-dimensional character

with displacement in the air ηi and that of the frame ηj . The energies of the oscillations in the material

can be written as follows:

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T = 12T11 · η̇21 + 1

2T22 · η̇22 + T12 · η̇1 · η̇2; (5.6)

F = 12F11 · η̇21 + 1

2F22 · η̇22 + F12 · η̇1 · η̇2; (5.7)

V = 12V11 · η21 + 1

2V22 · η22 + V12 · η1 · η̇2; (5.8)

Finally we get by performing the differentiations:

∑j

Tij η̈j +∑j

Fij η̇j +∑j

Vijηj = Ψj (i, j = 1, 2) (5.9)

We follow the reasoning given by Rayleigh in his book page 104. If we substitute expressions of T, F

and V and write D for the term ∂/∂t, we obtain a system of two equations which can be put in the

form:

e11 · η1 + e12 · η2 = Ψ1 (5.10)

e21 · η1 + e12 · η2 = Ψ2 (5.11)

where eij denotes the quadratic operator:

ers = TijD2 + FijD + Vij . (5.12)

(D denotes the operation ∂../∂t). These are reciprocal relation since e12 = e21 and a relation may exist

that: (Ψ′

1

η′2

)η′1=0

=

(Ψ′′

2

η′′1

)η′′2 =0

. (5.13)

where a single prime denotes the first or ”direct” experiment, while double primes denotes the second

or ”reciprocal” experiment.

It is thus necessary to bring the four differential equations for the motion of the porous layer in the

form of a Euler-Lagrange operator by eliminating the pressures out of these equations. This idea will

be elaborated in section 5.3.

5.2 Violation of the Lagrange symmetry of the Zwikker-Kosten equations

Janssen [3] observed already in 1958 that the Zwikker-Kosten equations did not satisfy the symmetry

property as given in Eqs. (5.10/5.11). For years a discussion was going on whether this is a necessary

consequence of the material properties of inhomogeneous materials or that it was an artefact of the

Zwikker-Kosten model. The well-known book of Pierce [12] (page 198), devotes some interesting

remarks in a footnote to this question. Curious enough Palthe had already resolved this question in

1967 by his more careful definition PA(8) of the porosity h !! The non-reciprocity was apparently not a

consequence of the material properties nor an artifact of the model.

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5.3 Reciprocity and the Palthe equations

We find the Euler-Lagrange equations from the Palthe equations:

−∂p∂x

= ω(ρs + µ) · u/hs − ωµ · v1. PA(5)

−∂p1∂x

= ω(ρ1 + µhs) · v1 − ωµ · u. PA(6)

ωp · hsK2

= −∂u∂x− (1− hs) ·

∂v1∂x

. PA(7)

−E∂v1∂x

= ω[p1 − (1− hs)p

]. PA(10)

We may reduce the number of equations by eliminating p1 and p so that equations are made up with

only the velocities v1 and u . Use ∂.../∂t = ω as frequency parameter and use ∂.../∂x = −γ as

propagation parameter.

Take Eq. PA(7) and differentiate with respect to the X-coordinate:

+∂p

∂x=

[−∂

2u

∂x2− (1− hs)

∂2v1∂x2

]· K2

ωhs; (5.14)

Take Eq. PA(10) and do the same:

−∂p1∂x

= +E

ω

∂2v1∂x2

− (1− hs) ·∂p

∂x. (5.15)

Insert Eq. (5.14) into Eq. PA(5):

[K2

ωhs· ∂

2u

∂x2− ω(ρs + µ)

hs· u

]+

[(1− hs)K2

ωhs· ∂

2v1∂x2

+ ωµ · v1

]= 0 (5.16)

and insert Eq. (5.15) using (5.14) into Eq. PA(6):

[(1− hs)K2

ωhs· ∂

2u

∂x2+ ωµ · u

]+[{

E

ω+

(1− hs)2K2

ωhs

}∂2v1∂x2

− ω(ρ1 + µhs) · v1

]= 0 (5.17)

Rewriting Eqs (5.16) and (5.17):

[γ2 · (1− hs) ·

K2

ωhs+ ωµ

]· v1 =

[−γ2 · K2

ωhs+ ω(ρs + µ)/hs

]· u (5.18)[

+γ2 · (1− hs) ·K2

ωhs+ ωµ

]· u =

[− Eωγ2 − (1− hs)γ2 ·

K2

ωhs+ ω(ρ1 + µhs)

]· v1. (5.19)

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This is a typical Euler-Lagrange system for small-amplitude vibrations in the following form:

a11 · v1 + a12 · u = 0

a21 · v1 + a22 · u = 0

and it is clear that

a12 = a21 =[+γ2(1− hs) ·

K2

ωhs+ ωµ

], (5.20)

hence there is symmetry or reciprocity in the Rayleigh coefficients.

