Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid

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Article Nonlinear Behavior of High-Intensity Ultrasound Propagation in an Ideal Fluid

Jitendra A. Kewalramani 1,* , Zhenting Zou 2 , Richard W. Marsh 3, Bruce G. Bukiet 4 and Jay N. Meegoda 1

1 Department of Civil & Environmental Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA; [email protected] 2 Dynamic Engineering Consultants, Chester, NJ 07102, USA; [email protected] 3 Department of Chemical & Materials Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA; [email protected] 4 Department of Mathematical Science, New Jersey Institute of Technology, Newark, NJ 07102, USA; [email protected] * Correspondence: [email protected]

Received: 2 February 2020; Accepted: 29 February 2020; Published: 3 March 2020

Abstract: In this paper, nonlinearity associated with intense ultrasound is studied by using the one-dimensional motion of nonlinear shock wave in an ideal fluid. In nonlinear acoustics, the wave speed of different segments of a waveform is different, which causes distortion in the waveform and can result in the formation of a shock (discontinuity). Acoustic pressure of high-intensity waves causes particles in the ideal fluid to vibrate forward and backward, and this disturbance is of relatively large magnitude due to high-intensities, which leads to nonlinearity in the waveform. In this research, this vibration of fluid due to the intense ultrasonic wave is modeled as a fluid pushed by one complete cycle of piston. In a piston cycle, as it moves forward, it causes fluid particles to compress, which may lead to the formation of a shock (discontinuity). Then as the piston retracts, a forward-moving rarefaction, a smooth fan zone of continuously changing pressure, density, and velocity is generated. When the piston stops at the end of the cycle, another shock is sent forward into the medium. The variation in wave speed over the entire waveform is calculated by solving a Riemann problem. This study examined the interaction of shocks with a rarefaction. The flow field resulting from these interactions shows that the shock waves are attenuated to a Mach wave, and the pressure distribution within the flow field shows the initial wave is dissipated. The developed theory is applied to waves generated by 20 KHz, 500 KHz, and 2 MHz transducers with 50, 150, 500, and 1500 W power levels to explore the effect of frequency and power on the generation and decay of shock waves. This work enhances the understanding of the interactions of high-intensity ultrasonic waves with fluids.

Keywords: power ultrasound; nonlinear wave propagation; shock; rarefaction

1. Introduction Ultrasound is a type of sound wave with frequencies ranging from 20 KHz up to several gigahertz. For practical applications, ultrasound is mostly generated by ultrasonic transducers and propagates into the subject material. Ultrasonics is a branch of acoustics dealing with the generation and use of inaudible sound waves [1]. Applications of ultrasonics are rigidly classiﬁed as being of either low intensity (popularly known as non-destructive applications) or high-intensity (also known as power ultrasonics) [1]. A low-intensity acoustic wave in a homogeneous medium propagates at a certain speed (relative to the medium), and deformations produced by the wave in the medium are purely elastic. Ultrasonic non-destructive testing and imaging used as means of exploration, detection,

Acoustics 2020, 2, 147–163; doi:10.3390/acoustics2010011 www.mdpi.com/journal/acoustics Acoustics 2019, 2 FOR PEER REVIEW 4 Acoustics 2020, 2 148 detection, and information (e.g., the location of a crack or determination of material properties), are some of the promising low-intensity applications. and informationHigh-intensity (e.g., acoustic the location wave of behaves a crack ordifferen determinationtly than oflow-intensity: material properties), they can are permanently some of the changepromising the low-intensityphysical, chemical, applications. or biological properties or, if intense enough, even destroy the medium to whichHigh-intensity it is applied acoustic [1]. At high wave intensity, behaves di thefferently amplitudes than low-intensity: of vibration used they are can sufficiently permanently high change that nonlinearthe physical, propagation chemical, or occurs. biological The properties uses of or,high if-power intense enough,ultrasonics even include destroy thecleaning, medium enhancing to which chemicalit is applied reactions, [1]. At highemulsification, intensity, the dispersion, amplitudes welding of vibration of metals used and are polymers, sufficiently machining high that and nonlinear metal formingpropagation in solids occurs. and The fluids, uses offood high-power processing, ultrasonics ultrasonic include agglomeration, cleaning, enhancing etc. The nonlinear chemical nature reactions, of intenseemulsification, acoustic dispersion, waves underlies welding many of metals applications and polymers, of high-intensity machiningand acoustic metal waves forming [2,3]. in solids The branchand fluids, of acoustics food processing, dealing with ultrasonic high-intensity agglomeration, acoustic etc.is called The nonlinearnonlinear natureacoustics. of intense acoustic wavesIn underliesordinary elastic many applicationsmedia, finite-amplitude of high-intensity sound acoustic propagation waves has [2 ,been3]. The intensively branch of studied acoustics to understanddealing with many high-intensity nonlinear acoustic acoustic is effects called nonlinearsuch as cumulative acoustics. wave distortion with propagation distanceIn ordinary as well as elastic the concept media, of finite-amplitude the parametric ar soundray [4]. propagation Nonlinear acoustics has been is intensively based on a studiednonlinear to theoryunderstand of elasticity many nonlinear [5]. Once acoustic the original effects sinusoidal such as cumulative finite-amplitude wave distortionwave propagates with propagation from an ultrasonicdistance as source well as into the concepta fluid medium, of the parametric the wave array will [be4]. exposed Nonlinear to acousticstwo effects is basedinfluencing on a nonlinear its time course:theory of(1) elasticitythe dissipation [5]. Oncearising the from original viscosity, sinusoidal heat conductivity, finite-amplitude and relaxation wave propagates processes from and (2) an theultrasonic nonlinearity source leading into a fluidto the medium, formation the of wave higher will harmonics be exposed to the totwo fundamental effects influencing frequency its of time the wavecourse: [6]. (1) Understanding the dissipation the arising basic frommechanisms viscosity, of heat the conductivity,nonlinear effects and of relaxation intense acoustic processes waves and (2)is criticalthe nonlinearity to studying leading the distribution to the formation of high-intensi of higherty harmonics acoustic waves to the in fundamental a system [7]. frequency The purpose of the of thiswave paper [6]. Understandingis to understand the and basic model mechanisms the nonlinea of ther behavior nonlinear of high-intensi effects of intensety acoustic acoustic waves. waves The is nonlinearitycritical to studying associated the with distribution intense ofultrasound high-intensity is studied acoustic by using waves the in one-dimensional a system [7]. The motion purpose of nonlinearof this paper shock is towave understand in an ideal and fluid. model the nonlinear behavior of high-intensity acoustic waves. The nonlinearity associated with intense ultrasound is studied by using the one-dimensional motion of 2.nonlinear Nonlinear shock Wave wave Propagation in an ideal and fluid. Shock Formation

