Applied Acoustics 104 (2016) 85–100

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Applied Acoustics

journal homepage: www.elsevier .com/locate /apacoust

Modeling of offshore pile driving noise using a semi-analyticalvariational formulation

http://dx.doi.org/10.1016/j.apacoust.2015.11.0020003-682X/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Room 820, Building A, Institute of Vibration, Shockand Noise, 800 Dongchuan Road, 200240 Shanghai, China. Tel.: +86 (0)2134206332x820.

E-mail address: wkjing@sjtu.edu.cn (W. Jiang).1 Present address: Institute of Vibration, Shock and Noise, State Key Laboratory of

Mechanical System and Vibration, Shanghai Jiao Tong University, 800 DongchuanRoad, 200240 Shanghai, China.

Qingpeng Deng a,1, Weikang Jiang a,⇑, Mingyi Tan b, Jing Tang Xing b

a Institute of Vibration, Shock and Noise, State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, 200240 Shanghai, Chinab Fluid Structure Interactions Group, FEE, University of Southampton, Southampton SO16 7QF, United Kingdom

a r t i c l e i n f o

Article history:Received 25 March 2015Received in revised form 5 September 2015Accepted 2 November 2015

Keywords:Structural–acoustic modelingUnderwater noisePile drivingCylindrical shell

a b s t r a c t

Underwater noise radiated from offshore pile driving got much attention in recent years due to its threatto the marine environment. This study develops a three-dimensional semi-analytical method, in whichthe pile is modeled as an elastic thin cylindrical shell, to predict vibration and underwater acousticradiation caused by hammer impact. The cylindrical shell, subject to the Reissner–Naghdi’s thin shell the-ory, is decomposed uniformly into shell segments whose motion is governed by a variational equation.The sound pressures in both exterior and interior fluid fields are expanded as analytical functions in fre-quency domain. The soil is modeled as uncoupled springs and dashpots distributed in three directions.The sound propagation characteristics are investigated based on the dispersion curves. The case studyof a model subject to a non-axisymmetric force demonstrates that the radiated sound pressure hasdependence on circumferential angle. The case study including an anvil shows that the presence ofthe anvil tends to lower the frequencies and the amplitudes of the peaks of sound pressure spectrum.A comparison to the measured data shows that the model is capable of predicting the pile driving noisequantitatively. This mechanical model can be used to predict underwater noise of piling and explorepotential noise reduction measures to protect marine animals.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

There is a rapid increase of offshore constructions over theworld in the recent years, such as wind turbine installations,petroleum and gas platforms, artificial islands and oversea bridges.These constructions produce severe underwater noise pollutionwhich has raised serious concerns from environmental protectionorganizations and academic world. Pile-driving noise is generallyconsidered as the most severe underwater noise which wouldharm or even kill marine animals inhabiting around piling sites,such as fishes, dolphins and whales [1–3].

It is necessary to establish an accurate and realistic mechanicalmodel to understand the mechanism and propagation characteris-tics of the underwater noise radiated from pile driving. In order topredict the sound radiation accurately, the complex fluid–structure

interactions between the pile wall and its surrounding fluid mustbe taken into consideration. The compressibility of the fluid altersthe effective stiffness of the system and the fluid exerts inertialloading on the shell, which results in significant changing of theresonant frequencies. The fluid-structural interaction problems ofcircular cylindrical shells partly or completely submerged and sub-jected to either external or internal fluid have caused interest toscientists. Green function and Fourier integral technique were usedby Stepanishen [4] to evaluate the radiated loading and radiatedpower from a non-uniform harmonically vibrating surface on aninfinite cylinder which was in contact with a fluid medium. Theeffect of fluid loading on the vibration and sound radiation of finitecylindrical shells has been studied by the authors [5,6] who usedin-vacuo shell modes to present the structural vibration fieldand, in order to compute the sound field, extended the finite shellsto infinity at both ends by rigid co-axial baffles. The low-frequencystructural and acoustic responses of a fluid-loaded cylindrical shellwith structural discontinuities, corresponding to ring stiffeners andbulkheads, were investigated by Caresta and Kessissoglou [7]. Thework was then extended to consider the sound radiation from amore detailed submarine structure in which the cylindrical shellis closed by truncated conical shells and circular plates [8]. Kwaket al. [9] investigated the natural vibration characteristics of a hung

Nomenclature

cx; ch; cr equivalent viscous soil damping coefficients along thethree coordinate directions

