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i

NASA/TM—2012–217455

Theoretical Consolidation of Acoustic Dissipation

M.J. Casiano and T.F. Zoladz

Marshall Space Flight Center, Huntsville, Alabama

March 2012

National Aeronautics and

Space Administration

Marshall Space Flight Center • Huntsville, Alabama 35812
ii

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076 –1320

443–757–5802

This report is also available in electronic form at

<https://www2.sti.nasa.gov/login/wt/>
iii

TABLE OF CONTENTS

1. INTRODUCTION ............................................................................................................. 1

2. FORMULATION DESCRIPTION ................................................................................... 3

3. DISSIPATION PARAMETER CONSOLIDATION ......................................................... 6

3.1 Linearized Resistance (Pressure and Mass Flow Rate) .................................................. 6

3.2 Pressure Damping Rate ................................................................................................ 6

3.3 Energy Damping Rate .................................................................................................. 8

3.4 Damping Ratio ............................................................................................................. 8

3.5 Linearized Resistance (Head and Flow Rate) ............................................................... 9

4. ABSORPTION COEFFICIENTS ...................................................................................... 11

4.1 Intrinsic Dissipation ..................................................................................................... 11

4.2 Boundary Dissipation ................................................................................................... 13

5. EXAMPLE ......................................................................................................................... 15

6. SUMMARY ....................................................................................................................... 17

REFERENCES ....................................................................................................................... 19
iv

LIST OF FIGURES

1. Sensor port example .................................................................................................... 15

2. Acoustic dissipation parameters .................................................................................. 16

3. Pressure transfer function for a sensor port using acoustic absorption:

(a) Gain of pB /pA and (b) phase of pB /pA .................................................................... 17

LIST OF TABLES

1. Fluid parameters for air with a temperature of 540 oR (80.33 oF)

and pressure of 4 torr (0.077347 psia) .......................................................................... 16

2. Acoustic dissipation relationship as a function of absorption coefficient ..................... 18
v

NOMENCLATURE

Symbols

A area (in2)

a0 sound speed (in/s)

aph phase speed (in/s)

a∞ frozen sound speed (in/s)

b damping-related coefficient (s/in2)

cp specific heat at constant pressure (lbf in/lbm-oR)

DH hydraulic diameter (in)

E0 energy amplitude (lbf in)

f Darcy friction factor (-); frequency (Hz)

fM molecular relaxation frequency (Hz)

gc constant of proportionality; for EE: (386.087 lbmin/lbf -s2), for SI: (1 kg m/N-s2) 

h head (in)

i imaginary unit, –1=i

2 (-)

k wave number (rad/in); spring constant-related coefficient (1/in2)

L length (in)

m mass-related coefficient (s2/in2)

m mass flow rate (lbm/s)

n number of end walls (-)

P perimeter (in)

P0 pressure amplitude (lbf/in2)

Pr Prandtl number (-)

p pressure (lbf /in2)

q flow rate (in3/s)
vi

NOMENCLATURE (Continued)

R radius (in)

RL linearized resistance (using pressure and mass flow rate) (lbf s/lbm-in2)

RL,W linearized resistance per unit length (using head and flow) (s/in3)

T period (s)

t time (s)

u velocity (in/s)

z axial location (in)

Greek Symbols

a total absorption coefficient (Np/in) 

aM intrinsic molecular relaxation absorption coefficient (Np/in)

awk wall thermal absorption coefficient (Np/in)

awm wall viscous absorption coefficient (Np/in)

ak intrinsic thermal absorption coefficient (Np/in)

am intrinsic viscous absorption coefficient (Np/in)

g propagation constant (1/in); specific heat ratio (-)

dk thermal acoustic boundary layer thickness (in)

dm viscous acoustic boundary layer thickness (in)

e acoustic energy density (lbf /in2)

ζ damping ratio (-)

k thermal conductivity (lbf /s-oR)

l wavelength (in)

m dynamic viscosity (lbm/in-s) 

mmax maximum value of absorption per wavelength (Np)

mV volume viscosity (lbm/in-s)
vii

NOMENCLATURE (Continued)

r density (lbm/in3)

σE energy damping rate (Np/s)

σp pressure damping rate (Np/s)

t shear stress (lbf /in2)

w angular frequency (rad/s) 

