# Acoustic metamaterial design and applications

Table of Contents 1. INTRODUCTION .................................................................................................................... 1 1.1. Metamaterial ..................................................................................................................... 1 1.2. Thesis Organization .......................................................................................................... 4 2 ACOUSTIC TRANSMISSION LINE ..................................................................................... 9 2.1 Introduction ....................................................................................................................... 9 2.2 Locally Resonant Sonic Materials .................................................................................. 10 2.3 Acoustic Circuits ............................................................................................................. 11 2.4 Reflection and Transmission ........................................................................................... 15 2.5 Absorption and Attenuation of Sound in Pipe ................................................................ 23 2.6 Acoustic Isotropic Metamaterial ..................................................................................... 36 2.7 Anisotropic Acoustic Metamaterial ................................................................................ 42 3 ULTRASOUND FOCUSING USING ACOUSTIC METAMATERIAL NETWORK .......... 51 3.1 Introduction ..................................................................................................................... 51 3.2 Negative Refractive Index Lens ...................................................................................... 53 3.3 hononic Crystal ............................................................................................................... 55 3.4 Ultrasound Focusing by Acoustic Transmission Line Network ..................................... 59 4 BROADBAND ACOUSTIC CLOAK FOR ULTRASOUND WAVES ................................ 86 4.1 Introduction ..................................................................................................................... 86 4.2 Optical Transformation ................................................................................................... 88 4.3 Acoustic Cloak ................................................................................................................ 95 4.4 Numerical Simulation of Acoustic Cloak Based on Transmission Line Model ............. 98 4.5 Irregular Transmission Line Network ........................................................................... 103 4.6 Experimental Study of Acoustic Cloak Based on Transmission Line Model ............... 110 5 SUMMARY AND FUTURE WORK ................................................................................. 130 5.1 Summary ....................................................................................................................... 130 5.2 Future Work .................................................................................................................. 131 APPENDIX A: LUMPED CIRCUIT MODEL .......................................................................... 133 APPENDIX B: FRESNEL LENS DESIGN BY ACOUSTIC TRANSMISSION LINE ........... 140 APPENDIX C: NEGATIVE INDEX LENS BASED ON METAL-INSULATOR-METAL (MIM) WAVEGUIDES .............................................................................................................. 148 APPENDIX D: SCATTERING FIELDS FROM THE CLOAK ................................................ 161 APPENDIX E: EXPERIMENTAL SETUP AND DATA ACQUISITION ................................. 174 APPENDIX F: CIRCUIT MODELING ..................................................................................... 179

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1 INTRODUCTION 1.1 Metamaterial Over the past eight years, metamaterials have shown tremendous potential in many disciplines of science and technology. The explosion of interest in metamaterials is due to the dramatically increased manipulation ability over light as well as sound waves which are not available in nature. The core concept of metamaterial is to replace the molecules with man-made structures, viewed as “artificial atoms” on a scale much less than the relevant wavelength. In this way, the metamaterial can be described using a small number of effective parameters. In late 1960s, the concept of metamaterial was first proposed by Veselago for electromagnetic wave 1 . He predicted that a medium with simultaneous negative permittivity and negative permeability were shown to have a negative refractive index. But this negative index medium remained as an academic curiosity for almost thirty years, until Pendry et al 2,3 proposed the designs of artificial structured materials which would have effectively negative permeability and permittivity. The negative refractive index was first experimentally demonstrated at GHz frequency. 4,5

It is undoubtedly of interest whether we can design metamaterial for the wave in other systems, for example, acoustic wave. The two waves are certainly different. Acoustic wave is longitudinal wave; the parameters used to describe the wave are pressure and particle velocity. In