2. NonlinearIn linearWave wave Propagation motion, disturbances and Shock Formationin a medium can be defined as the result of three independent modes: the acoustic, entropy, and vorticity modes [2,8]. If wave intensity increases, the In linear wave motion, disturbances in a medium can be deﬁned as the result of three independent disturbances in a medium becomes large enough such that nonlinear wave propagation starts [3,7]. modes: the acoustic, entropy, and vorticity modes [2,8]. If wave intensity increases, the disturbances in The nonlinear elastic behavior of the materials becomes progressively more important at high- a medium becomes large enough such that nonlinear wave propagation starts [3,7]. The nonlinear amplitude wave excitation [3,9]. As the speed of the disturbance (or particle velocity) (𝑢) increases, elastic behavior of the materials becomes progressively more important at high-amplitude wave linear wave motion ceases, and therefore the superposition principle is no longer valid. Thus, the (u) disturbanceexcitation [3 ,cannot9]. As thebe speeddescribed of the in disturbancethe form of (orthe particle three independent velocity) modesincreases, [8]. linearThese wave modes motion start interactingceases, and with therefore one another. the superposition As shown principle in Figure is 1, no the longer following valid. three Thus, interactions the disturbance between cannot these be modesdescribed are inactivated the form due of theto high-intensities: three independent the modessound-sound, [8]. These sound-vorticity modes start interactingand sound-entropy with one interactionsanother. As shown[2,8]. inDue Figure to1 , thethe followingsound-sound three interactionsinteractions, between harmonic these modesgenerations are activated and self- due demodulationsto high-intensities: occur the within sound-sound, the acoustic sound-vorticity mode [2,8]. andThen, sound-entropy the acoustic interactionsmode starts [to2, 8interact]. Due towith the thesound-sound other two modes, interactions, and this harmonic interaction generations induces andacoustic self-demodulations heating and generation occur within of hydrodynamic the acoustic flowmode [8]. [2 ,8]. Then, the acoustic mode starts to interact with the other two modes, and this interaction induces acoustic heating and generation of hydrodynamic ﬂow [8].

Figure 1. Study of nonlinear acoustics [Adopted based on a ﬁgure in Sapozhnikov (2015)].

As aFigure consequence 1. Study of of nonlinearity,nonlinear acoustics the local [Adopted sound ba speedsed onc adepends figure in onSapozhnikov the particle (2015)]. velocity u (c is function of u), cu, and also the local wave propagation velocity depends on the particle velocity u [2]. Thus, there are two independent sources of the acoustic nonlinearity:

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As a consequence of nonlinearity, the local sound speed 𝑐 depends on the particle velocity 𝑢 Acoustics 2020, 2 149 (𝑐 is function of 𝑢), 𝑐, and also the local wave propagation velocity depends on the particle velocity 𝑢 [2]. Thus, there are two independent sources of the acoustic nonlinearity: •Change of the wave propagation speed due to drift with velocity u. • Change of the wave propagation speed due to drift with velocity 𝑢. •Change in local sound speed from c0 to cu. • Change in local sound speed from 𝑐 to 𝑐. This transition from the linear regime to a nonlinear corresponds to a change in the local wave This transition from the linear regime to a nonlinear corresponds to a change in the local wave propagation from c0 to cu + u, as shown in Figure2. The sections of the waveform with higher u propagation from 𝑐 to 𝑐 +𝑢, as shown in Figure 2. The sections of the waveform with higher 𝑢 propagate faster than those with lower u. This accumulation of the nonlinear effects in the waveform propagate faster than those with lower 𝑢.This accumulation of the nonlinear effects in the waveform leads to variation (or deviation) in the propagation speed of different segments of the waveform leads to variation (or deviation) in the propagation speed of different segments of the waveform (Figure2c) resulting in distortion of the sinusoidal wave and increasing steepness of the wavefront (Figure 2c) resulting in distortion of the sinusoidal wave and increasing steepness of the wavefront (Figure2d). Eventually, these distortions result in a shock formation represented by an N-wave (Figure 2d). Eventually, these distortions result in a shock formation represented by an N-wave (Figure2e) [ 3,7,10]. At the shock, the jump condition forms, and the medium undergoes an abrupt and (Figure 2e) [3,7,10]. At the shock, the jump condition forms, and the medium undergoes an abrupt nearly discontinuous change in the pressure, density, and temperature [2,4,10]. The shock formation and nearly discontinuous change in the pressure, density, and temperature [2,4,10]. The shock corresponds to the first appearance of infinitely steep waveform regions associated with the N-wave. formation corresponds. to the first appearance of infinitely steep waveform regions associated with The critical distance X is the distance propagated by an acoustic wave in an ideal medium when the the N-wave. The critical distance 𝑋 is the distance propagated by an acoustic wave in an ideal infinite slope first appears in the waveform [7]. medium when the infinite slope first appears in the waveform [7].

Figure 2. Schematic representation of acoustic distortion and shock formation during nonlinear propagation [Adopted based on figures in Blackstock and Hamilton (1998) and Leighton (2007)]. 3. NonlinearFigure Interactions 2. Schematic Within representation the Acoustic of acoustic Mode distortion and shock formation during nonlinear propagation [Adopted based on figures in Blackstock and Hamilton (1998) and Leighton (2007)]. In order to analytically study the nonlinear behavior of intense ultrasound waves, an ideal model is to3. consider Nonlinear simulating Interactions a sinusoidal Within wave the Acoustic propagating Mode in one-dimension. The motion of the wave can be modeled as the flow of an ideal fluid in an infinitely long tube extending along the x-axis, with one end having aIn moving order to piston analytically and the study other the end nonlinear being either behavior open orof havingintense aultrasound fixed wall, waves, where an there ideal is nomodel viscosityis to orconsider thermal simulating conductivity a sinusoidal [10]. In such wave a case, propagatingv = (u, 0, 0 )inand one-dimension. all variables depend The motion only on ofx theand wavet. canIn abe gas-filled modeled as tube, the ifflow a pistonof an ideal is moved fluid in into an infinitely it or if a long receding tube extending piston is along stopped the ax-axis, shock with (discontinuities) is generated that moves away from the piston [10]. Similarly, a forward-moving Acoustics 2019, 2 FOR PEER REVIEW 4 one end having a moving piston and the other end being either open or having a fixed wall, where there is no viscosity or thermal conductivity [10]. In such a case, 𝑣=(𝑢, 0,0) and all variables depend Acousticsonly on2020 x and, 2 t. 150 In a gas-filled tube, if a piston is moved into it or if a receding piston is stopped a shock (discontinuities) is generated that moves away from the piston [10]. Similarly, a forward-moving rarefaction wave is sent into the medium when the piston recedes away from the fluid [10]. Rarefactions rarefaction wave is sent into the medium when the piston recedes away from the fluid [10]. and shocks have finite speed such that the regions they influence grow over time. A pulse high-intensity Rarefactions and shocks have finite speed such that the regions they influence grow over time. A ultrasonic wave can be conceived as one complete cycle of piston motion in a tube. That is, the piston pulse high-intensity ultrasonic wave can be conceived as one complete cycle of piston motion in a moves from a stationary or a rest position at a constant speed until it reaches the maximum point of tube. That is, the piston moves from a stationary or a rest position at a constant speed until it reaches displacement, stops, and reverses direction, moving in the opposite direction at the same speed as the maximum point of displacement, stops, and reverses direction, moving in the opposite direction earlier until it reaches its original position and stops again. Figure3 shows the generation of shock and at the same speed as earlier until it reaches its original position and stops again. Figure 3 shows the rarefaction waves, where (a) a shock is produced by a piston moving with constant velocity into the generation of shock and rarefaction waves, where (a) a shock is produced by a piston moving with fluid at rest, (b) the piston reaches its maximum displacement, changes its direction of propagation, constant velocity into the fluid at rest, (b) the piston reaches its maximum displacement, changes its and the forward-moving rarefaction wave is sent into the compressed fluid behind the shock, and (c) direction of propagation, and the forward-moving rarefaction wave is sent into the compressed fluid when the piston comes to rest, another forward moving shock is produced. behind the shock, and (c) when the piston comes to rest, another forward moving shock is produced.