Other Symbols

O order of magnitude

p mean component

p′ oscillatory component

Re real part

∆ change in

〈 〉	 time average over an acoustic period, 

1

T ( )dt

0

T

∫
viii
1

TECHNICAL MEMORANDUM

THEORETICAL CONSOLIDATION OF ACOUSTIC DISSIPATION

1. INTRODUCTION

In many engineering problems, the effects of dissipation can be extremely important. Dissipation can be represented by several parameters depending on the context and the models that 

are used. Some examples of dissipation-related parameters are damping ratio, viscosity, resistance, absorption coefficients, pressure drop, or damping rate. This Technical Memorandum (TM) 

describes the theoretical consolidation of the classic absorption coefficients with several other dissipation parameters including linearized resistance. The primary goal of this TM is to theoretically 

consolidate the linearized resistance with the absorption coefficient. As a secondary goal, other 

dissipation relationships are presented. Table 2 in section 6 presents all the dissipation relationships 

in this TM.

Reducing the linearized resistance parameter as a function of acoustic absorption coefficient is a primary goal because of its importance in impedance or lumped parameter modeling of 

the dynamics in pipe systems without flow. Examples of its use would be for dead-headed cavities, 

closed pipes, or sensor ports when the flow resistance is reduced to zero. It is well known that pressure drop is a function of flow rate, and linearized resistance follows directly from a Taylor series 

expansion of Δp(m ) as shown in equation (1):

Δp(m ) = Δp(m ) +

∂Δp

∂m m

(m − m ) +O m ′

2 ( ) . (1)

The dependent variables in this TM are made up of a steady part and an oscillatory part; e.g., 

p = p + p′, so in oscillatory form, equation (1) can be written as equation (2) after dropping 

the higher order terms, O m ′

2 ( ):

Δp(m )′ = ∂Δp

∂m m

m ′ . (2)

Linearized resistance can be defined as the linear relationship between oscillatory pressure 

drop and mass flow rate and is shown in equation (3):

RL ≡ ∂Δp

∂m m

. (3)
2

In a system without flow, the usually dominant form of resistance due to steady flow viscous 

effects, orifice pressure drops, or other dissipative components, is equal to zero. To model the effects 

of dissipation in a fluid system without flow, acoustic absorption coefficients can be used. The 

absorption coefficients, described in further detail in section 4, are used classically as estimates of 

wave propagation dissipation in a fluid. They are useful because they can be estimated directly with 

known properties of the fluid and also can be related to a linearized resistance so they can be used 

in a dynamic system without flow. 

A resistor, inductor, and capacitor (RLC) circuit is commonly used as an analogy in fluid 

systems. There are several RLC fluid analogies, however one of the most commonly used analogy 

considers the RL in series with a shunt C for each component or lump. Reference 1 (chp. 12), for 

example, in dynamic fluid systems analysis, uses this formulation. Reference 2 (pp. 270–275) also 

discusses the circuit analogy in terms of modeling the feed system components in the combustion 

stability problems. Reference 3 (pp. 18–22) describes this fluid analogy along with others including 

the forms of their equations. Recognizing the dissipation formulation and wave equation used is an 

important step in consolidating dissipation parameters.
3

2. FORMULATION DESCRIPTION

The key to consolidating the dissipation terms is to develop identical theoretical formulations while incorporating generic forms of the dissipation parameters. By casting each theoretical 

formulation as a damped wave equation, each dissipation parameter can be directly compared. 

Keeping the dissipation terms generic from the start of the theoretical development, rather than 

substituting in known parameters, provides a good approach. In this section, a damped wave equation is developed with linearized resistance and one is also provided with the acoustic absorption 

coefficient.

A force balance in a differential section of pipe can be simplified to equation (4). This is 

similar to that described in reference 1 (p. 21, equation (2.1)) except that the pipe slope is ignored. 