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electromagnetism (EM), both electric and magnetic fields are transverse wave. However, the two wave systems have the common physical concepts as wavevector, wave impedance, and power flow. Moreover, in a two-dimensional (2D) case, when there is only one polarization mode, the electromagnetic wave has scalar wave formulation. Therefore, the two sets of equations for acoustic and electromagnetic waves in isotropic media are dual of each other by the replacement as shown in Table 1.1 and this isomorphism holds for anisotropic medium as well. Table1.1 presents the analogy between acoustic and transverse magnetic field in 2D under harmonic excitation. From this equivalence, the desirable effective density and compressibility need to be established by structured material to realize exotic sound wave properties. The optical and acoustic metamaterial share many similar implementation approaches as well. The first acoustic metamaterial, also called as locally resonant sonic materials was demonstrated with negative effective dynamic density. 6 The effective parameters can be ascribed to this material since the unit cell is sub-wavelength size at the resonance frequency. Furthermore, by combining two types of resonant structural, acoustic metamaterial with simultaneous negative bulk modulus and negative mass density was numerically demonstrated. 7

Recently, Fang et al. 8 proposed a new class of acoustic metamaterial which consists of a 1D array of Helmholtz resonators which exhibits dynamic effective negative modulus in experiment.

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Table 1.1 Analogy between acoustic and electromagnetic variables and material characteristics Acoustics Electromagnetism (TMz) Analogy xx ui x P ωρ−= ∂ ∂

yy ui y P ωρ−= ∂ ∂

Pi y u x u y x ωβ−= ∂ ∂ + ∂ ∂

yy z Hi x E ωμ−= ∂ ∂

xx z Hi y E ωμ= ∂ ∂

zz x y Ei y H x H ωε−= ∂ ∂ − ∂ ∂

Acoustic pressure P

Electric field z E

PE z ↔−

Particle velocity x u y u Magnetic field yx HH,

xy uH − ↔ yx uH ↔

Dynamic density x ρ y ρ Permeability x μ y μ

yx μ ρ ↔ xy μ ρ ↔

Dynamic compressibility β Permittivity z ε

β ε ↔ z

The concept of metamaterial extends far beyond negative refraction, rather giving enormous choice of material parameters for different applications. One of the most interesting examples is the invisible cloak by transformation optics. 9,

10 . A simpler version of two-dimensional (2D) cloak was implemented at microwave frequency. 11 Later on, this new design paradigm is extended theoretically to make an “inaudible cloak” for sound wave. 12,13,14,15 The sound wave is directed to flow over a shielded object like water flowing around a rock. Because the waves reform their original conformation after passing such a shielded object, the object becomes

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invisible to the detector. The cloak for surface wave is also proposed in hertz frequency range. 16

Since the inception of the term metamaterials, acoustic metamaterials have being explored theoretically but there has been little headway on the experimental front. The development of acoustic metamaterial will yield new insight in material science and offer great opportunities for several applications.

1.2 Thesis Organization The central theme of this thesis is to design and characterize the acoustic metamaterial for potential application in ultrasound imaging and sound controlling. This dissertation is organized into four chapters. Besides the current chapter which intends to give a brief introduction of the acoustic metamaterial and the motivations of this dissertation, the other three chapters organized as following: The second chapter describes the approach to build an acoustic metamaterial based on the transmission line model. The basic concept and derivation of lumped acoustic circuit is introduced. In the lumped circuit model, the aluminum is assumed as acoustically rigid, considering the acoustic impedance of aluminum is around eleven times of that of water. A more careful analysis including elasticity of the solid suggested that at low frequency the majority of acoustic energy can be predominantly confined in the fluid, when such an excitation

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originates from the liquid. 17 On the other hand, aluminum does participate in the wave propagation, and may increase the loss. 18 Moreover, the loss and limitation of current lumped circuit model are discussed. The third chapter deals with one of the most promising application of acoustic metamaterials, obtaining a negative refractive index lens which can possibly overcome the diffraction limit. An acoustic system is simulated by the analogous lumped circuit model in which the behavior of the current resembles the motion of the fluid. Based on this lumped network, an acoustic negative index lens is implemented by a two-dimensional (2D) array of subwavelength Helmholtz resonators. The experimental studies are presented, demonstrating the focusing of ultrasound waves through the negative index lens. The fourth chapter is to construct an anisotropic cloak for sound waves at kHz range. Relying on the flexibility of the transmission line approach, an acoustic metamaterial exhibiting effective anisotropic density and bulk modulus is proposed to construct the cloak. An object can be shielded inside the cloak and thus becomes invisible to the detector. Given the simulation results, the sound-shielding capability is explored experimentally by measuring the scattered pressure field.