Esoil elastic modulus of soilI the total number of shell segmentsIe the number of the shell segments in contact with

exterior fluidj imaginary unit: j ¼

ffiffiffiffiffiffiffi�1

pkx; kh; kr equivalent soil spring coefficients along the three

coordinate directionsm orders of axial displacement functionsM the highest order of the axial displacement functionsn circumferential orders of displacement functions and

pressure modesN the highest order of the circumferential displacement

functionsp index of the wavenumbers of the sound pressurexi minimal vertical coordinate of fluid–structure interface

on the ith submerged shell segmentxiþ1 maximal vertical coordinate of fluid–structure

interface on the ith submerged shell segment

xs1 minimal vertical coordinate of shell-soil interface oneach shell segment beneath the seabed

xs2 maximal vertical coordinate of shell-soil interface oneach shell segment beneath the seabed

lsoil Poisson ratio of soil materialqsoil density of soilf the ratio of the impedance of the seabed to characteris-

tic impedance of the fluidCe;C i;Csoil coupling matrixes introduced by exterior fluid, interior

fluid and soil, respectivelyF(x) generalized force vectorK main structural stiffness matrixKk stiffness matrix incorporated by interface forces and

momentKj stiffness matrix incorporated by least-squares

weighted parametersM structural mass matrixq(x) generalized coordinate vectord( ) the variation of ( )( )T the transpose of ( )

86 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

clamped-free cylindrical shell partially submerged in fluid basedon the added virtual mass approach. Detailed discussions of thedynamics of coupled fluid-shells can be found in [10–13].

The sound radiation of cylindrical pile has been getting increas-ing attention in recent years. The Finite Element Method (FEM) is acommon method for establishing pile-driving models. Axisymmet-ric FEM models (two-dimensional models) are frequently usedbecause they are computationally efficient. Axisymmetric FEMmodels were established to analyze underwater noise of pile driv-ing, in which perfectly matched boundary conditions for water andsediment were applied to truncate the domain and emulate theSommerfeld radiation condition [14,15]. The discretization of thesound field at long distances by FEM models is impractical becausethe total number of unknowns of the system is limited due to com-putational reasons. It is common to use a hybrid scheme consistingof a discretized FEM model for pile and its surroundings combinedwith an approximate propagation model for long ranges. A combi-nation of a FEM model and a parabolic wave equation (PE) modelwas used by Reinhall and Dahl [16]. A similar combination wasalso established by Kim et al. [17] who used a modified versionof the Monterey-Miami Parabolic Equation (MMPE) to predict longrange noise propagation. Lippert et al. [18] combined a FEM modelwith a head wave model to forecast the noise of pile driving. Intheir subsequent publication the significance of soil parameteruncertainties for the underwater noise is investigated based onthe wave number integration scheme combined with an axis-symmetric FE model [19]. The model of Fricke and Rolfes [20]included a FEM model for the computation of the vibration andthe radiated sound of the pile, a parabolic model for the long-range propagation and an analytical model for hammer force esti-mate. Semi-analytical models are also used in predicting offshorepile driving noise. A model of a semi-infinite tubular pile was pre-sented by [21] to predict the peak pressure of the radiated soundpulse by linking with both depth-independent vibration anddepth-dependent vibration. The model was then extended to com-pute the radiated sound of a pile with finite length in combinationwith the Ray theory which deals with the sound propagation inwater and soil in a simple manner [22]. Another semi-analyticalmodel was established by Tsouvalas and Metrikine [23] whoexpressed the shell displacements as the summation of the

in-vacuo shell modes and expressed the sound pressure in fluidfield as the summation of a set of analytical functions.

In this study a linear semi-analytical model is developed to pre-dict pile-driving noise based on a modified variational methodol-ogy, in which the shell is divided into several sub-shells alongthe axial direction and the effect of water and soil is taken intoaccount as virtual work acting on the sub-shells. The mechanicalmodel presented in the paper is similar to the previous one intro-duced by Tsouvalas and Metrikine [23]. What the authors do in thispaper is to propose a new approach to solve the responses of themodel. In addition, we expand upon sound propagation character-istics, pressure distributions and reveal the influence of the non-axisymmetric impact force and the anvil on the sound radiation.Compared to the approach used in [23], the effort to seek in-vacuo shell modes is avoided, which makes it capable of dealingwith piles with more complex boundary conditions in which, forexample, the pile anvil, cushions and the pile footing (usually aconical shell) are connected to the pile ends.

The paper is arranged as follows. In the next section, thedescription and basic assumptions of the model are given. In Sec-tion 3, the basic theory of the methodology is introduced and thegoverning equations of the motion of the pile are derived. Section 4gives a description of the sound propagation characteristics basedon the dispersion relation curves of the pressure modes, followedby a numerical case with an axisymmetric hammer force in Sec-tion 5 and a numerical case with a non-axisymmetric hammerforce in Section 6. The influence of the pile anvil on the sound radi-ation is investigated in Section 7. Section 8 gives a comparisonbetween modeling values and tested data, before the conclusionsare summarized in Section 8.