The significant observation in this formulation is that the dissipation is represented by shear stress 

rather than viscous terms as in the Navier-Stokes equation:

A∂p

∂z

+ ρA

gc

∂u

∂t

+ u

∂u

∂z

⎛

⎝

⎜ ⎞

⎠

⎟ + τP = 0 . (4)

As described in reference 1 (p. 21), the usage of shear stress, a steady flow resistive force, is an 

assumption typically used in transient flow calculations and forms the basis for the unsteady flow 

equations. Also described in reference 1 (p. 290) and reference 4 (p. 231), it is recognized that the 

wall shear stress is not in phase with the mean velocity in pulsatile flow and the inertia term must 

be modified. Additionally, using wall friction as a static function of the mean velocity underestimates the wave attenuation at moderate and high frequencies (ref. 1, p. 290). Considering that the 

advective acceleration term, u×∂u/∂z, is small in low Mach number unsteady flows, it is eliminated 

from the equation; equation (4) becomes the following momentum equation:

A∂p

∂z

+ ρA

gc

∂u

∂t

+ Pτ = 0 . (5)

In steady flow, a force balance gives equation (6), except now it is assumed that this relationship 

also applies in transient flow, thus the dependent variables contain an oscillatory component:

ΔpA = τLP . (6)

In steady state, equation (6) reduces to Δ pA = τ PL. Therefore, equation (6) can be reduced to 

oscillatory form as equation (7):

Δ p′A = τ′LP . (7)
4

In oscillatory form, equation (5) can be written as equation (8):

A∂p′

∂z

+ ρA

gc

∂u′

∂t

+ Pτ ′ = 0 . (8)

The relationship, Δ p′ = RLm ′, holds from equation (2). After substituting the oscillatory mass 

flow rate, m ′ = ρAu′, into this relationship, the oscillatory pressure drop, Δ p′, can be substituted 

into equation (7). The result is shown as equation (9) and is the oscillatory shear stress. Recall the 

hydraulic diameter, DH=4A/P. 

τ ′ = ρADHRL

4L

u′ . (9)

Equation (9) can be substituted directly into equation (8). This is the final form of the oscillatory 

momentum equation while keeping the linearized resistance as a generic parameter:

∂p′

∂z

+ ρ

gc

∂u′

∂t

+ ρARL

L

u′ = 0 . (10)

The approach used by reference 1 is to obtain shear stress specifically from the DarcyWeisbach equation:

Δ p = f L

DH

ρu

2

2gc

. (11)

And then by substituting equation (11) into equation (6), the shear stress relationship in equation (12) is obtained. Since the shear stress will be used in oscillatory flow, the absolute value 

on the velocity term ensures that the shear stress will always oppose the direction of velocity. 

τ = ρ fu u

8gc

. (12)

Equation (12) is then linearized to obtain equation (13):

τ ′ = f ρu

4gc

u′ . (13)

At this point, equation (13) is used directly in equation (8) to obtain a final form of the oscillatory 

momentum equation, which does not contain the generic linearized resistance parameter as shown 

in equation (14). This form of the oscillatory momentum equation is not helpful for determining 

linearized resistance since it has already been eliminated in the equation.
5

∂p′

∂z

+ ρ

gc

∂u′

∂t

+ f ρu

DH gc

u′ = 0 . (14)

As an aside, equation (11) with equation (3) can also be used to obtain the linear resistance in 

a pipe with flow shown as equation (15):

RL = f L

DH

m

ρA2gc

. (15)

The next step in obtaining a damped wave equation with a general linearized resistance term 

is to bring in the continuity equation. Also considering that the advective term is small, the oscillatory continuity equation can be written simply as equation (16) and is also limited to low Mach 

number flows:

∂u′

∂z

+ gc

ρa0

2

∂p′

∂t = 0 . (16)

By differentiating equation (10) with respect to z, differentiating equation (16) with respect to t, 

and combining to eliminate the velocity terms, the following wave equation is obtained. This is 

a damped wave equation that incorporates the linearized resistance:

∂2

p′

∂z

2 − 1

a0

2

∂2

p′

∂t

2 − ARLgc

a0

2

L

∂p′

∂t = 0 . (17)

Reference 5 provides the development of a damped wave equation for waves in a viscous 

stationary medium. The linear momentum equation is derived by incorporating the pressure drop 

per unit length due to visco-thermal friction and can be written as equation (18):

∂p′

∂z

+ ρ

gc

∂u′

∂t

+

2ραa0

gc

u′ = 0 . (18)

And similarly combining with continuity, the damped wave equation becomes equation (19):

∂2

p′

∂z

2 − 1

a0

2

∂2

p′

∂t

2 − 2α

a0

∂p′

∂t = 0 . (19)

By direct comparison of the damped wave equations, equations (17) and (19), the relationship 

between linear resistance and absorption coefficient can be deduced and is described in section 3. 
6

3. DISSIPATION PARAMETER CONSOLIDATION

Several dissipation parameters are consolidated in this section. The linearized resistance 

(using pressure and mass flow rate), pressure damping rate, energy damping rate, damping ratio, 

and linearized resistance (using head and flow rate) are all related to the acoustic absorption coefficients. The relationship between each individual parameter can also be found by comparison 

of the parameters in table 2 in section 6. 