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References

1 Veselago,V. G.,“The electrodynamics of substances with simultaneously negative values of µ and ε,” Sov. Phys. Usp., Vol. 10, No. 4,509(1968)

2 J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart. “Magnetism from conductors and enhanced nonlinear phenomena.” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).

3 J. B. Pendry, A. J. Holden, W. J. Stewart and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures.” Phys. Rev. Lett. 76, 4773–4776 (1996).

4 Smith, D. R. et al. “Composite medium with simultaneously negative permeability and permittivity”. Phys. Rev. Lett. 84, 4184–4187 (2003).

5 Shelby, R. A., Smith, D. R. & Schultz, S., “Experimental verification of a negative index of refraction”, Science 292, 77–79 (2001).

6 Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, “Locally Resonant

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Sonic Materials”, Science 289, 1734 (2000).

7 Y. Ding, Z. Liu, C. Qiu, and J. Shi, “Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density”, Phys. Rev. Lett. 99, 093904 (2007).

8 N. Fang, D. J. Xi, J. Y. Xu, M. Ambrati, W. Sprituravanich, C. Sun, and X. Zhang, “Ultrasonic Metamaterials with Negative Modulus”, Nat. Mater. 5, 452 (2006).

9 J. B. Pendry, D. Schurig, D. R. Smith,” Controlling Electromagnetic Fields”, Science 312, 1780 (2006)

10 U. Leonhardt, “Optical Conformal Mapping”, Science 312, 1777 (2006).

11 D. Schurig et al, “Metamaterial Electromagnetic Cloak at Microwave Frequencies.”, Science 314, 977-980(2006)

12 Milton G W, Briane M and Willis J R, “On cloaking for elasticity and physical equations with

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a transformation invariant form”, New J. Phys. 8, 248 (2006)

13 S. A. Cummer and D. Schurig, “One path to acoustic cloaking”, New J. Phys . 9, 45 (2007).

14 Torrent D and Sánchez-Dehesa J, “Anisotropic mass density by two dimensional acoustic metamaterials”, New J. Phys. 10 023004 (2008)

15 Daniel Torrent and José Sánchez-Dehesa1, “Acoustic cloaking in two dimensions: a feasible approach”, New J. Phys. 10 063015 (2008)

16 M. Farhat et al, “Broadband Cylindrical Acoustic Cloak for Linear Surface Waves in a Fluid”, Phy.Rev.Lett 101, 134501 (2008)

17 C.R. Fuller and F. J. fahy, Journal of Sound and Vibration 81(4), 501-518, (1982) 18 H. LAMB, Manchester Literary and Philosophical Society-Memoirs and Proceedings, 42(9), (1898)

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2 ACOUSTIC TRANSMISSION LINE 2.1 Introduction Recently, there is a new research field that is known under the generic term of metamaterials. Metamaterial refers to materials “beyond” conventional materials, which dramatically increased our ability to challenge our physical perception and intuition. The exponential growth in the number of publications in this area has shown exceptionally promising to provide fruitful new theoretical concepts and potentials for valuable applications. The physical properties of conventional materials are determined by the individual atoms and molecules from which they are composed. There are typically billions of molecules contained in one cubic wavelength of matter. The macroscopic wave fields, either electromagnetic or acoustic wave, are averages over the fluctuating local fields at individual atoms and molecules. Metamaterials extend this concept by replacing the molecules with man-made structures, viewed as “artificial atoms” on a scale much less than the relevant wavelength. In this way the metamaterial properties described using effective parameters are engineered through structure rather than through chemical composition. 1 The restriction that the size and spacing of this structure be on a scale smaller than the wavelength distinguishes metamaterials from photonic/phononic crystals. Photonic/phononic crystal is another different class of artificial material with periodic structure on the same scale as the wavelength. Therefore photonic/phononic crystals usually have a complex response to wave radiation that cannot be