2. Model description

The schematic diagram of the pile-driving model is shown inFig. 1. The model assumptions are similar with those introducedin [23]. The assumptions are listed as following:

(1) The pile is modeled as a thin elastic cylindrical shell withfinite length and constant thickness. The motion of the shellis subject to Reissner–Naghdi’s thin shell theory [24].

Fig. 1. Pile-driving model.

Q. Deng et al. / Applied Acoustics 104 (2016) 85–100 87

(2) The seawater inside and outside the pile is assumed to becompressible and inviscid fluid with constant sound velocityand density. There is no flow in the fluid field.

(3) The influence of the air on the vibration of the pile isneglected.

(4) The sea surface (fluid-air interface) is assumed to be apressure release boundary on which the sound pressurevanishes.

(5) The shell and the fluid field are required to satisfy the nor-mal velocity continuity condition on the shell-fluid couplinginterface.

(6) The seabed is assumed to be either an acoustic rigidboundary or an acoustic impedance boundary.

(7) The soil surrounding the pile is modeled as uncoupleddistributed springs and dashpots in three directions.

(8) The impact force of the hydraulic hammer is modeled asdistributed force acting on the circumference of pile top.

The constants E;l;g;qs; L;h and R correspond to the modulus ofelasticity, Poisson’s ratio, structural loss factor, density, length,thickness and radius of the mid-surface of the cylindrical shell,respectively. c and qf are the sound velocity and density of the sea-water, respectively. The pile is filled by interior fluid at xis < x < xband surrounded by exterior fluid at xes < x < xb.

3. Basic theory

A linear 3D semi-analytical mechanical model is established, inwhich the circular cylindrical tubular pile is divided into a numberof shell segments along the symmetrical axis. A modified variationmethodology is used to derive the mathematic equations govern-ing the motion of the pile. The displacements of the shell segmentsare expressed as summation of a set of admissible displacementfunctions. The sound pressure in both exterior fluid domain andinterior fluid domain is expressed as the summation of a set of ana-lytical functions weighted by the generalized coordinates of theshell displacements. The soil is modeled as uncoupled distributedsprings and dashpots in three directions. The coupling effects offluid and soil on the pile are taken into consideration by incorpo-rating corresponding virtual work into the variation equation.Linear equations in terms of generalized displacement coordinatescan eventually obtained by seeking the stationary statement of thetotal functional.

This methodology is of advantage in several aspects. Firstly,compared to the FEM models, the discretization of the fluid

volume, the approximate treatment on the truncated artificialboundaries and approximate modeling for sound propagation infar acoustic field are altogether avoided. Secondly, a semi-analytical model is generally more computationally efficient com-pared to a non-axisymmetric three-dimensional FEM model forhigh frequency response and for longer ranges, because a largeamount of fluid elements require large storage space and longcomputational time. Thirdly, the analytical expression of the soundpressure makes it possible to examine the influence of someparameters on the sound radiation in a qualitative manner. Addi-tionally, compared to the method using in-vacuo shell modes toexpress structural displacements, the effort to seek the modes isavoided and the variational methodology leads to a solution thatis generally easier to obtain than by solving the differential govern-ing equations directly. At last, this methodology is applicable topiles with more complicated boundary conditions or attachments,such as the pile anvil, cushions and the pile footing. However, thepresented model fails to model the pressure wave propagating inthe bottom or layered bottoms due to the simplification treatmenton the soil, and to evaluate the effects of the pressure wave passingthrough the water–soil interface due to the rigid seabed orimpedance seabed assumption.

3.1. Shell decomposition

A structure decomposition method with high-order of accuracyand small requirement of computational effort [25,26] is employedhere to obtain the governing equations of shell vibration, in whichthe cylindrical shell is divided into several shell segments along itsaxisymmetric axis. As shown in Fig. 2, the pile is divided equallyinto I sub-shells with Ie sub-shells coupled with exterior fluid andIs coupled with soil. The method involves seeking the solution ofthe equation given in Eq. (1), with subscript i ¼ 1;2; . . . ; I denotingthe sequence numbers of the shell segments. Pkj is the interfacepotential defined on the common boundary of two adjacent shellsegments, which incorporates the interface continuity conditionsinto equation. The geometrical boundaries on the two pile endsare treated as special interfaces [26]. This incorporation of the termdPkj provides a great flexibility for the selection of admissible dis-placement functions, since the interface continuity conditions andgeometrical boundary conditions are no longer imposed on the dis-placement functions, as their eventual satisfaction is implied in thevariation statement. The kinetic energy Ti and strain energy Ui areall formulated based on Reissner–Naghdi’s thin shell theory [24]and their expressions are given by Eqs. (3) and (4), where

I segmentsIe

Is

Fluid

Soil

r

x

l

wi

uivi

wi+1

ui+1vi+1

Fig. 2. Pile decomposition.