Figure 3. Generation of shock and rarefaction waves [Adopted based on a figure in Courant, (1948)]. Figure 3. Generation of shock and rarefaction waves [Adopted based on a figure in Courant, (1948)]. The one-dimensional flow eliminates the vorticity mode, and the absence of dissipations makes the processThe one-dimensional adiabatic. That flow is, one eliminates may exclude the vortic theity entropy mode, modes and the as absence well. Therefore, of dissipations a 1-D modelmakes describesthe process only adiabatic. the nonlinear That is, behavior one may of exclude the acoustic the entropy mode [4mo]. Thedes as propagation well. Therefore, of intense a 1-D acoustic model wavesdescribes in one-dimensional only the nonlinear motion behavi canor be of mathematically the acoustic mode described [4]. Th bye usingpropagation the conservation of intense equations acoustic ofwaves mass, in momentum, one-dimensional and energy motion [11 ]:can be mathematically described by using the conservation equations of mass, momentum, and energy [11]:      ρ  𝑚 m     2    𝑚  m + P  𝜌  m  + .  ρ  = 0 (1)  ⎛ ∇ +𝑃 ⎞   e  𝜌  m (e + P)  𝑚 +∇.⎜ t ρ ⎟ =0 (1) 𝑒 𝑚 where ρ is the density of the fluid, m is the momentum, (𝑒+𝑃) P is the pressure, . is the gradient, and e is the ⎝ 𝜌 ⎠ ∇ energy per unit volume which is expressed as: where 𝜌 is the density of the fluid, 𝑚 is the momentum, 𝑃 is the pressure, ∇. is the gradient, and 𝑒 is the energy per unit volume which is expressed as:ρu 2 e = ρε + (2) 𝜌𝑢 2 𝑒=𝜌𝜀+ (2) where u is the particle velocity, and specific internal2 energy ε is given by:

P ε = (3) ρ(γ 1) − Acoustics 2020, 2 151 where γ is a speciﬁc heat ratio and with shock speed U, the conservation of mass can be written as:

ρ (u U) = ρ (u U) (4) 1 1 − 0 0 − The subscripts 0 and 1 represent the constant states on each side of the shock. The conservation of momentum for the shock can be expressed as:

ρ (u U)2 + P = ρ (u U)2 + P (5) 1 1 − 1 0 0 − 0 The conservation of energy for the shock can be written as:

(u U)(e + P ) = (u U)(e + P ) (6) 1 − 1 1 0 − 0 0 By using the speciﬁc volume, τ = 1/ρ, Equations (4)–(6) can be reduced to the Hugoniot relation [12]: γ0τ0P0 γ1τ1P1 (P0 P )(τ0 + τ ) = − 1 1 (7) γ 1 − γ 1 2 0 − 1 − Equations (4)–(6) describe the nonlinear behavior of the acoustic mode alone. Riemann showed these equations can have an exact general solution in the form of shock, rarefaction, and contact wave across which density and pressure changes [10]. The solution would consist of a right propagating wave, a left propagating wave, and a contact wave [12]. The right (or left) wave can be a shock or a rarefaction. The intermediate region can be connected to left or right regions by following Riemann invariants and using the isentropic law [12]:

uL cL u c + = ∗ + ∗ 2 γL 1 2 γ 1 uR cR− u ∗c− (8) + = ∗ + ∗ 2 γR 1 2 γ 1 − ∗−

γL γ PLρL− = P ρ − ∗ ∗ ∗ (9) γR γ PRρR− = P ρ − ∗ ∗ ∗ where * denotes the region inside the wave or mid-region. The equations for shock or rarefaction waves can be found by eliminating U and ρ1 from Equations (4)–(6). Since the left and right regions connect to regions with the same velocity and pressure, the equations can be expressed in terms of velocity as a function of pressure [12]. Newton’s iteration scheme is used to solve the equation for the velocity and pressure of the mid-region. Bukiet, 1988, presented the following equations, relating the intermediate state of the wave to the side state [12]: Right shock:

AR(αR 1) du αRDR + 3γR 1 u = uR + − , = − (10) ∗ 1/2 dP p 3/2 (DRαR + ER) 2ρRPR(DRαR + ER)

Right rarefaction: B 2CR αR R 1 du 1 u = u + − = R , J (11) ∗ ER dP HRαR R

With: q q P 2PR γR 1 γRPR αR = P ;AR = ρ ;BR = 2γ− ;CR = ρ ;DR = γR + 1;ER = γR 1; R R p R R − HR = γRPRρR JR = (γR + 1)/2γR where R denotes the region of ﬂuid to the right or left side of mid-region. Equations (4) and (7) can be used to compute mid-region density and wave speed for the shock, while Equations (8) and (9) can be used for rarefactions. Acoustics 2020, 2 152

3.1. Head Shock With the one-dimensional model, as the intense ultrasound wave is sent into a medium, using the piston model analogy, a shock will be generated. As mentioned earlier, a nonlinear wave propagates in one direction only, assuming the direction of propagation is to the right, the right shock Equation (15) can be used to calculate the pressure jump across the shock. To demonstrate the calculation, the following parameters for an ultrasound transducer were assumed: a transducer of 1500 W with 80% efficiency, source diameter of 12.7 mm, frequency of 2 MHz, and an ideal fluid where ρ = 1.225kg/m3 and c = 343 m/s. For the given parameters, the amplitude of displacement, A, will be 1.69 10 5 m and 0 × − the displacement speed u will be 212.39 m/s. (*Calculations for ρr1 and u are shown in AppendixA). As shown in Figure3 the fluid particles in the front of the head shock are in a calm state; the particle speed is 0 m/s, and behind the shock, the particle speed (or piston) is 212.39 m/s. Equation (10) is used to calculate the pressure behind the jump. In the equation below the subscripts “1” and “r1” represent the fluid ahead of and behind the shock, respectively. Therefore, u1 and c1 are equal to 0 and 343 m/s, respectively, as the fluid in front of the head shock is undisturbed, P1, the atmospheric 3 fluid pressure, is equal to 101,325 Pa, ρ1 is equal to the 1.225 kg/m , and the specific heat ratio γ for the fluid is 1.4. The particle speed behind the head shock is ur1 is 212.39 m/s. The unknown Pr1 can be calculated using Equation (10):

q 2P1 Pr1 1 ρ P1 u = u + 1 − r1 1 hn o i1/2 (γ + 1) Pr1 + (γ 1) P1 − q 2 101325 Pr1 ∗1.225 101325 1 212.39 = 0 + − h i1/2 (1.4 + 1) Pr1 + (1.4 1) 101325 − Pr1 = 229, 029 Pa or 40, 530 P

For the propagation direction of the piston as right, the particle speed ur1 and sound speed cr1 are in the same direction and ur1 + cr1 is greater than u1 + c1(= 343 m/s). Therefore, Pr1 > P1, so Pr1 is equal to 229,029 Pa. By applying the Hugoniot relationship, the density behind the head shock ρr1 is equal to 2.16 kg/m3. Using the piston speed of particles as 212.39 m/s and the law of conservation, ** the head shock speed U1 is 490.65 m/s . Shocks always move at supersonic speed as observed from ahead of the shock, and subsonic speed as observed from behind the shock [10]. (** Calculations for ρr1 and U1 are shown in AppendixB).