The same procdure to proof ”reciprocity” can be applied to the modified Zwikker-Kosten equations

as presented in the original Palthe memorandum: PA(12), (13), (14) and (15).

5.4 Matrix coefficients for a slice of material

It is an interesting exercise to find the coefficients for the matrix from the original Palthe equivalent

circuit which describes a slice of material as an eight-pole circuit, see Fig. 4. This might be important

to find reciprocal relations between pairs of terminals. The idea is to express the acoustic pressures p,

p+ dp, p1 and p1 + pdp1 as a linear combination of the velocities u, u+ du, v1 and v1 + dv1, as given

below:

p = a11 · u+ a12 · (u+ du) + a13 · v1 + a14 · (v1 + dv1)

p+ dp = a21 · u+ a22 · (u+ du) + a23 · v1 + a24 · (v1 + dv1)

p1 = a31 · u+ a32 · (u+ du) + a33 · v1 + a34 · (v1 + dv1) (5.21)

p1 + dp1 = a41 · u+ a42 · (u+ du) + a43 · v1 + a44 · (v1 + dv1)

Fig. 4 Schematic of a slice material considered as 8 - pole circuit

In order to obtain the coefficients we use the circuit with loops and nodes as given in Fig. 3. There are

a number of loops indicated by the sumbols JI, JII, · · · , JV I. The second law of Kirchhoff stated

that the sum of the voltages around a loop (or mesh) must be zero: Σu = 0. For a number of loops

equations can be composed from which the coefficients can be calculated after some lengthy algebraic

manipulation. We leave this this arithmetic to the reader. As can be expected the coefficients, for

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example, a14 = (±)a41 are symmetric and this proofs that the circuit is a reciprocal circuit. This is not

surprising since the circuity is linear and passive.

Note that the matrix is anti-symmetric, due to the fact the currents on the right hand side have been

taken going out the circuit. It is reciprocal as expected, since amn = (−)anm . An interesting reciprocal

relation might be for example:

First experiment (’):

p′ = a11 · u′ + a12 · (u+ du)′

(p+ dp)′ = a21 · u′ + a22 · (u+ du)′

Second experiment (”):

p′′ = a11 · u′′ + a12 · (u+ du)′′

(p+ dp)′′ = a21 · u′′ + a22 · (u+ du)′′

This leads to:

a12 = a21 =

[(p+ dp)′

u′

](u+du)′=0

=

[p′′

(u+ du)′′

]u′′=0

(5.22)

with the assumption that v1 = v1 +dv1 = 0. This proofs that this kind materials are ”reciprocal”, since

it relates the transmission of sound from position 1 to position 2 (direct experiment) to the transmission

of sound from position 2 to position 1 (reciprocal experiment).

6 Determinant for the propagation constants

The differential equations for the propagation of sound in the porous material can be seen as a set

of dispersion equations in the frequency ω . A propagation constant Γ = γ/ω , where γ = ω/c is

introduced. In fact is Γ2 = 1/c2 . The system is then a set of four equations with four unknown, viz.

p, p1, u and v1 . For a viable solution the determinant must be equal to zero. In this way a second-

order equation in is obtained. The determinant is shown below. With respect to the Palthe equations

the following determinant can be formulated:

p p1 u v1∣∣∣∣∣∣∣∣∣∣Γ 0 −(ρs + µ)/hs +µ

0 Γ +µ −(ρ1 + µ · hs)−hs/K2 0 Γ (1− hs) · Γ

(1− hs)/E −1/E 0 Γ

∣∣∣∣∣∣∣∣∣∣= 0

or [A]Γ4 +

[B]Γ2 +

[C]= 0 (6.1)

After the usual elaboration of the determinant with the aid of the rule of Sarrus we obtain the coefficients

for the terms of the second-order equations.