88 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

K ¼ Eh=ð1� l2Þ and D ¼ Eh3=½12 1� l2

� �� are the membrane stiff-ness and bending stiffness of the shell, respectively. The modifiedterm Pkj is referred to Qu’s work [25]. The term dWi is the virtualwork of external force and is given in Eq. (2), where dui; dv i; dwi

are corresponding virtual displacements of the ith shell segmentin x; h and r direction, respectively. f ui ; f v i

; f wiare the distributed

forces acting on the ith shell in the three directions, respectively.The impact force of the hydraulic hammer, the forces of springsand dashpots and the sound pressure of the fluid are all consideredas external forces. Therefore, the effects of the fluid and soil on thepile are all included in the equation in the form of virtual work.

In Sections 5 and 6, the pile head is subject to free boundarycondition, with the axial stress, circumferential and lateral shearstresses, and the bending moment being zero. The axial stress‘equals zero’ in a sense that no geometrical constraint is appliedin axial direction, but the axial stress does exist as the hammerimparts force to the pile head. Strictly speaking, the pile toe is acomplicated boundary connected to springs and dashpots. How-ever, it can be treated as a free boundary here because the effectsof the springs and dashpots are included in the form of externalvirtual work. In Section 7, the geometrical boundary condition onthe pile head are a little different due to the presence of the anvil,which will be discussed further later.Z t1

t0

XI

i¼1

ðdTi � dUi þ dWiÞdt þZ t1

t0

Xi;iþ1

dPkjdt ¼ 0 ð1Þ

dWi ¼Z Z

Si

ðf uidui þ f v idv i þ f wi

dwiÞRdxdh ð2Þ

Ti ¼ 12

Z ZSi

qhðu2i þ v2

i þw2i ÞRdxdh ð3Þ

Ui ¼ K2

Z ZSi

@ui

@x

� �2

þ 2lR

@ui

@x@v i

@hþwi

� �þ 1R2

@v i

@hþwi

� �"

þ1� l2

@v i

@xþ 1R@ui

@h

� �2#Rdxdhþ D

2

Z ZSi

@2wi

@x2

!224

�2lR2

@2wi

@x2@v i

@h� @2wi

@h2

!þ 1R4

@v i

@h� @2wi

@h2

!2

þ1� l2R2

@v i

@x� 2

@2wi

@x@h

!235Rdxdh ð4Þ

3.2. Admissible displacement functions

The displacement components ui;v i;wi can be expanded interms of admissible displacement functions weighted bygeneralized coordinates. Due to the incorporation of the interfacepotentials, the admissible displacement functions of each shellsegment are not constrained to satisfy any continuity conditionsor geometrical boundary conditions. They are only required to belinearly independent, complete and differentiable, which createsconsiderable flexibility in the selection of displacement functions.Compared with the analytical method, the effort to seek the in-vacuo natural modes is avoided, which significantly simplifiesthe solution process.

Good convergence has been observed when following fourkinds of functions are adopted as axial displacement functions[26]: (a) Chebyshev orthogonal polynomials of the first kind;(b) Chebyshev orthogonal polynomials of the second kind; (c)Legendre orthogonal polynomials of the first kind; (d) Hermiteorthogonal polynomials [27]. In this model, Fourier series andChebyshev orthogonal polynomials of the first kind are employedto expand the displacement variables in circumferential and axialdirections, respectively. The displacement components of eachsub-shell can be given as Eqs. (5)–(7). TmðxÞ is the mth orderChebyshev polynomial. M is the highest order of the Chebyshevpolynomials taken in the computation. U �x; hð Þ;V �x; hð Þ andW �x; hð Þ are the displacement function vectors in the threedirections. ui xð Þ;v i xð Þ and wiðxÞ are corresponding generalizedcoordinate vectors, with x being the radian frequency and jbeing the imaginary unit. It should be noted that the Chebyshevpolynomials are complete and orthogonal series defined on�x 2 ½�1;1� interval while the actual axial coordinate in eachsub-shell is defined on x 2 0; l½ �. Therefore, the coordinate trans-formations in Eq. (8) are introduced.