3.2. Rarefaction As an acoustic wave propagates, fluid particles adjacent to the transducer (or piston) are vibrating back and forth in the direction of propagation around its original position. The reiteration of fluid particles can be modeled as the piston moving backward. In the second half of the cycle, the piston speed is the same as that it had in the forward (first half) cycle, but it becomes negative as the direction reverses and a forward propagating rarefaction wave is generated. A rarefaction is a smooth fan zone with continuously changing pressure, density, and velocity. In the equation below, the subscripts r1 and r2 represent the face and tail of rarefaction. The particle velocity at the rarefaction face is ur1 = 212.39 m/s. As the direction of piston reverses the velocity of the piston becomes negative, ur = 212.39 m/s. Using Equation (11) for the right rarefaction, u = 212.39 m/s, representing 2 − r1 Acoustics 2020, 2 153

3 the speed attained by a particle at the face of piston with Pr1 = 229, 029 Pa and ρr1 = 2.16 kg/m . Pr2, the pressure at the tail of rarefaction can be calculated as:

r  γ 1   −  γ Pr1  Pr2 2γ  2 ∗  1 ρr1  Pr1 −  ur2 = ur1 + γ 1   q − 1.4 1  P 2 1.4−  2 1.4 229029  r2 ∗ 1 ∗2.16  229029 −  212.39 = 212.39 + 1.4 1 − − Pr2 = 40, 055 Pa

Equation (9) is used to determine density at the tail of the rarefaction wave,

γR γ Pr2ρr2− = Pr1ρr1− ∗ 1.4 1.4 40055 ρr2− = 229029 2.16− ∗ ×3 ρr2 = 0.62 kg/m

For the rarefaction wave, the particle speed ur2 and sound speed cr2 are in opposite directions, and thus, Pr2 < P1. As mentioned before, the transition from the linear to the nonlinear regime corresponds to a change in propagation velocity from c to u + c . Using this and the data from the shock speed 0 u q 1.4 229029 calculations, the wave velocity at the rarefaction face is equal to ur1 + cr1 = 212.39 + ∗2.16 = 212.39 + 385.69 = 597.68 m/s, which is faster than the speed of the head shock. Similarly, the wave q 1.4 40055 speed at the tail of the rarefaction wave will be ur + cr = 212.39 + ∗ = 212.39 + 300.74 = − 2 2 − 0.62 − 88.34 m/s. With the knowledge of the state at the front and tail of the rarefaction, the intermediate states through the rarefaction can be computed [13]. To compute the pressure throughout the rarefaction, following constants k1, k2, and k3 can be assumed: dx k1 = = u + c (12) dt ∗ ∗ ur1 cr1 u c k2 = = ∗ ∗ (13) 2 − γ 1 2 − γ 1 − − γ γ k3 = Pr1ρr1− = P ρ − (14) ∗ ∗ The values of k1, k2, and k3 can be related as follows, let k1 minus two k2 gives: s P k1 2 k2 = 6 1.4 ∗ (15) − ∗ × ρ ∗ From Equation (14), P = k3 ρ1.4 (16) ∗ ∗ ∗ Substituting P* from Equation (16) in Equation (15) to obtain:

 2.5 k1 2k2 2  −   6  ρ =   (17) ∗  1.4k3 

Here k2 and k3 are constant throughout the rarefaction. Therefore, knowing the state at the front rarefaction, and by using diﬀerent values of k1, the density throughout the rarefaction can be found. Then, using Equations (17) and (9), the pressure throughout the rarefaction can be calculated as:

1.4 Pr1 ρ P = × ∗ (18) ∗ 1.4 ρr1 Acoustics 2020, 2 154

Due to the smooth change in velocity through the rarefaction, k1 has its own velocity range between the tail rarefaction velocity of 88.3m/s and the face rarefaction velocity 597.7m/s. Consequently, k2 and k3 can be calculated as:

ur1 Cr1 γ k2 = = 857k3 = P ρ − = 77944 2 − γ 1 − r1 r1 −

The rarefaction wave can be divided into two parts separated by a sound wave where u + c = c0 [10]. ∗ ∗ Let ρr and Pr represent the density and pressure at the separation boundary, respectively. Using k1 = ur + cr = 343 m/s, and Equations (17) and (18) to find the physical state at separation boundary produces:

2.5  343+2 857 2   ∗   6  3 ρr =   = 1.21 kg/m  1.4 77944  ∗

229029 1.21.4 Pr = × = 101, 161.4 Pa 2.161.4 3 Thus, the rarefaction can be divided into two zones, where Pr = 101, 161.4 Pa and ρr = 1.21 kg/m at the boundary.

3.3. Tail Shock At the end of the pulse wave, as the piston (and similarly) ﬂuid particles come to rest, the displacement speed will change from 212.39 to 0 m/s, and other forward-moving shocks will be − generated. Performing calculations similar to the head shock pressure P2, the density behind tail shock will be 99,954 Pa. Similarly, by applying the Hugoniot relation, the density ρ2 behind the tail, 3 *** shock equals 1.38 kg/m and the tail shock speed U2 is 173.27 m/s . (*** Calculations for P2, ρ2, and U2 are shown in AppendixC).

4. Decaying of N-Wave As shown in Figure4 a pulse of an ultrasonic wave consists of two shocks and rarefaction between them. The rarefaction can be divided into two parts separated by the sound wave, where u + c = c0. ∗ ∗ The velocity of the head shock is subsonic relative to the local wave speed behind it, i.e., at the face of the rarefaction, the wave speed ur1 + cr1 is faster than the head shock speed, U1. When the rarefaction meets the shock, the shock is weakened, causing the shock speed to decrease. Similarly, the tail shock velocity is supersonic relative to the local sound speed ahead of it, and the tail shock will thus overtake the rarefaction wave. The forward part of the rarefaction weakens the head shock, and the tail shock reduces by the backward part of the rarefaction [10]. The speed of the waveform at the face of the rarefaction is ur1 + cr1 = 597.7 m/s, whereas the speed of head shock is U1 = 490.8 m/s. Thus, after some time, the rarefaction will overtake the head shock causing the head shock to weaken. The rarefaction fan impinges on the head shock decaying it further. Similarly, the speed of the waveform at the tail of the rarefaction is ur2 + cr2 = 88.3 m/s, while the speed of the tail shock is U2 = 172.4 m/s. Thus, the tail shock will overtake the backward face of the rarefaction wave, as shown in Figure4. In the 1950s, extensive experimental investigations were performed at the Institute of Aero physics, the University of Toronto (which was earlier known as Institute of Aerospace Studies) on the ﬂow ﬁeld resulting from one-dimensional wave interactions. Two studies namely ‘On the One-Dimensional Overtaking of a Shock wave by Rarefaction waves’ by Glass, et al., 1959 and another study ‘On the One-Dimensional Overtaking of Rarefaction waves by a Shock wave’ by Bermner et al., 1960 are used here to study the interaction between the head shock and the forward part of rarefaction wave and the tail shock and the backward region of rarefaction, respectively [14–16]. AcousticsAcoustics2020 2019,,2 2 FOR PEER REVIEW 1554