Terms without the factor Γ2:

De Bruijn-Janssen


(ρ1 + µ · hs)(ρs + µ)

K2 · E− µ · µ · hsK2 · E

=

ρ1 · ρs + ρ1 · µ+ ρs(µ · hs)K2 · E

[= C

]. (6.2)

Terms with the factor Γ2:

−µ · (1− hs)E

+−(ρ1 + µ · hs)

E+− (1− hs)

E· (1− hs) ·

(ρs + µ)

hs

− hsK2· (ρs + µ)

hs+

1− hsE

· µ =

−2 · (1− hs)E

· µ− (ρ1 + µ · hs)E

+− (1− hs)E

· (1− hs) ·(ρs + µ)

hs− hsK2· (ρs + µ) =

−2 · (1− hs)E

· µ− −(ρ1 + µ · hs)E

−[ (1− hs)2

hs · E+hsK2

]· (ρs + µ)

[= B

]. (6.3)

A = 1 (6.4)

There are in fact four solutions: two pairs of sound speed of waves in the positive and the negative

directions. The last direction is not important for our problem. There are two independent waves with

different sound speeds. These waves cannot be related to one in the air and the other in the frame. Both

are waves in the total structure. Solutions can be easily found by considering the standard second-order

equation:

Γ2 = −B/2±√B2/4− C (6.5)

It is rather simple to find numerically the solution for Γ2 = 1/c21,2 for a number of characteristic

parameters like material density, porosity, tortuosity etc. An interesting parameter is the incompressibility

B = 1/(ρ1 · c2) of the porous material. This quantity can be calculated as a function of the frequency.

The quantity ρ1 denotes the density of the solid material as given by the mass of solid material per unit

volume of porous material. (Dont confuse B with the constant B ).

This can also be carried out for the Zwikker-Kosten equations. We refer to page 55 in the book SAM[1]:

Γ4 − Γ2[hρ1 + S

hK1+ρ2 + S

hK2+ρ2(1− h) + S

hK1K2· (1− h)(K2 − Po)

h

]+ρ1ρ2 + (ρ1 + ρ2)S

hK1K2= 0. (6.6)

Another interesting equation is the characteristic equation with the aid of the original Zwikker-Kostenequations

with the correction term derived by Palthe. We obtain then:

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Fig. 4 Incompressibility B as a function of frequency in the complex plane for a certain porous material.

[1− (1− h)2P0

hK1

]Γ4 − Γ2

[1− hh· PoSK1K2

+hρ1 + S

hK1+ρ2 + S

hK2+ρ2(1− h) + S

hK1K2· (1− h)(K2 − Po)

h

]+

ρ1ρ2 + (ρ1 + ρ2)S

hK1K2= 0 (6.7)

Three solution results have been depicted in the graph below, cf. Fig 4:

• The original Zwikker-Kosten characteristic equation in the book [1], (Eq. 6.6).

• The characteristic equation of Zwikker-Kosten completed with the Palthe addition (Eq.6.7).

• The characteristic equation obtained from the pure Palthe equations. (Eqs. 6.1, .2, .3,.4).

For the relevant parameters the data of Fig. 21 in the book [1] on page 64 have been borrowed:

hs =0.95, k = 7, K1 = 0.24 · 105 Pa K2 = 1.4 · 105 Pa, ρ1 = 120.0 kg/m3, ρ2 =1.18 kg/m3,

σ = 2 · 104 mks units.

The results agree in fact very closely. The dots representing the results of the Zwikker-Kosten + Palthes

addition agree principally with the results for the unique Palthe equations. The original Zwikker-Kosten

results differ slightly from the improved equations. This fact was already expected by Palthe in his

memorandum. It is a reassuring idea that the numerical results in the book are essentially sufficiently

accurate for no need of correction.

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7 Conclusions

1. D.W. Van Wulfften Palthe derived in 1967 a set of four differential equations for the propagation

of plane sound waves in elastic and porous materials and designed a corresponding electric circuit

for an infinitesimal slice. It is shown that this circuit concept can be reworked to his differential

equations. The circuit is per definition reciprocal and may generate interesting reciprocal relations.

2. It has been proven that Palthe’s equations lead to a symmetric, reciprocal description of the

acoustic behaviour of porous materials, thus satisfying the Rayleigh reciprocity theorem require-

ments, this in contrast to the set by Zwikker and Kosten.

3. The two sets of equations are nearly identical, however, the main difference being a small but

essential extra term in the air motion equation.

4. Palthe expected that the effect of this term is negligible for practical design applications. This

is in accordance with the example of the compressibility calculated for the Zwikker-Kosten set

in comparison with that for the Palthe set. This, in combination with the refutation of Rosin’s

criticism regarding the two Zwikker-Kosten equations of continuity results into ongoing reliability

of the Zwikker-Kosten theory.

5. Probably, Palthe’s results are unique. None of the references which we have investigated, men-

tioned the possible reciprocal properties of their model. Neither an equivalent electric circuit was

found in the literature. Did other researchers overlook the physical necessity of reciprocity and

its very useful application in checking computation?