3.1 Linearized Resistance (Pressure and Mass Flow Rate)

By comparing equations (17) and (19), an equality can be written as equation (20):

ARLgc

a0

2

L = 2α

a0

. (20)

Solving for the linear resistance gives the following relationship:

R L a

Ag L

c

= 2 α 0 . (21)

Equation (21) gives the linearized resistance as a function of the absorption coefficient. Absorption 

coefficient is in the form of dissipation per unit length with units of neper per unit distance. The 

absorption coefficient can be evaluated as described in detail in the next section.

3.2 Pressure Damping Rate

Another common parameter used for dissipation is the damping rate. The damping rate can 

be in two forms and care must be taken when using this parameter. The energy damping rate 

is twice that of the pressure damping rate because of the relationship between total energy and 

pressure. This will be described further below. 

Because this derivation is usually discussed only in part in literature,4,6,7 it is shown here 

in completion. To represent damping, the frequency or the wavenumber can be regarded as complex. For damping rate, the frequency is regarded as complex where ω→ω– iσp. In this form, 

σp>0 indicates that the wave decays as is seen in equation (22): 

p(t) = P0e

−i ωt−iσp ( t−kz ) = P0e−iωt

e

ikz

e

−σpt . (22)

The acoustic energy density of a plane wave can be represented as equation (23):
7

ε(t) = p(t)

2

gc

ρa0

2 . (23)

Since the complex form of equation (22) is normally used as a convenience in representing 

a harmonic signal in acoustics, the complex conjugate must be incorporated in the evaluation 

of p(t)2. Also assuming that the fractional change of amplitude is small in one cycle of oscillation, |σp 

/ω|<<1, the quantity e

−2σp t

 can be regarded as constant in time. The evaluation of p(t)2

becomes equation (24):

p(t)

2 = 1

2

Re P0

2

e

−2i ωt−iσp ( t−kz ) + P0

2

e

−2σp ⎛ t

⎝

⎜

⎞

⎠

⎟ . (24)

And the energy density can be simplified to equation (25):

ε(t) = P0

2

e

−2σp t

gc

2ρa0

2 (1+ cos(2ωt − 2kz)) . (25)

Continuing with the small fractional change in amplitude assumption, the quantity e

−2σp t

 can 

again be regarded as constant in time and factored out of the integral in equation (26). The period 

is over one cycle and can be written as T=2p/w. Parseval’s theorem can also be applied directly.

ε = 1

T

ε(t) dt =

0

T

∫ 1

T

p(t)

2

gc

ρa0

2

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

dt

0

T

∫ = P0

2

e

−2σ p t

gc

2ρa0

2 . (26)

It can be shown that the following equality is satisfied by evaluating:

σp = 1

2 ε

d ε

dt . (27)

For the absorption coefficient, the relationship in equation (28) holds. In this case, the wave 

number is regarded as complex where k→k–ia. In this form, a>0 indicates that the waves decay 

as is seen in equation (28):

p(t) = P0e−iωte

ikz

e−αz = P0e−iωte

ikz

e−αa0t . (28)

Using equation (28) and following the same procedure gives equation (29):

α = 1

2a0 ε

d ε

dt . (29)
8

And so the following relationship is obtained by comparing equations (27) and (29):

σ α p = a0 . (30)

3.3 Energy Damping Rate

Acoustic energy can be described as the energy density multiplied by volume. Assuming the 

oscillatory part of acoustic energy follows the form E e Et

0 , it is seen from equation (25) that 

the energy damping rate is given by equation (31):

σ E = 2σ p . (31)

Or in terms of absorption coefficient, it is given by equation (32):

σ α E = 2 a0 . (32)