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simply described by effective parameters .However, the structural elements which make up a metamaterial is not necessary periodic. Metamaterial with negative refractive index and the application in superlens has initiated the beginning of this material research. Early in late 1960s, metamaterial was first proposed by Veselago for electromagnetic wave. 2 He predicted that a medium with simultaneous negative permittivity and negative permeability were shown to have a negative refractive index. The first experimental demonstration of metamaterial with negative refractive index is reported at microwave frequency. 3,4 This metamaterial composed of a cubic lattice of artificial meta-atoms with split ring resonators and metallic wires. However, metamaterial with negative index is not the only possibility. Most recent developments explore new realms of anisotropic metamaterial that can produce novel phenomena such as invisibility 5,6,7 and hyperlens. 8,9 Moveover, it is of great interest to extend the metamaterial concept to other classical waves, such as acoustic wave. 10,11,12 Since the analogy between light and sound waves, the electromagnetic and acoustic metamaterials have been sharing much same design freedom while there has been less headway on the experimental front of acoustic wave. 2.2 Locally Resonant Sonic Materials Locally resonant sonic materials, which are a major step towards acoustic metamaterial, are designed by including a resonant unit into the building block of phononic crystal. The key difference between this sonic material and phononic crystal is that the individual unit cell is in deep-subwavelength range compared with the resonant frequency, enabling effective properties

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as mass density and bulk modulus to be assigned to this material. Although the static elastic modulus and density need to be positive to maintain stable structure, these dynamic effective acoustic properties are dispersive in nature and can turn negative at resonance. When the resonance-induced scattered field prevails over the incident fields in background, the volume change can be out of phase with applied dynamic pressure, implying negative bulk modulus. On the other hand, the acceleration can be out of phase with the dynamic pressure gradient, showing negative mass density effect. Liu 13 experimentally demonstrated the localized resonance structure by coating heavy spheres with soft silicon rubber and encasing the coated spheres in epoxy .Negative effective density was obtained due to a dipolar resonance at low sonic frequency. Those anomalous phenomena resulted from strong coupling of the traveling elastic wave in the host medium with the localized resonance rather than Bragg scattering. In the long wavelength limit, the effective medium approach can be employed to offer a good estimation and give an intuitive understanding of this complex system. 14,15,16 It was demonstrated that an acoustic metamaterial can possess simultaneous negative bulk modulus and mass density by combining two types of structural units. While the monopolar resonances give rise to the negative bulk modulus, the dipolar resonances lead to the negative mass density. 17

2.3 Acoustic Circuits A close analogy can be established between the propagation of sound in pipes or chambers and electrical circuits. When the dimensions of the region in which the sound propagates are much

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smaller than the wavelength, a lumped-parameter model is appropriate. The essential thing here is that the phase is roughly constant throughout the system. (Appendix F) 2.3.1 Acoustic Impedance of a Pipe

Figure 2-1 A tube with rigid side wall terminated with acoustic impedance Z l

Assume a hollow cylindrical tube, open at one end and close another end with impedance Z l . The origin of coordinates is chosen to be coinciding with the position of the open end of the tube. We shall assume the diameter of the tube is sufficiently small so that the waves travel down the tube with plane wave fronts. In order to make this true, the ratio of the wavelength of the sound wave to the diameter of the tube must be greater than about 6. If an initial wave traveling in the positive x direction p ai , when the wave propagates at point x=l, a reflected wave traveling in the negative x direction will in general be produced p ar , the corresponding particle velocity can be written as

(2-1)

(2-2)

Where

,

The total pressure in the tube at any point is l

Z 0 Z l

x=0

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(2-3) The total particle velocity is

=

(2-4) So the general expression for the acoustic impedance includes the reflected wave is

(2-5) So we know the impedance at x=0, l as

(2-6)

(2-7) Combine (2-6) and (2-7), we can express the impedance at the open end x=0 as a function of the impedance

, 18

(2-8)

2.3.2 Acoustic Inductance Consider the water in a tube of length l and area S. Assume the tube is acoustically rigid and open on both ends. Since all quantities are in phase when the dimension of the tube is much smaller than the corresponding wavelength, it moves as a whole with displacement under the action of an unbalanced force. The whole part moves without appreciable compression because of the open ends. Substitute (2-8) with Z Al =0, 19

tan (2-9)

Since

replace d Substit u When per cen t

Figure acousti c

As a re s the tub e 2.3.3 A If the t u

is much s d by the Ta y u te (2-10) i n

, we c t error. So w 2-2 A pi p c capacitor s ult of the r e added by a A coustic C a u be is rigidl s maller tha n y lor series f n to (2-9) yi e c an keep o n w e can defi n