FigureFigure 4.4. DecayingDecaying ofof N-waveN-wave [Adopted[Adopted basedbased onon aa figurefigure inin Courant,Courant, (1948)].(1948)]. 4.1. Shock Wave Overtaken by a Rarefaction Wave 4.1. Shock Wave Overtaken by a Rarefaction Wave The problem of the overtaking of a shock by a rarefaction wave has been considered by Courant and The problem of the overtaking of a shock by a rarefaction wave has been considered by Courant Friedrichs (1948) [10]. The possible wave systems that can result from such an interaction are as follows: and Friedrichs (1948) [10]. The possible wave systems that can result from such an interaction are as I.follows: A transmitted shock and a reflected rarefaction wave II.I. A transmittedA transmitted shock shock and and a reflected a reflected rarefaction compression wave wave that steepens into a shock wave III.II. A transmittedA transmitted shock rarefaction and a reflected wave and compression a reflected wave rarefaction that steepens wave into a shock wave IV.III. A transmittedA transmitted rarefaction rarefaction wave wave and and a reflectedreflected compression rarefaction wave wave that steepens into a shock wave IV. A transmitted rarefaction wave and reflected compression wave that steepens into a shock wave It is also possible to have the limiting cases where either or both the reflected or transmitted wave are MachIt is Waves.also possible to have the limiting cases where either or both the reflected or transmitted waveIn are all Mach these situations,Waves. the transmitted wave decays with interactions, its strength decreases, and the entropyIn all these change situations, across thethe transmittedtransmitted wavewave diminishes.decays with Thus,interactions, there will its strength be a region decreases, of entropy and change,the entropy i.e., achange contact across region the between transmitted the reflected wave diminishes. and transmitted Thus, waves there [ 14will,15 be]. Aa region Mach wave of entropy is the envelopechange, i.e., of the a contact wavefront, region traveling between at thethe soundreflecte speedd and propagated transmitted from waves an infinitesimal[14,15]. A Mach disturbance wave is inthe supersonic envelope speed.of the Glasswavefront, et al., 1959traveling presented at the the sound algebraic speed expressions propagated that from give an the infinitesimal final wave strengthdisturbance and in states supersonic in terms speed. of the Gl initialass et wave al., 1959 strengths presented and specific the algebraic heat ratio expressions [15]. that give the finalThe wave strength strength of aand shock states (or in rarefaction) terms of the wave initial is defined wave strengths as the ratio and of specific fluid pressure heat ratio ahead [15]. of and behindThe the strength wave. Theof a finalshock waves (or rarefaction) and its states wave resulting is defined from as the the interaction ratio of fluid in a pressure one-dimensional ahead of flowand ofbehind a perfect the gaswave. are dependentThe final waves on the and relative its strengthsstates resulting of the initialfrom interactingthe interaction shock in( Pa1 /one-P5) anddimensional rarefaction flow waves of a( Pperf5/Pect4) [gas14,15 are]. Fordependent interaction, on the overtakingrelative strengths wave (P of5/ theP4) initialcan be interacting defined as weakshock or (𝑃 strong,/𝑃) and based rarefaction on the ratio waves of the (𝑃 pressure/𝑃) [14,15]. jump For (or relativeinteraction, strength the overtaking of) across thewave overtaken (𝑃/𝑃) wavecan be(P defined1/P5) and as overtakingweak or strong, waves based(P5/ Pon4) ,the i.e., ratio(P1/ ofP5 the)/( Ppre5/ssureP4) = jumpP1/P 4(or. This relative ratio strength can also beof) denotedacross the as overtakenP14 [15]. Thewave overtaking (𝑃/𝑃) and wave overtaking in this system waves of(𝑃 interactions/𝑃), i.e., (𝑃 is/𝑃 considered)/(𝑃/𝑃)=𝑃 to be/𝑃 a weak. This shockratio can if the also ratio be denoted is less than as 𝑃 or equal [15]. The to 1, overtaking(P 1), wave and is in considered this system as of strong interactions if the ratiois considered is more 14 ≤ thanto be 1, a (weakP14 > shock1). if the ratio is less than or equal to 1, (𝑃 ≤1), and is considered as strong if the ratioFigure is more5 showsthan 1, (A),(𝑃 the>1 overtaking). of a shock by a weak rarefaction wave and the flow field resultingFigure from 5 shows this interaction (A), the overtaking can result inof (B) a shock a transmitted by a weak shock rarefaction wave with wave a reflected and the rarefaction flow field orresulting (C) a transmitted from this interaction shock with can a reflected result in wave(B) a transmitted that can be ashock shock, wave with with a contact a reflected region rarefaction between theor (C) reflected a transmitted and transmitted shock with waves. a reflected As this wave transmitted that can wave be a shock, decays with with a interactions, contact region its strengthbetween the reflected and transmitted waves. As this transmitted wave decays with interactions, its strength

Acoustics 20192020,, 2 FOR PEER REVIEW 1564 decreases, and the entropy diminishes. Glass et al., 1959 presented algebraic equations relating the pressuredecreases, ratio and across the entropy the transmitted diminishes. and Glass reflecte et al.,d wave 1959 presented(at the end algebraic of interactions) equations to relatingthe known the valuespressure of ratio pressure across theratios transmitted across the and incident reflected waveshock (at and the endovertaking of interactions) rarefaction to the wave known (before values interactions)of pressure ratios [15]. When across the overtaking incident shock rarefaction and overtaking wave is weak, rarefaction the interaction wave (before results interactions) in a reflected [15]. rarefactionWhen the overtaking wave, the quasi- rarefactionsteady wave flow isinweak, regions the (𝑃 interaction) and (𝑃 results) can be in evaluated a reflected using rarefaction expressions wave, Equationsthe quasi-steady (19) and flow (20) in [14]. regions Here,(P 2the) and reflected(P3) can compression be evaluated wave using (case expressions II, Figure Equations5C) was tactically (19) and assumed.(20) [14]. Here,Hence, the it reflectedwould not compression influence the wave termin (caseal states II, Figure and5 C)can was be neglected tactically assumed.[15]. Case Hence,III and caseit would IV mentioned not influence above the terminalcorrespond states to the and strong can be overtaking neglected [ 15rarefaction]. Case III wave and case and IV they mentioned do not pertainabove correspond to our discussion. to the strong Thus, overtaking in the case rarefactionof a weak rarefaction wave and they wave do overtaking not pertain a toshock our discussion.wave, the resultingThus, in theflow case field of or a weakterminal rarefaction state will wave consis overtakingt of a transmitted a shock shock wave, and the resultinga reflected flow rarefaction field or waveterminal (case state I, Figure will consist 5B). The of apressure transmitted jump shock across and it can a reflected be evaluated rarefaction using Equations wave (case (19) I, Figure and (20).5B). The pressure jump across it can be evaluated using Equations (19) and (20). ( ) 𝛽𝑃 𝛼+𝑃 1−𝑃 1−𝑃 s " + +2# (𝑃 ) − (𝑃 𝑃 ) −1 P ( + P ) (1+𝛼𝑃β15) α 𝑃15 (𝛼+𝑃1 P)15 𝛼𝑃 1+1P21 β β (19) − + − + 2(P45) (P15P21) 1 = 0 (19) (1 + αP15) √P15(α + P15) √αP21 + 1 − − =0 (P ) = P P P (20) (𝑃) =𝑃34𝑃21𝑃15 54 (20) γ1 γ+1 where, 𝛽= − and 𝛼= . where, β = 2γ and α = γ 1 . −