3.4 Damping Ratio

The damping ratio is used at times and is commonly used in finite element modeling because 

of analytic features designed to couple acoustics and structures. It is also representative of the 

damping parameter used in a system with a single degree of freedom. Analogous to a mass-springdamper system, the wave equation can be rearranged and written as equation (33). The parameters 

m, b, and k do not represent the actual mass, spring constant, and damping coefficient. For the 

parameters to be representative, the wave equation would be written as a force balance, however 

doing this would only introduce a linear factor that would be eliminated when rewriting in a form 

for an analogous equation comparison:

m

∂2

p′

∂t

2 + b ∂p′

∂t

+ kp′ = 0 . (33)

The definition of damping ratio of a damped harmonic oscillator can be represented by the damping coefficient, equation (34):

b = 2 kmζ = 2mωζ . (34)

To provide a direct comparison, the lossless wave equation parameters for m and k, in the form 

of equation (33), are taken from equation (19). Also, by incorporating equation (34) using the 

newly defined parameter for m, the damped wave equation with damping ratio can be written as 

equation (35):

− 1

a0

2

⎛

⎝

⎜

⎞

⎠

⎟

∂2p′

∂t

2 + − 2ω

a0

2 ζ ⎛

⎝

⎜

⎞

⎠

⎟

∂p′

∂t

+

∂2

∂z

2

⎛

⎝

⎜

⎞

⎠

⎟ p′ = 0 . (35)
9

By direct comparison of equations (19) and (35), the equality shown as equation (36) is obtained:

−2α

a0

= −2ωζ

a0

2 . (36)

Damping ratio as a function of absorption coefficient can be related as equation (37):

ζ α

ω = a0 . (37)

3.5 Linearized Resistance (Head and Flow Rate)

Care must also be taken for resistance terms in various formulations. For example, it is common to normalize resistance to density or length. Additionally, the linearized resistance may be 

based on head and flow relationships. For example, in reference 1, the linear resistance is based on 

head and flow rate, rather than pressure and mass flow rate and is also normalized to length. Their 

momentum equation is given by equation (38):

∂h′

∂z

+

1

gA

∂q′

∂t

+ RL,Wq′ = 0 . (38)

With h′ = gc

ρg

p′ and q′ = Au′, equation (38) can be written as equation (39):

∂p′

∂z

+ ρ

gc

∂u′

∂t

+

ρgARL,W

gc

u′ = 0 . (39)

And by comparing the momentum equations, equations (10) and (39), the equality given by equation (40) is obtained:

ρARL

L = ρgARL,Wgc

. (40)

Thus, the relationship between the Wyle & Streeter1 form of linearized resistance (using head 

and flow rate) and the linearized resistance (using pressure and mass flow rate) is described by 

equation (41):

RL,W = RL

L

gc

g . (41)

It is also clear that the Wyle & Streeter1 form of linearized resistance is normalized to length. 
10

In terms of absorption coefficient, the Wyle & Streeter1 linearized resistance relationship is given 

as equation (42):

R a

gA L W, = 2α 0 . (42)
11

4. ABSORPTION COEFFICIENTS

Sources of dissipation can be broken into two main categories—those intrinsic to the 

medium and those associated with the boundary of the medium. The losses intrinsic to the medium 

can be further divided into three basic types: (1) viscous losses, (2) heat conduction losses, and 

(3) losses associated with internal molecular processes. At the boundary there are viscous losses 

and losses to heat conduction between the fluid and the surface. Without derivation the absorption coefficients are provided in this section. Most acoustics textbooks discuss these absorption 

coefficients in further detail.3,4 For the dissipation associated with the boundary, the absorption 

coefficients are usually derived for standing waves. However, reference 7 gives the formulation of 

damping rate that includes nonresonant conditions. The forms in reference 7 are provided in this 

TM to account for both resonant and nonresonant conditions. 

The total absorption coefficient can be regarded as the sum of the absorption coefficients 

for the individual loss mechanisms. Superposition is typically justified in practice, and true when 

losses are small; however, in general, they do have interactions with each other. The unit for the 

absorption coefficient is neper per unit length where a neper is analogous to a bel, but represents 

the natural logarithmic scale rather than the decadic logarithmic scale.

α = αi

i

∑ = αμ + ακ + αM +αwμ +αwκ + ... (43)

4.1 Intrinsic Dissipation

The losses intrinsic to the medium can be further divided into three basic types: (1) viscous 

losses, (2) heat conduction losses, and (3) losses associated with internal molecular processes. The 

complete absorption coefficient is defined as equation (43), but the combination of equations (44) 

and (45) under Stokes’ assumption is regarded as the classical absorption coefficient.