(a) p e with (a) o respectivel y r adiation i m a correctio n a pacitance y closed at L A

l n wavelen g f orm, t e lds n ly the firs t n e the acou s

o pen end a n y

m pedance, t h n factor. ′

one end, s u 14 g th,

t an

t term and n s tic inducta

n d (b) rigid

h e l in (2-1 2

u bstitute

is a ver y

n eglect the nce for an o

end is ana l

2 ) should b e 0.85,

∞ in ( 2 y small val u

+… higer orde r o pen end tu b (b) l ogous to a n

e replaced b y where i s 2 -8) 19 C A

V u e, the tan g

r terms wi t b e as n acoustic i n

y an effecti v s the radius g ent can b e (2-10 ) (2-11 ) t hin about 5 n ductor an d (2-12 ) v e length o of the tube e

)

)

5

d

)

f .

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cot (2-13 ) For small value of , the cotangent can be replaced by the equivalent-series form cot

(2-14 ) Equation (2-13) becomes

(2-15 )

is valid within 5 percent for l up to

series as a combination of an acoustic inductance and capacitance. Furthermore, if the second term is small enough, we may neglect it, such that the impedance of the cavity can be expressed as an acoustic capacitance.

(2-16 ) 2.3.4 Helmholtz Resonator A typical Helmholtz resonator as in Figure 2-3 can be presented as a series of inductance and capacitance. The fluid inside the cavity is much easier to be compressed compared with that in the neck part. Moreover, the pressure gradient along the open neck is much greater than that inside the large cavity. Therefore the cavity displays capacitive property and leaves the smaller neck as an acoustic inductor. 2.4 Reflection and Transmission When an acoustic wave traveling in one medium encounters the boundary of a second medium, reflected and transmitted waves are generated. For normal incidence, solids obey the same equations developed for fluids, which is greatly simplified. The only modification needed is that

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the speed of sound in the solid must be the bulk speed of sound, relying on both bulk and shear module. The characteristic acoustic impedances and speeds of sound in two media and the angle of incident wave determine the ratios of the pressure amplitudes and intensities of the reflected and transmitted waves to those of the incident wave. For fluids, the characteristic acoustic impedance is defined as .

Figure 2-3 Helmholtz resonator

2.4.1 Normal Incidence

Figure 2-4 Normal incidence

0

17

Let the boundary 0 be the boundary between two fluids with characteristic acoustic impedance

and

. A plane wave traveling in the direction,

(2-17 ) when the incident wave strikes the boundary, generates a reflected wave and a transmitted wave

(2-18)

(2-19) Where

,

, is the angular frequency and

,

are the speed of sound. The particle velocities are

(2-20 )

(2-21)

(2-22)

,

,

(2-23) The boundary conditions require the continuity of pressure and the normal component of the particle velocities must be equal at both sides of the boundary. The first condition implies that there is no net force on the boundary plane separating the media. The continuity of the normal component of velocity requires that the media remain in contact. So at 0

(2-24)

(2-25)

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Substitute (2-17)-(2-23) into (2-24) and (2-25), we can obtain the reflection and transmission coefficients.

(2-26)

(2-27)

(2-28)

(2-29) Where

,

The acoustic intensity of a harmonic plane progressive wave is defined as

. The intensity reflection and transmission coefficients are calculated.

(2-30)

1

(2-31)

Figure 2-5 Oblique incidence

0

19

In the limit when

,

1,

1,

2,

0.The wave is reflected with amplitude equal to the incident wave and no change in phase. The transmitted wave has pressure amplitude twice that of the incident wave. The normal particle velocity of the reflected wave is equal to but 180

out of phase with that of the incident wave. Therefore the total normal particle velocity is zero at the boundary. The boundary with

is termed rigid. In fact, such total reflection caused standing wave pattern in medium 1 and the boundary is the node for the particle velocity and antinode for the pressure. While there is no acoustic wave propagates in medium 2 since the particle velocity is zero and the pressure is static force. Given one example, the density and speed of sound of aluminum are 2700kg/m 3 and 6420m/s. While the density and speed of sound of water is 1000 kg/m 3 and 1500m/s. So the acoustic impedance of aluminum is around 12 times of that of water. Therefore when acoustic wave travels through water in an aluminum tube, the boundary can be assumed as rigid. 2.4.2 Oblique Incidence Assume that the incident, reflected and transmitted waves make the respective angles

,

,

.