ui x; h; tð Þ ¼X1a¼0

XMm¼0

X1n¼0

Tm �xð Þ cos nhþ ap2

� ��umna tð Þ

¼ U �x; hð Þui xð Þejxt ð5Þ

v i x; h; tð Þ ¼Xi

a¼0

XMm¼0

X1n¼0

Tm �xð Þ sin nhþ ap2

� ��vmna tð Þ

¼ V �x; hð Þv iðxÞejxt ð6Þ

Q. Deng et al. / Applied Acoustics 104 (2016) 85–100 89

wiðx; h; tÞ ¼X1a¼0

XMm¼0

X1n¼0

Tmð�xÞ cos nhþ ap2

� ��wmnaðtÞ

¼ Wð�x; hÞwiðxÞejxt ð7Þ

x ¼ l2

�xþ 1ð Þ; �x ¼ 2lx� 1 ð8Þ

3.3. The virtual work of fluid

Based on the assumptions given in Section 2, the motion of thefluid in frequency domain is described by the velocity potential£ðx; h; r;xÞ which satisfies Eq. (9). The fluid pressure in a circularcylindrical coordinate system, with certain regular boundary con-ditions, can be expanded as the summation of a set of analyticalfunctions by applying variable separation technique. Each analyti-cal function is considered as a pressure mode. The fluid-air inter-face is assumed to be a pressure released boundary and theseabed is considered as either a perfectly rigid boundary or a localimpedance boundary. For the fluid field outside the pile, the fluidpressure and the velocity component normal to the surface ofthe shell can be given as Eq. (10) [23] and Eq. (11) [23].

r2£ x; h; r;xð Þ þ xc

� �2£ x; h; r;xð Þ ¼ 0 ð9Þ

~pe x; r; hð Þ ¼ �jxqf

X1a¼0

X1n¼0

X1p¼0

DanpHð2Þn krpr� �

sin kxpðx� xesÞ

� cos nhþ ap2

� �ð10Þ

~ver x; r; hð Þ ¼X1a¼0

X1n¼0

X1p¼0

DanpH0 2ð Þn krpr� �

sin kxpðx� xesÞ

� cos nhþ ap2

� �ð11Þ

In Eqs. (10) and (11), qf is the density of the fluid. Hð2Þn is the

Hankel function of the second kind of order n and H0 2ð Þn denotes

its derivative with respect to r. Danp are unknown coefficientswhich are dependent on the radial displacements of the shell seg-ments in contact with the fluid.

The terms kxp and krp are wave numbers in x direction and rdirection respectively and depend on radian frequency x andboundary conditions on the sea surface and the seabed. The valuesof kxp can be obtained according to Eq. (12), in which

p ¼ 0;1;2; . . . ;1, and eZsoil xð Þ ¼ qcf is the acoustic impedance ofthe seabed and can be measured with standard acoustic techniques[28]. It is obvious that kxp are real values when the seabed isassumed to be perfectly rigid and complex values when the seabedis assumed to be an acoustic impedance boundary.

krp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix=cð Þ2 � k2xp

qare the wave numbers in radial direction and

must satisfy the condition that ReðkrpÞ P 0 and ImðkrpÞ 6 0 dueto the requirement of radiation condition at r ! 1.

kxp ¼ 2pþ1ð Þp2 xb�xesð Þ ; Rigid seabedð Þ

tan kxp xb � xesð Þ þ kxp~Zsoil xð Þjxqf

¼ 0 Impedance seabedð Þð12Þ

By making use of the normal velocity continuity condition onthe fluid–structure interface and the orthogonality property ofpressure modes, the unknown coefficient Dnp can be obtained.Then the sound pressure in the exterior fluid field can be expressed

as the summation of a set of analytical functions weighted by theradial displacement coordinates of the shell, i.e. wiamn, as shownin Eq. (13).

~pe x; r; h;xð Þ ¼XIei¼1

X1a¼0

XMm¼0

X1n¼0

X1p¼0

wiamn � ePe;iamnp with :

ePe;iamnp ¼ x2qf p

Hð2Þn krpr� �

H0 2ð Þn krpR� � � sin kxp x� xesð Þ � cos nhþ a

p2

� ��Z xiþ1

xi

Tm x� xið Þ � sin kxp x� xesð Þdx ð13Þ

and

f p ¼xb�xes

2 ðrigid seabedÞxb�xes

2 � 14kxp

sin 2kxp xb � xesð Þ ðimpedance seabedÞ

(The effect of the fluid is considered as fluid loading acting on the

pile. The virtual work done by the sound pressure on the shell isgiven as Eq. (14), with ~pe x;R; h;xð Þ being the sound pressure onthe exterior surface of the shell and dw denoting the virtual dis-placement of the shell in radial direction. The derivation procedureof the virtual work of the interior fluid is completely analogous asthat of exterior fluid pressure.