FigureFigure 5. 5. OvertakingOvertaking of of a a shock shock wave wave by by a a weak weak rarefaction rarefaction wave wave [Adopted [Adopted based based on on a a figure figure in in Glass Glass etet al al (1959)]. (1959)]. 4.2. Shock Wave Overtaking a Rarefaction Wave 4.2. Shock Wave Overtaking a Rarefaction Wave Bermner et al., 1960 presented an analytical solution for a 1-D shock wave overtaking a rarefaction wave.Bermner Analytical et al., solutions 1960 presented are presented an suchanalytical that ifsolution the initial for strength a 1-D ofshock the rarefactionwave overtaking wave and a rarefactionovertaking wave. shock Analytical wave strength solutions are known, are presented it is possible such tothat predict if the theinitial strengths strength of of the the reflected rarefaction and wavetransmitted and overtaking wave after shock the interaction wave strength as well asare the known, properties it is ofpossible the newly-formed to predict quasi-steadythe strengths regions. of the reflectedAsin and the transmitted previously wave described after case,the interaction here, the resulting as well as wave the properties pattern depends of the onnewly-formed the relative quasi-steadystrength of the regions. initial interacting shock wave and the rarefaction wave. Figure6 shows a schematic representationAs in the previously for a relatively described weak shockcase, here, overtaking the resulting a rarefaction wave wave,pattern i.e., depends(P 1on), wherethe relative (A) a 14 ≤ strengthshock of knownof the initial strength interacting overtaking shock a rarefaction wave and wave, the rarefaction (B) the flow wave. field resultingFigure 6 fromshows the a interactionschematic representationcan result in a reflectedfor a relatively shock weak wave shock with aovertaking reflected rarefaction a rarefaction or wave, (C) the i.e., reflected (𝑃 ≤1 and), where transmitted (A) a shockwaves of both known being strength rarefaction overtaking waves. Bermner a rarefactio et al.,n 1960wave, presented (B) the algebraicflow field equations resulting relating from the interactionpressure ratio can across result the in transmitteda reflected shock and reflected wave with wave a (atreflected the end rarefaction of interactions) or (C) to the knownreflected values and transmitted waves both being rarefaction waves. Bermner et al., 1960 presented algebraic equations

Acoustics 2019, 2 FOR PEER REVIEW 4

Acousticsrelating2020 the, 2 pressure ratio across the transmitted and reflected wave (at the end of interactions) to 157the known values of pressure ratios across the incident rarefaction and overtaking shock wave (before interactions) [16]. For a weak incident shock wave, (a case when it completely attenuates to a Mach of pressure ratios across the incident rarefaction and overtaking shock wave (before interactions) [16]. wave), the reflected wave is a shock wave and the quasi-steady flow in the regions (𝑃) and (𝑃) in ForFigure a weak 6 can incident be evaluated shock wave, using (a Equations case when (21) it completely and (22) [16]. attenuates. Here the to a reflected Mach wave), rarefaction the reflected wave ( ) ( ) wave(Figure is a 6C) shock was wave tactically and the assumed. quasi-steady Hence, flow it in wo theuld regions not influenceP2 and Pthe3 interminal Figure6 statescan be and evaluated can be usingneglected Equations [14]. Thus, (21) andin case (22) of [ 16a weak]. Here shock the wave reflected overtaking rarefaction a rarefaction wave (Figure wave,6C) the was resulting tactically flow assumed.field or terminal Hence, itstate would will not consist influence of a the transmitted terminal statesrarefaction and can and be a neglected reflected [ 14shock]. Thus, wave. incase The ofpressure a weak jump shock across wave them overtaking can be aevaluated rarefaction using wave, Equations the resulting (21) and flow (22). field or terminal state will consist of a transmitted rarefaction and a reflected shock wave. The pressure jump across them can be / (21) evaluated1−(𝑃 using) Equations(𝑃) (21)( and𝑃 (22).−1) 𝑃(𝛼+𝑃) (1−𝑃) + / + / 𝛽 1+𝛼(𝑃) "1+𝛼𝑃 #1/21+𝛼𝑃 1 (P )β(P )β (P 1) P (α + P ) (1 P ) − 34 45 + 45 − + 45 45 − 34 = 0 (21) p 1/2 1 + αP 1/2 =0β [1 + α(P45)] 45 [1 + αP34] and and 𝑃 P21 (22) (𝑃 ) =(P34) = (22) P45P51 𝑃𝑃 γ 1 γ+1 − where,where,β 𝛽== 2γ and andα 𝛼== γ 1 . . −

Figure 6. Overtaking of a rarefaction wave by a weak shock wave [Adopted based on a figure in Glass Figure 6. Overtaking of a rarefaction wave by a weak shock wave [Adopted based on a figure in Glass et al (1959)]. et al (1959)]. Based on the above discussion, for a head shock overtaken by the forward part of rarefaction P Based on the above discussion, for a head shock overtaken by the forward part of rarefaction 1 1 , the resulting flow field can be evaluated using Equations (19) and (20): Pr ≤≤1, the resulting flow field can be evaluated using Equations (19) and (20): uv   tβ P1 α + P1  1 P1 1 Pr12  !β !β Pr1 Pr1  Pr1 P1  Pr Pr12 P1 𝑃  𝑃 − + 𝑃 −  + 2 𝑃 1 = 0 P1  q q  1 𝛽+ α 𝛼 + P1 ⎡ P1 1− Pr12  1−Pr1 − ⎤ P1 Pr1 − 𝑃Pr1  𝑃 α + 𝑃α + 1 𝑃 Pr1 ⎢ Pr1 P1 + ⎥ 𝑃 ⎢ 𝑃 𝑃 𝑃 ⎥ and 1 + 𝛼 𝛼 + 𝛼 +1 𝑃 ⎣ 𝑃P 𝑃P P P 𝑃 ⎦ r13 = r12 1 r1 Pr P1 Pr1 Pr Using the above expression,𝑃 Pr12 = 101,𝑃 269𝑃 Pa and Pr13 = 101, 269 Pa. +2 − −1=0 𝑃 𝑃 𝑃 and

Acoustics 2019, 2 FOR PEER REVIEW 4

𝑃 𝑃 𝑃 𝑃 = 𝑃 𝑃 𝑃 𝑃 Using the above expression, 𝑃 = 101,269 Pa and 𝑃 = 101,269 Pa. Similarly, using Equations (21) and (22), evaluating the flow field resulting from the interaction of the tail shock wave overtaking the backward part of the rarefaction wave can be shown by: 𝑃 𝑃 𝑃 1− 𝑃 𝑃 𝑃 −1 + 𝛽 𝑃 / 1 + 𝛼 Acoustics 2020, 2 𝑃 158 𝑃 𝑃 / 𝑃 𝛼 + 1 − Similarly, using Equations (21)𝑃 and (22), evaluating𝑃 the ﬂow ﬁeld𝑃 resulting from the interaction of + / the tail shock wave overtaking the backward𝑃 part of the rarefaction𝑃 wave can be shown by: 1+𝛼 1 + 𝛼 𝑃 𝑃 β β  1/2 Pr23 P2 P2 P2 + P2 Pr23 1 P P P 1  P α P  1 P and − 2 r2 + r2 − +  r2 r2  − 2 p h i1/2  P  h i1/2 P2  2  Pr23 β 1 + α 1 + α P 1 + α Pr2 r2 P2 𝑃 𝑃 𝑃 𝑃 and = P P P P 𝑃 r23𝑃 =𝑃r22𝑃 r r2 Using the above expression, 𝑃 = 100,527P2 PaPr 𝑎𝑛𝑑Pr2 P 𝑃 2 = 100,527 Pa. UsingFigure the 7 shows above the expression, resultingP pressurer12 = 100, across 527 Pa statesand P(Pr13r12=–P100,r23) and 527 transmitted Pa. shocks (P1 and P2). AfterFigure the interaction,7 shows the the resulting state behind pressure the head across sh statesock and (P r12ahead–Pr23 of) andforwarding transmitted rarefaction shocks have (P1 and the Psame2). After pressure. the interaction, Thus, the theforward state behindrarefaction the headhas attenuated shock and the ahead head of shock forwarding to a Mach rarefaction wave, just have as theits sametail reaches pressure. the Thus,head shock. the forward Similarly, rarefaction in the hascase attenuated of the tail theshock head overtaking shock to athe Mach backward wave, justrarefaction, as its tail states reaches (Pr the22–P headr23) and shock. (P2) Similarly, are similar in thepressure. case of Thus, the tail the shock incident overtaking tail shock the backward has been rarefaction,attenuated to states a Mach (Pr22 wave,–Pr23) just and as (P it2) reaches are similar the head pressure. of the Thus, backward the incident rarefaction tail shock[16]. Thus, has beenafter attenuatedthe interaction to a Mach pressure wave, justjump as it reachesacross thehead head shock of the backwardbecomes rarefactionnearly 1 (101,269/101,325 [16]. Thus, after the = interaction0.99~1) which pressure before jump the acrossinteraction head shockwas 2.26 becomes (229,029/101.325) nearly 1 ( 101,, 269similarly/101, 325 pressure= 0.99 jump1) whichacross ∼ beforetail shock the interaction becomes wasunity2.26 ((99,954/100,527229, 029/101.325) =, similarly 0.9943~1) pressure which jump before across the tail interaction shock becomes was unity2.46 (99,954/40,055)( 99, 954/100, 527. = 0.9943 1) which before the interaction was 2.46 (99, 954/40, 055). ∼