(2-32)

(2-33)

(2-34)

(2-35)

(2-36)

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(2-37) Continuity of pressure and normal component of particle velocity at 0 yields

(2-38)

(2-39) Since

(2-39) must be true for all y, this means sin

sin

(2-40)

(2-41) Equation (2-41) is the statement of Snell’s law So (2-39) can be further simplified as

(2-42)

(2-43) So the reflection and transmission coefficients are

(2-44)

(2-45) Where

,

Where the Snell’s law reveals cos

1 sin

1

sin

/ (2-46) If

and

, define sin

/

,

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cos

sin

1

/ (2-47) cos

becomes pure imaginary. The transmitted pressure is

(2-48)

sin

1

/ (2-49) The transmitted pressure field decays perpendicular to the boundary and propagates in the y direction, parallel to the boundary. For incident angle greater than the critical angle, the incident wave is totally reflected and in the steady state, no energy propagates away from the boundary into the second medium. Even though the transmitted wave possessed energy, but it propagates parallel to the boundary. As an example, the speed of sound in aluminum is greater than the one in water. As a result, as a plan wave propagates in water inside an aluminum tube and the incident angle in the solid/fluid interface is almost 90 o , the wave is totally reflected from the aluminum and confined inside the water. 2.4.3 Reflection from the Surface of Solid Define the normal specific acoustic impedance as

·

(2-50) Where

is the unit vector perpendicular to the interface. So the pressure reflection coefficient can be written as

(2-51)

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Solids can support two types of elastic waves: longitudinal and shear. If the transverse dimensions of an isotropic solid are much larger than the wavelength of the acoustic wave, the appropriate phase speed for the longitudinal waves is

(2-52) Where and are bulk and shear modulus of the solids and is the density. For the case of normal incidence, the transmission and reflection coefficient between solid and fluid is the same as those with two fluids interface. However, when plane wave obliquely incident on the surface of a solid, the wave transmitted into the solids might be refracted in three different cases. The wave may propagate along the surface of the solids. Another possibility is that the wave can propagate in a manner similar to two-fluid interface. Moreover, the wave may be converted into two waves, a longitude wave and a transverse wave. For most solids, the normal specific acoustic impedance has two parts, resistance and reactance, respectively.

. The pressure reflection coefficient can be revised as

(2-53) This means the reflected wave at the boundary may either lead or lag the incident wave by certain angle. When

90

,

approaches unity.

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2.5 Absorption and Attenuation of Sound in Pipe The previous sections are under the assumption that all losses of acoustic energy could be neglected .There are two kinds of loss in acoustic wave .The first source is associated with the boundary conditions and the other type of loss is intrinsic to the medium. The losses in the medium can be further subdivided into three basic types: 20 viscous losses, heat conduction losses and losses associated with internal molecular processes. Viscous losses occur when there is relative motion between adjacent portions of the medium. Heat conduction losses are caused by the conduction of thermal energy from high temperature condensations to lower temperature rarefactions. Losses resulted from molecular processes is by converting kinetic energy of the molecular into stored potential energy ,rotational and vibration energies and energies of association and dissociation between different ionic species and complexes in ionized solutions. On the other hand, the loss due to the boundary is more significant when the volume of the fluid is small in comparison with the area of the walls, as when the pipe is narrow. The acoustic velocity amplitude increases from zero at the wall to the maximum value in the center of the pipe. Therefore there exit dissipative forces due to the shearing viscosity of the fluid. In addition to these viscous losses, heat conduction between fluid and the solid wall also causes energy loss. Usually it was assumed that the condensations and rarefactions in fluid are adiabatic and resulting in temperature change. However, for solid wall, the temperature is nearly constant, thus causing the tendency for heat to be conducted from the fluid medium to the solid walls during condensation and vise verse during rarefaction. The heat transfer increases the entropy of the