4.1.1 Intrinsic Absorption Due to Viscosity

The intrinsic absorption in a fluid due to viscosity is given by the intrinsic viscous absorption coefficient, equation (44). In Stokes’ assumption, the bulk volume viscosity is zero. Substantial 

theoretical development in the field has occurred since Stokes’ pioneering work that has introduced 

the bulk volume viscosity parameter into the absorption formulation. The sound speed dispersion 

for intrinsic viscous effects is very small and usually too small to measure.

αμ = ω2

2ρa0

3

4

3

μ + μ ⎛ V

⎝

⎜ ⎞

⎠

⎟ . (44)
12

4.1.2 Intrinsic Absorption Due to Thermal Conduction

The intrinsic absorption in a fluid due to thermal conduction is given by the intrinsic thermal absorption coefficient, equation (45). Variations in temperature occur between the slightly 

warmer compressed portion and slightly colder expanded portion of the wave. The variation in 

temperatures gives rise to heat flow as the fluid medium tries to reestablish temperature equilibrium. The heat flow reduces the amount of energy available for the sound wave and represents an 

acoustic loss. The wave due to this absorption mechanism is shown to be nondispersive. For most 

gases the absorption due to intrinsic viscosity and intrinsic thermal conduction is comparable; for 

most liquids the intrinsic absorption due to thermal conduction is usually negligible. 

ακ = ω2

2ρa0

3

(γ − 1)κ

cP

. (45)

4.1.3 Intrinsic Absorption Due to Molecular Thermal Relaxation

The intrinsic absorption in a fluid due to molecular thermal relaxation takes into account 

the internal structure of the molecules and the interactions between them that lead to internal 

vibrations, rotations, ionizations, and short-range ordering. In this case, the pressure is not just 

dependent on the local instantaneous values of density and pressure; additionally, it has a dependence on the rate at which density and temperature are changing. The molecular thermal relaxation 

time is associated with the time needed for a new equilibrium to be established. Transformation 

of the mechanical energy into heat represents a loss to the sound wave. The absorption is most 

significant at the molecular relaxation frequencies of the fluid. The fluid may have several relaxation frequencies in which the total molecular thermal relaxation is the sum of all the absorption 

coefficients calculated separately for each of the relaxations. The sound speed transitions from 

the equilibrium sound speed, a0, at low frequencies that are well below the relaxation frequency, 

to the frozen sound speed, a∞, at high frequencies that are well above the relaxation frequency. 

Experimental data for a particular fluid are needed to obtain the parameters, μmax and f

M, which 

are the maximum value of absorption per wavelength and the relaxation frequency, respectively. 

These parameters are both functions of temperature. This form of absorption can be responsible 

for most of the intrinsic sound absorption in the audio and low frequency regions. The absorption 

coefficient due to molecular relaxation is given by equation (46) and the phase speed dispersion for 

molecular relaxation is given as equation (47):

αM = 2μmax

a0

fM

1+ fM

f

⎛

⎝

⎜ ⎞

⎠

⎟

2 (46)

and

aph = a0

1+

a∞

2 f

a0

2 f

M

⎛

⎝

⎜

⎞

⎠

⎟

2

1+

a∞ f

a0 f

M

⎛

⎝

⎜

⎞

⎠

⎟

2 . (47)
13

4.2 Boundary Dissipation

Boundary dissipation is an entirely separate mode of absorption due to the passage of 

a wave over a surface boundary. The surface exerts a frictional shear force on the overlying fluid 

while thermal conduction allows heat transfer to take place between the fluid and the surface. Both 

result in loss of energy from the wave.

Similar to a boundary layer thickness in steady flow, acoustic waves have an analogous 

acoustic boundary layer thickness. It is much thinner because the boundary layer only has a period 

of half a cycle to develop. For the formulas to hold, the acoustic wavelength must be much greater 

than the acoustic boundary layer thickness   . The viscous and thermal acoustic boundary 

layer thicknesses are given as equations (48) and (49), respectively:

δ μ = 2μ

ρω

(48)

and

δκ = 2μ

ρω Pr . (49)

For the wavelength, λ =2πa0/ω, to be much greater than the acoustic boundary layer thickness, 

equations (50) and (51) must be satisfied:

1

πa0

μω

2ρ

 1 (50)

and

1

πa0

μω

2ρPr

 1. (51)

The combination of the wall viscous absorption coefficient, equation (52), and the wall thermal conduction absorption coefficient, equation (54), is regarded as the combined absorption

coefficient for wall losses. Normally, the formulation of the following boundary layer absorption 

coefficients are derived for resonant conditions, however, reference 7 gives the formulation of damping rate that includes nonresonant conditions. The absorption coefficients at general conditions are 

provided so that both resonant and nonresonant conditions can be estimated. For other pipe shapes, 

the radius can be replaced with a hydraulic diameter term, R=DH /2. Reference 8 compares experimental data, to a very fine degree, to the combined absorption coefficient theoretical formulation.

4.2.1 Viscous Losses at a Rigid Wall

The boundary layer absorption in a fluid due to viscosity for a cylindrical pipe is given by 

the wall viscous absorption coefficient, equation (52). This absorption coefficient accounts also 
14

for nonresonant conditions. The phase speed dispersion for wall viscous effects is given as equation (53). It shows that the sound speed is slightly slower.

αwμ = 1

a0R

μω

2ρ

1− sin(2kL)

2kL

⎛

⎝

⎜ ⎞

⎠

⎟ (52)

and

aph = a0 1−

αwμa0

ω

⎛

⎝

⎜

⎞

⎠

⎟ . (53)

4.2.2 Thermal Conduction at an Isothermal Wall

The boundary layer absorption in a fluid due to thermal conduction for a cylindrical pipe is 

given by the wall thermal absorption coefficient, equation (54). This absorption coefficient accounts 

also for nonresonant conditions, but also end-wall effects that are important in thermal conduction. The sound speed dispersion for wall thermal effects is given as equation (55). It shows that 

the sound speed is slightly slower.

αwκ = 1

a0R

μω

2ρ

γ −1

Pr

⎛

⎝

⎜ ⎞

⎠

⎟ 1+

sin(2kL)

2kL

+ n

R

L

⎛

⎝

⎜ ⎞

⎠

⎟ (54)

and

aph = a0 1− αwκa0

ω

⎛

⎝

⎜ ⎞

⎠

⎟ . (55)
15

5. EXAMPLE

As an example, a 5-in-long senseline with an inner diameter of 0.5 in is examined. A schematic is shown in figure 1. The contents are air at a temperature of 540 °R and pressure of 4 torr.

A

D=0.5 in

B

L=5 in

Figure 1. Sensor port example.

An expression for complex pressure ratio, pB /pA, is needed where A is the open end and B

is the closed end. From reference 1, for a simple pipe closed at one end, this complex pressure ratio 

is described with the hyperbolic secant as equation (56):

pB

pA

= sech(γ L) . (56)

And the propagation constant, γ , can be written as equation (57):

γ = − ω

a0

⎛

⎝

⎜

⎞

⎠

⎟

2

+ i

ωgARL,W

a0

2

⎛

⎝

⎜

⎞

⎠

⎟ . (57)

In this case, the acoustic dissipation is dominated by the viscous and thermal losses at the senseline 

wall. Since the wall effects dominate in this problem, the acoustic absorption coefficient is found by 

substituting the fluid parameters in table 1 and the geometric parameters into equations (43), (52), 

and (54). 
16

Table 1. Fluid parameters for air with a temperature of 540 oR (80.33 oF)

and pressure of 4 torr (0.077347 psia).

Parameters Variable Value Units

Viscosity (dynamic) μ 1.0373 x 10–6 lbf/in-s

Density ρ 2.2367 x 10–7 lbm/in3

Sound speed a0 13,671.6 in/s

Specific heat ratio γ 1.4 –

Prandtl number Pr 0.70641 –

Additionally, substituting these parameters into equation (42) to obtain the linearized resistance term, using equation (57) to obtain the propagation constant, and then lastly combining into 

equation (56), the complex pressure transfer function can be obtained. Equation (42) allows access 

of the absorption coefficient in the form of linearized resistance and thus can be applied directly 

into the complex pressure ratio model. Figure 2 shows both acoustic absorption coefficient and 

damping ratio as a function of frequency. Figure 3 shows the gain and phase of the complex pressure transfer function between location A and B due to acoustic dissipation.