dWp ¼Z 2p

0

Z xb

xs

�Reð~pe x;R; h;xð ÞejxtÞ � dwRdxdh ð14Þ

3.4. The virtual work of soil

A complete and absolutely exact evaluation of the influence ofsoil on the vibration and sound radiation of the pile is not easyto realize because the interactions between pile and its surround-ing soil are very complex. In this work, the soil effect is incorpo-rated into the model in a simple manner, in which the soil ismodeled as uniformly distributed elastic springs and dashpots inall directions, as shown in Fig. 1. The effect of springs and dashpotson the pile shell is taken into consideration by incorporating theirvirtual work into the variation equation. The virtual works done bythe springs and dashpots are given as Eqs. (15) and (16), respec-tively. The spring coefficients and radiation damping are intro-duced to the model in a simple manner presented by Gazetasand Dobry [29]. The derivation of the radiation damping is basedon the simplification treatment that the circular pile cross sectionis replaced by a square section having the same perimeter. It is fur-ther assumed that each of the four sides of the square emits only inthe associated truncated quarter-plane and each of the fourquarter-planes vibrates independently of the three others. Addi-tionally, it is assumed that there is no tangential slip on the pile-soil interface. For detailed derivation of the radiation damping,the reader is referred to [29]. The derivation process of the springcoefficients is completely analogous as that of the radiation damp-ing. The material damping of the soil is expressed with reference toan equivalent damping ratio of the soil throughout the frequencyrange for a certain strain rate [30]. The detailed expressions ofthe spring and damping coefficients are given in Appendix A.

The introduction of springs and dashpots is of practical signifi-cance from an engineering point of view, but it presents a chal-lenge in the realistic evaluation of the coefficients and createsuncertainty to the calculations. The influence of the chosen springand dashpot coefficients on sound radiation was discussed in theprevious work [23] in which the similar simplification treatmentwas used. A complete modeling of the soil effect should take a vari-ety of factors into consideration, such as the wave propagation inthe soil field, the inhomogeneity and material nonlinearity of thesoil, slippage and separation behavior of the pile-soil interface,

90 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

etc., which requires theoretical and experimental supports. Thiswould be a significant research topic to follow by the authorsand readers.

dWspring;i ¼ �ZS

dui � kxui þ dv i � khv i þ dwi � krwið Þds ð15Þ

dWdashpot;i ¼ �ZS

dui � cx @ui

@tþ dv i � ch @v i

@tþ dwi � cr @wi

@t

� �ds ð16Þ

3.5. Impact force of the hammer

A practical and accurate evaluation of the impact force is ofgreat importance for the prediction of the underwater noisebecause the force contains all the input energy of the interactionsystem. Analytic models [31,32] were used to predict the pile-driving force for piles which transmit longitudinal wave. In thesemodels the pile is simplified as a dashpot with constant impe-dance. For tubular piles the wave propagates along the pile withdispersion and the prediction of the impact force becomes morecomplicated. Reinhall and Dahl [16] got the approximate impactforce as a step-exponential function by using the Finite Element(FE) impact analysis with the effect of the cushion, water and sed-iment neglected. Zampolli et al. [15] adopted the similar functionand obtained the peak force and decay time constant of the func-tion by fitting the step-exponential pulse with the forcing curveobtained from a publicly available tool known as IHCWave [33].Tsouvalas and Metrikine [23] used an impact force taken from areal case as the input of the interaction system. The amplitudeand wave form of the exciting force of the hydraulic hammer canbe influenced by many factors, such as the categories of the impactpile drivers [34], the properties of the anvil and cushion, the size ofthe pile and the precise boundary condition on the pile top. Sys-tematic investigations and further discussions are needed for theevaluation of impact forces in different situations.

In this work, the impact force is a fictitious one and it isassumed to distribute on the intersecting line of the neutral layerof the shell and the cross section on the pile top and is definedas Eq. (17). F0ðhÞ is the distributed force dependent on h and its unitis N m�1. HðtÞ is the waveform function in terms of time and itspeak value is 1.

F tð Þ ¼ HðtÞZ 2p

0F0ðhÞRdh ð17Þ

3.6. Governing equations of the motion of the pile

In this work, the derivation of the equations is implemented infrequency domain. Therefore, Fourier transform pairs with respectto time and frequency [27] are introduced herein to transform vari-ables between time domain and frequency domain. By performingthe variation operation to Eq. (1) with respect to the generalizedcoordinate vectors u;v ;w and substituting the virtual works ofthe hydraulic impact force, fluid pressure, spring and dashpotforces into the equation, the linear governing equations of themotion of the pile can be obtained as Eq. (18).