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8 References

[1] C. Zwikker, C.W. Kosten, Sound Absorbing Materials,

Elseviers Publishing Company, Inc. 1949, reprint issued by the Netherlands Acoustical Society, 2012.

[2] D.W. van Wulfften Palthe, Sound waves in porous materials with flexible frames,

unpublished hand-written memorandum. Delft, June 1967.

[3] J.H. Janssen, A note on reciprocity in linear passive acoustical systems, Acustica 8 [1958], 76-78.

[4] G.S. Rosin, Oscillations induced in porous materials with an elastic matrix by sound waves at normal

incidence, Sov. Physics-Acoustics 19 [1973], 60-64.

[5] J. F. Allard, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, Chapman

& Hall, London, 1993.

[6] L.L. Beranek, Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets,

J.Acoust.Soc.Amer. 19 [1947], 556-568.

[7] W. Lauriks, A. Cops, Akoestische impedantie van poreuze materialen,

Publicatie nr. 92, Nederlands Akoestisch Genootschap, 1988, 25-36.

[8] J.W.S. Rayleigh, The Theory of Sound, Vol I and II,

Dover publications, New York, 1945, paragraphs 82, 107.

[9] L.M. Lyamshev, A question in connection with the principle of reciprocity in acoustics,

Sov. Physics-Doklady, 3 [1959, 406-409].

[10] H. Goldstein, Classical Mechanics,

Addison-Wesley Publishing Company, Inc, 1974

Other useful books:

[11] H.J. Carlin, A.B. Giordano, Network Theory: An Introduction to Reciprocal and Nonreciprocal

Circuits,

Prentice-Hall, Inc, 1964.

[12] A.D Pierce, Acoustics, An introduction to its physical principles and applications,

McGraw-Hill Book Company, 1981. [12] L. Brillouin, Wave Propagation in Periodic Structures,

Dover Publications, Inc, 1953.

[13] J. Kaashoek, T. Poorter, Elektriciteitsleer, in Technisch Repertorium II Natuurkunde, hoofdstuk

5.5H,

Evenwichtsvergelijkingen van een netwerk (in Dutch), Elsevier 1961.

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Appendix

The Zwikker-Kosten equations and some critical notes

Equations SAM(3.04) and (3.05) have been criticized in the literature.

Severe criticism was put forward by Rosin [1A]. Later Depollier, Allard &Lauriks extended this criti-

cism. In a short Letter to the Editor of the JASA [2A], they attempted to prove that especially SAM4

Eq. (3.05) contains the superfluous parameter Po (barometric pressure) which should be left out. They

based their reasoning on the work of Biot and referred also to Rosin [1A], who based his rejection of

the two continuity equations on a mistake and on a misinterpretation of the SAM elucidation, however.

Biot [3A],[4A],[5A] published various papers in which he elaborated an extensive theory for porous

materials; the fluid in the pores can be a highly viscous and also the elastic properties of the frame

is assumed to be either isotropic or anisotropic. His equations have been simplified for typical flexible

sound absorbing materials, like glasswool. They will be denoted mostly as the simplified Biot equa-

tions. These latter equations have been used by Depollier et al. [2A] to prove the incorrectness of the

Zwikker-Kosten equations.

However, it is shown in a short memorandum of the present authors [6A] that their arguments are

unjustified and that the SAM equations are identical to the simplified Biot equations.

Addional References

[1A] G.S. Rosin, Oscillations induced in porous materials with an elastic matrix by sound waves at

normal incidence, Sov. Physics-Acoustics 19(1973) 60-64.

[2A] C. Depollier, J.F. Allard, W. Lauriks, Biot theory and stress-strain equations in porous sound-

absorbing materials,

Letter to the editor, J. Acoust. Soc. Am. 84(1988) 2277-2279.

[3A] M.A. Biot, Theory of propagation of elastic waves in a fluid-filled saturated porous solid. I; Low

frequency range. J. Acoust. Soc. Am. 28(1956) 168-179

[4A] M.A. Biot, idem. II Higher frequency range, J. Acoust. Soc. Am. 28)1956 179-191;

[5A] M.A. Biot, Generalised theory of acoustic propagation in porous dissipative media, J. Acoust. Soc.

Am. 34(1956) 179-191.

[6A] A. de Bruijn, J.H. Janssen, Justification of the Zwikker-Kosten equations for sound-absorption by

flexible porous materials,

Published on the website www. academia.edu (2016).

4. SAM stands for Sound Absorbing Materials (Zwikker & Kosten)

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Documents

Reciprocity and the Zwikker-Kosten equations for porous sound absorbing materials