0.125

0.1

0.075

0.05

0.025

0

0.25

0.2

0.15

0.1

0.05

0

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000

Damping Ratio ( ) (–)

Absorption Coefficient

Damping Ratio

Frequency (Hz)

Absorption Coefficient ( ) (Np/in)

Figure 2. Acoustic dissipation parameters.
17

Frequency (Hz)

Gain of pB/pA (–) Phase of pB/pA (deg)

Frequency (Hz)

(a)

(b)

5

4

3

2

1

0

200

100

0

–100

–200

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000

0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000

Figure 3. Pressure transfer function for a sensor port using acoustic absorption:

(a) Gain of pB /pA and (b) phase of pB /pA.
18

6. SUMMARY

This TM describes the theoretical consolidation of the classic absorption coefficients with 

several dissipation parameters including linearized resistance. Table 2 provides the various relationships developed in this TM in terms of absorption coefficient. This allows engineering problems 

that use various dissipation parameters to account for acoustic dissipation by easily estimating and 

incorporating the needed parameters into their particular models. This TM summarizes the consolidation of several dissipation parameters and gives a simple example of a senseline problem. 

Table 2. Acoustic dissipation relationship as a function of absorption coefficient.

Linearized resistance (pressure/mass flow rate)

RL = ∂Δp

∂m m =m

R

L a

Ag L c

= 2  0

Pressure damping rate σ p = αa0

Energy damping rate σ E = 2αa0

Damping ratio 

 = a0

Linearized resistance per unit length (head/flow rate)

R

L

h

q

L W

q q

, = 

 =

1 RL,W = 2αa0

gA
19

REFERENCES

1. Wylie, E.B.; and Streeter, V.L.; Suo, L. (ed.): Fluid Transients in Systems, Prentice-Hall, Inc., 

Upper Saddle River, NJ, 1993.

2. Harrje, D.T.; and Reardon, F.H.: “Liquid Propellant Rocket Combustion Instability,” NASA, 

NASA SP-194, Washington, DC, 1972.

3. Blackstock, D.T.: Fundamentals of Physical Acoustics, John Wiley & Sons, Inc., New York, 

2000.

4. Kinsler, L.E.; Frey, A.R.; Coppens, A.B.; and Sanders, J.V.: Fundamentals of Acoustics, John 

Wiley & Sons, Inc., New York, 2000.

5. Munjal, M.L.: Acoustics of Ducts and Mufflers, John Wiley & Sons, New York, 1987.

6. Culick, F.E.C.: Unsteady Motions in Combustion Chambers for Propulsion Systems, NATO 

Research and Technology Organisation, RTO AGARDograph AG-AVT-039, 2006.

7. Searby, G.; Nicole, A.; Habiballah, M.; and Laroche, E.: “Prediction of the Efficiency of 

Acoustic Damping Cavities,” Journal of Propulsion and Power, Vol. 24, No. 3, pp. 516–523, 

2008.

8. Quinn, T.J.; Colclough, A.R.; and Chandler, T.R.D.: “A New Determination of the Gas 

Constant by an Acoustical Method,” Philosophical Transactions of the Royal Society, Vol. 283, 

pp. 367–420, 1976.
20

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Theoretical Consolidation of Acoustic Dissipation

M.J. Casiano and T.F. Zoladz

George C. Marshall Space Flight Center

Huntsville, AL 35812

National Aeronautics and Space Administration

Washington, DC 20546–0001

Unclassified-Unlimited

Subject Category 71

Availability: NASA CASI (443–757–5802)

Prepared by the Propulsion, Structural, Thermal, and Fluid Analysis Division, Engineering Directorate

M–1331

Technical Memorandum

NASA/TM—2012– 217455

acoustic dissipation, absorption coefficient, linearized resistance, damping rate, unsteady fluid dynamic 

systems with no flow

01–03–2012

UU 32

NASA

U U U

In many engineering problems, the effects of dissipation can be extremely important. Dissipation can 

be represented by several parameters depending on the context and the models that are used. Some 

examples of dissipation-related parameters are damping ratio, viscosity, resistance, absorption coefficients, 

pressure drop, or damping rate. This Technical Memorandum (TM) describes the theoretical consolidation of the classic absorption coefficients with several other dissipation parameters including linearized 

resistance. The primary goal of this TM is to theoretically consolidate the linearized resistance with the 

absorption coefficient. As a secondary goal, other dissipation relationships are presented.

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