�x2M þ K � Kk þ Kjð Þ � Ce þ C i þ jxCsoil

qðxÞ ¼ FðxÞ ð18ÞIn the equation, M and K are the disjoint mass matrix and stiff-

ness matrix of the shell, respectively. Kk and Kj are the generalizedinterface stiffness matrices incorporated by the Lagrange multipli-ers and interface least-squares weighted residual method, respec-tively [25]. The expressions of the matrices M;K;Kk;Kj andgeneralized force vector FðxÞ are given in the Appendix B. MatricesCe;C i and Csoil are coupling matrices introduced by the exterior

sound pressure, the interior sound pressure and the soil in contactwith the pile, respectively. The details of the matrices Ce;C i andCsoil are given in Appendix C. The vector q xð Þ is the generalizedcoordinate vector consisting of the displacement coordinate vec-tors of the shell segments and its expression is given in Eq. (19).The sound radiation of the pile can be obtained by summing thepressure modes which are weighted by the radial generalizeddisplacement coordinates.

q xð Þ ¼ uT1;vT

1;wT1;u

T2;vT

2;wT2; . . . ;u

TI ;vT

I ;wTI

T ð19Þ

4. Dispersion relation and sound propagation characteristics

The analytical description of the sound pressure makes it possi-ble to show the dispersion relation and examine the influence ofsome parameters on the sound radiation in a qualitative manner.As mentioned above, the wavenumbers in radial directionkrpðp ¼ 0;1;2; . . .Þ depend on the radian frequency and thewavenumbers in vertical direction kxp which are dependent onthe acoustic boundary condition on the seabed. In this section,the propagation characteristics of exterior sound radiation for theperfectly rigid seabed are discussed based on the dispersion rela-tion (x� kr curves). The depth of the exterior seawater is assumedto be xb � xes ¼ 8 m and the sound velocity is c ¼ 1500 ms�1.

For the rigid seabed, the wavenumbers in vertical direction arereal and the wavenumbers in radial direction are either real (corre-sponding to propagating modes) or imaginary (correspondingto evanescent modes). The dispersion curves for the first ten pres-sure modes are shown in Fig. 3 for radian frequencies up to6000 rad s�1. As the radian frequency increases, the curvature ofeach curve decreases gradually and the phase velocity approachesto the defined sound velocity 1500 m s�1. Each mode has a crossingpoint with the frequency axis which divides the frequency axis intoa low frequency range and a high frequency range. In the low fre-quency range, radial wavenumber kr is imaginary and the mode isevanescent. In the high frequency range, the wavenumber is realand the mode works as a propagating mode. Therefore, a cut-offfrequency xc for propagating modes can be found at the crossingpoint of the frequency axis and the curve of the first pressure mode(corresponding to p ¼ 0). Below the cut-off frequency, all modesbecome evanescent and no sound can propagate effectively awayfrom the pile. By substituting the first equation of Eq. (12) into

the expression krp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixc=cð Þ2 � k2xp

q¼ 0, the cut-off frequency

can be obtained as Eq. (20). It is obvious that the cut-off frequencyis inversely proportional to the depth of the exterior fluid, whichmeans low-frequency pile-driving noise in shallow waters isharder to propagate away than that in deep waters.

For the impedance seabed, the wavenumbers in both verticaldirection and radial direction are complex-valued and dependenton the impedance of the seabed. In this situation, a clear cut-offfrequency no longer exists. For a more detailed analysis of theinfluence of the seabed impedance on the sound propagation, thereader is referred to [23].

xc ¼ c � p2 xb � xisð Þ ð20Þ

5. The case with an axisymmetric impact force

A pile with a certain geometry and material parameters ischosen for the axisymmetric numerical case. The seabed isassumed to be a perfectly rigid acoustic boundary. The materialproperties of the shell, geometry and parameters of the modelare summarized in Table 1. The impact force is assumed to be

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

1000

2000

3000

4000

5000

6000

Wavenumber kr (rad m-1)

Rad

ian

frequ

ency

ω

ω

(rad

s-1

)

cp=0

p=1

p=2

p=3

p=4

p=5

p=6

Fig. 3. Dispersion relation for rigid seabed.

Table 1Model parameters (reference values) for the examined case.