Figure 7. Flow ﬁeld resulting from interactions. Figure 7. Flow field resulting from interactions. Both the shocks continue to decay during these interactions with the rarefaction waves, and their strengthBoth (the the pressure shocks continue jumps across to decay it) diminishes.during these Thus, interactions at some with distance the rarefaction from the source,waves, and both their the shockstrength at head(the pressure and tail ofjumps the N-waveacross it) get diminishes. attenuated Thus, to Mach at some waves. distance As entropyfrom the across source, the both shock the continuesshock at head to decay and due tail to of interactions the N-wave with get theattenuated rarefaction, to Mach there iswaves. a reduction As entropy in the waveformacross the speedshock and amplitude. This reduction is due to the dissipation of ultrasound energy. The conservation of energy implies this dissipated energy is transferred from one form to another. In this case, it becomes heat. A similar set of calculations is performed for transducers of power levels of 50, 150, 500, and 1500 W and for frequencies of 20, 500 kHz and 2 MHz, as shown in Tables1 and2 and results are presented in Figure8. Acoustics 2019, 2 FOR PEER REVIEW 4 continues to decay due to interactions with the rarefaction, there is a reduction in the waveform speed and amplitude. This reduction is due to the dissipation of ultrasound energy. The conservation of energy implies this dissipated energy is transferred from one form to another. In this case, it becomes heat. A similar set of calculations is performed for transducers of power levels of 50, 150, 500, and 1500 W and for frequencies of 20,500 kHz and 2 MHz, as shown in Tables 1 and 2 and results are presented in Figure 8.

Table 1. Vibration speed of fluid particle in acoustic field of varying power level and frequency. Acoustics 2020, 2 159 Power Level, W Frequency, 𝐟 Intensity, I Amplitude, A Particle Speed, V (watt) (kHz) (W/m2) (10−6 m) (m/s) Table 1.1500Vibration speed of ﬂuid20 particle in9,477,702.96 acoustic ﬁeld of varying1691 power level and212.39 frequency. 1500 500 9,477,702.96 67.6 212.39 Power Level, W Frequency, f Intensity, I Amplitude, A Particle Speed, V 1500 2000 9,477,702.962 16.9 6 212.39 (watt) (kHz) (W/m ) (10− m) (m/s) 500 20 3,152,941.98 976.3 122.61 1500 20 9,477,702.96 1691 212.39 500 500 3,152,941.98 39.1 122.61 1500 500 9,477,702.96 67.6 212.39 1500500 20002000 9,477,702.963,152,941.98 9.76 16.9 122.61 212.39 500150 2020 3,152,941.9894,772.6 534.7 976.3 67.22 122.61 500150 500500 3,152,941.9894,772.6 21.4 39.1 67.2 122.61 500150 20002000 3,152,941.9894,772.6 5.34 9.76 67.22 122.61 150 20 94,772.6 534.7 67.22 50 20 315,924.2 308.7 38.78 150 500 94,772.6 21.4 67.2 15050 2000500 94,772.6315,924.2 12.3 5.34 38.78 67.22 5050 202000 315,924.2315,924.2 3.08 308.7 38.78 38.78 50 500 315,924.2 12.3 38.78 50 2000Table 2. Variation 315,924.2 in resulting flow field. 3.08 38.78

Head Particle Table 2. Variation in resulting ﬂow ﬁeld. P1 Pr1 Pr Pr2 P2 Pr12 = Pr13 Pr22 = Pr23 Shock, Speed, V Head Particle Speed,(Pa) P (Pa) P (Pa) P (Pa)P P(Pa) P =(Pa)P P =(Pa)P U1 (m/s) 1 r1 r r2 2 r12 r13 r22 r23 Shock, U V(m/s) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) 1 (m(m/s)/s) 212.39212.39 101,325 101,325229,029 229,029 101,161 101,161 40,055 40,055 99,954 99,954 101,269101,269 100,527100,527 490.8 490.8 122.61122.61 101,325 101,325164,678 164,678 102,005 102,005 60,027 60,027 101,058101,058 102,004102,004 101,152101,152 421.7 421.7 67.22 67.22 101,325 101,325132,853 132,853 102,220 102,220 76,414 76,414 101,281 101,281 102,228102,228 101,294101,294 383 383 38.78 101,325 118,629 102,258 86,225 101,317 102,260 101,319 364 38.78 101,325 118,629 102,258 86,225 101,317 102,260 101,319 364

Figure 8. ParticleParticle speed vs. head head shock. shock. 5. Discussions of Results

In an ideal fluid, the displacement amplitude of an intense ultrasonic wave decreases with an increase in frequency, as amplitude is inversely proportional to the intensity of the disturbance or vibration. However, the vibration speed remains constant for a given power level for different frequencies as shown in Table1. The speed of the head shock and thereby the strength of the shock depends on the disturbance speed of particle as shown in Figure8, where the higher the vibration speed, the stronger the discontinuity. For example, for a particle speed of 212.39 m/s, the strength of the head shock is Pr1/P1 which equals 2.26, whereas for a particle speed of 122.61 m/s, Pr1/P1 is 1.62. Similarly, it can be seen from the pressure distribution in the resulting flow field (before the interaction) that the strength of the wave will also increase with an increase in the power level. After the interaction Acoustics 2019, 2 FOR PEER REVIEW 4

5. Discussions of Results In an ideal fluid, the displacement amplitude of an intense ultrasonic wave decreases with an increase in frequency, as amplitude is inversely proportional to the intensity of the disturbance or vibration. However, the vibration speed remains constant for a given power level for different frequencies as shown in Table 1. The speed of the head shock and thereby the strength of the shock depends on the disturbance speed of particle as shown in Figure 8, where the higher the vibration speed, the stronger the discontinuity. For example, for a particle speed of 212.39 m/s, the strength of

Acousticsthe head2020 ,shock2 is 𝑃/𝑃 which equals 2.26, whereas for a particle speed of 122.61 m/s, 𝑃/𝑃 is160 1.62. Similarly, it can be seen from the pressure distribution in the resulting flow field (before the interaction) that the strength of the wave will also increase with an increase in the power level. After of the rarefaction wave and the shock, the shock strength (the pressure jumps across it) diminishes. the interaction of the rarefaction wave and the shock, the shock strength (the pressure jumps across Forit)instance, diminishes. for For particle instance, speed for particle of 212.39 speedm /ofs the212.39 pressure m/s the jump pressure across jump the across head the shock head was shock2.26 beforewas the2.26 interaction before the and interaction reducesto and near reduces atmospheric to near after atmospheric the interaction. after Thisthe intersectioninteraction. This causes dissipationintersection of ultrasoundcauses dissipation energy of into ultrasound heat. Figure energy9 summarizes into heat. Figure this observation 9 summarizes graphically. this observation Figure 9 showsgraphically. the initial Figure and ﬁnal9 shows conditions the initial of and the N-Wave,final conditions where (a)of the in a N-Wave, pulse of intensewhere (a) ultrasonic in a pulse waves, of twointense shocks ultrasonic will form waves, with a rarefactiontwo shocks wave will inform between with duea rarefaction to the nonlinear wave in wave between motion due of theto the wave, (b)nonlinear the rarefaction wave motion can be dividedof the wave, into (b) two the parts rarefaction separated can by be the divided sound into wave, twou parts+ c separated= c0, in which, by ∗ ∗ forwardthe sound part wave, of rarefaction 𝑢∗ +𝑐∗ =𝑐 wave, in traverses which, forward the head part shock, of rarefaction the tail shock wave traverses traverses the thebackward head shock, part, andthe (c) tail the shock interaction traverses results the backward in decaying part, of and shocks (c) th toe ainteraction Mach wave. results in decaying of shocks to a Mach wave.

Figure 9. Initial and final conditions of N-wave. Figure 9. Initial and final conditions of N-wave. 6. Summary and Conclusions 6. Summary and Conclusions In linear acoustics, small disturbances always propagate at a constant speed relative to the medium In linear acoustics, small disturbances always propagate at a constant speed relative to the and do not influence the fluid properties or behavior. This situation changes dramatically if the wave medium and do not influence the fluid properties or behavior. This situation changes dramatically if intensity is increased. Familiar laws like the principle of superposition do not apply for intense waves. the wave intensity is increased. Familiar laws like the principle of superposition do not apply for The local sound speed changes from c0 to cu + u. Thus, due to acoustic nonlinearity, propagation intense waves. The local sound speed changes from 𝑐 to 𝑐 +𝑢. Thus, due to acoustic nonlinearity, velocitiespropagation in di velocitiesfferent regions in different of a waveform regions of a are waveform different, are and, different, shocks and, (discontinuities) shocks (discontinuities) are formed withinare formed the waveform. within the waveform. InIn this this research, research, the the one-dimensional one-dimensional motion of of an an ideal ideal fluid fluid is is used used to to study study the the nonlinear nonlinear propagationpropagation of of intense intense acoustic acoustic waves. waves. With With a a pulse pulse of of high-intensity high-intensity waves,waves, twotwo shocks are formed at theat boundaries the boundaries and aand rarefaction a rarefaction zone zone of varying of varyin velocitiesg velocities is formed is formed between between them. them. The The strength strength of the shock will depend on the power level of the source, the higher the power level stronger the discontinuity.

The rarefaction can be separated into two zones separated by the sound wave, cu + u = c0. The leading edge of the rarefaction zone is traveling faster than that of the shock at the front, while the second shock at the back is travelling at a faster speed than that of the trailing edge of the rarefaction zone. The leading edge of rarefaction will overtake the shock at the front, while the tail shock overtakes the trailing edge of the rarefaction wave. The pressure jump across the shocks will continue to decay during these interactions. This results in the dissipation of the shock’s energy. Since energy is conserved, it is transferred from wave energy to heat. This research helps to understand the eﬀect of the nonlinearity of intense ultrasonic waves and can be used to model the processes involving the propagation of ultrasonic waves. Acoustics 2020, 2 161

7. Suggestions for Future Work In this research, an attempt is made to understand the formation of shocks within intense waves and the decaying of acoustic waves due to sound-sound interactions. However, in this study, the medium of propagation is an ideal fluid. It is proposed to extend the scope of this research to include the propagation of intense acoustic waves in non-ideal media, such as water. This will help to better understand the interactions of the wave within fluids that can be used to study the distribution of high-intensity acoustic wave in system such as sonochemical reactors and the effects of parameters such as frequency, acoustic intensity, etc. In a pulse cycle, the shock will not appear immediately adjacent to the source but will form at some distance from it, and the shock will continue to decay along the direction of propagation as the interaction with the rarefaction occurs. Determining the distance from the source at which the shock appears and until when its decays will be essential to improving the applications of intense ultrasound.

Author Contributions: Conceptualization, J.A.K.; formal analysis, Z.Z.; investigation, J.A.K., R.W.M., and J.N.M.; methodology, B.G.B. and J.N.M.; supervision, B.G.B. and J.N.M.; validation, Z.Z., B.G.B., and J.N.M.; writing—original draft, J.A.K.; writing—review and editing, Z.Z., R.W.M., and J.N.M. All authors have read and agreed to the published version of the manuscript. Funding: This research was sponsored by the US National Science Foundation Award #1634857 entitled “Remediation of Contaminated Sediments with Ultrasound and Ozone Nano-bubbles.” Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A The following equation can be used to calculate the maximum displacement.

s s 3 2 2I 2 (8 1500)/(3.14 25 (12.7 10− ) 5 A = = × × × × × = 1.69 10− m (ρc)ω2 (1.225 343) 125600002 × × ×

ω = 2 π f = 2 3.14 2 106 = 12560000 Hz × × × × × 5 u = 2 A f π = 1.69 10− 12560000 212.39 m/s × × × × × ≈ Using Equations (4) and (7) to calculate the shock speed and density behind the shock.

Appendix B Using Hugoniot Equation (7):

1 1 1.4 ( ) 101325 1.4 ρ 229029 × 1.225 × × r1 × 1.4 1 1.4 1 − − − (101325 229029) ( 1 )+ 1 − 1.225 ρr1 = 2

3 ρr1 = 2.16 kg/m With the conservation of mass, Equation (4):

2.16(212.39 U ) = 1.225(0 U ) − 1 − 1

U1 = 490.65 m/s

Appendix C Equation (15) is used to calculate the pressure jump across the tail shock. As shown in Figure4, the subscripts “r2” and “2” represent the ﬂuid ahead of and behind the shock, respectively. Where u2 Acoustics 2020, 2 162

is the particle velocity behind the tail shock, which is 0 m/s, ur is 212.39 m/s, P is 40, 055 Pa, ρr is 2 − r1 2 0.62 kg/m3, and γ for normal air equals 1.4. Substituting these values into Equation (15), the unknown P2 can be calculated: q 2Pr2 P2 1 ρr Pr2 u = u + 2 − 2 r2 hn o i1/2 P2 (γr + 1) + (γ 1) Pr2 − q 2 40055 P2 ×0.62 40055 1 0 = 212.39 + − − h P i1/2 (1.4 + 1) 2 + (1.4 1) 40055 − P2 = 99, 954 or 13, 818 Pa

For the propagation direction, right, the particle speed ur2 and sound speed cr2 are in opposite direction and ur + cr is smaller than c . Therefore, Pr < P , and P has the value 99,954 Pa. − 2 2 2 2 2 2 Equations (7) and (10) can be used to calculate the shock speed and density behind the shock. Using the Hugoniot expression as in Equation (10) to calculate the density behind the shock gives.

1 1 1.4 ( ) 40055 1.4 ρ 99954 × 0.62 × × 2 × 1.4 1 1.4 1 − − − (40055 99954) ( 1 )+ 1 − 0.62 ρ2 = 2

3 ρ2 = 1.38 kg/m With the conservation of mass, Equation (7) shock speed can be ﬁnd:

1.38(0 U ) = 0.62( 212.39 U ) − 1 − − 2

U2 = 173.27 m/s

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