Parameters Value Unit

E 2:1� 1011 N m�2

l 0.28 –qs 7800 kg m�3

g 0.002 –L 28 mR 1 mh 0.02 mc 1500 m s�1

qf 1000 kg m�3

xes ¼ xis 10 mxb 18 mEsoil 5:0� 107 N m�2

lsoil 0.4 –qsoil 1600 kg m�3

Q. Deng et al. / Applied Acoustics 104 (2016) 85–100 91

axisymmetric and uniformly distributed on the circumference ofthe pile top. For this case displacement functions and pressuremodes for only circumferential order n ¼ 0 can be excited andthe responses of the model are also axisymmetric. The distributedforce is independent of angle h, given as F0 hð Þ ¼ 1� 106 N m�1. Thetime function of the force is assumed to be a half sinusoid wavewith a very small duration s ¼ 2:5 ms and is given in Eq. (21).

H tð Þ ¼ sin pts

� �0 < t < s

0 t < 0 or t > s

(ð21Þ

5.1. Truncation of the pressure modes and the number of sub-shells

In the analytical expression, the sound pressure equals the sum-mation of unlimited number of pressure modes consisting of bothpropagating modes and evanescent modes. It is important toinclude the evanescent modes in the solution scheme of thefluid-pile-soil interaction problem, especially for low frequencies.For each excitation frequency, there are a set of radial wavenum-bers krp ¼ 0;1;2; . . . ;1 in the pressure expression, each of whichcorresponds to a pressure mode. Therefore, it is inevitable to trun-cate the modes in the computational program. As mentioned inSection 3.3, the wavenumbers in radial direction must satisfy thecondition ReðkrpÞ P 0 and ImðkrpÞ 6 0 because of the requirementof radiation condition at r ! 1. It is noticed that evanescent

modes are dramatically damped due to eIm krpð Þ�r term. Therefore,

only the modes with eIm krpð Þ�R P e are retained, with R being theradius of the pile and e denoting a small parameter

ðe ¼ 1� 10�4Þ. The modes with eIm krpð Þ�R < e are disregardedbecause the contribution of the modes is negligible and the trunca-tion leads to insignificant loss of accuracy.

Due to the accuracy requirement of high-frequency computa-tion, the pile should be decomposed into enough number of shellsegments with enough orders of admissible displacement func-tions adopted in each. Theoretically, increase of both the numberof shell segments and the terms of the displacement functionscan improve the accuracy of the model. However, excessively highorders of displacement functions should be avoided for two rea-sons. Firstly, there exists numerical integration of displacementfunctions in the variational equation and the integration of highorders of displacement functions is relatively inefficient. Secondly,incorporation of excessively high orders of displacement functionsleads to badly conditioned structural mass matrixes and stiffnessmatrixes. Therefore, a better settlement for high-frequency com-putation is to increase the number of shell segments. The soundpressure responses of an observing position located 3 m abovethe seabed and 16 m away from the pile surface are calculatedusing different numbers of sub-shells, i.e. I ¼ 4, 6, 8, 10. A set ofdisplacement functions with the highest order M ¼ 8 is adoptedin each sub-shell. The curves of the time histories are given inFig. 4. Good convergence can be observed with such few sub-shells used. The curves of I ¼ 8 and I ¼ 10 are almost overlappedcompletely, which indicates a decomposition of 10 sub-shellswould be accurate enough to predict the sound response of thiscase. In subsequent computation, the pile is decomposed into 12sub-shells with 8 displacement functions in each. Further discus-sions about Fig. 4 are given in Section 5.3.

5.2. Acoustic responses

Tp x; r;xð Þ ¼ p x; r;xð ÞF xð Þ ð22Þ

f ring ¼1

2pR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

qs 1� l2ð Þ

sð23Þ

The force to pressure transfer functions, defined as Eq. (22), forthree observed points located 4 m below the sea surface are shownin Fig. 5. The ordinate values in dB are given by 20 logðjTpj=T0Þ, with

the reference value T0 being 1 lPa N�1. The distances from the pilesurface to each observed point are 8 m, 25 m and 50 m, respec-tively. It is evident that the difference between the three curvesis depedent on frequency. The transfer function of a nearerobserved position is generally higher than that of a fartherobserved position. An obvious trough can be observed at the ringfrequency of the pile (f ring ¼ 860 Hz), defined as Eq. (23). At thering frequency, the quasi-longitudinal wavelength in the shell wallequals the shell circumference and an axisymmetric, n ¼ 0,‘breathing’ or ‘hoop’, resonance occurs [11]. It was shown that atrough at this frequency also appeared on the frequency spectraof the radial pile displacement [21].

An obvious difference between the transfer functions isobserved below the cut-off frequency f c ¼ 46:8 Hz. In this fre-quency range the amplitutes are very small compared with thosein the relatively high frequency range. This phenomenon can beascribed to the fact that below the cut-off frequency there existonly evanescant modes which attenuate dramatically as radial dis-tance increases. Below the cut-off frequency, the transfer functionof the observed position at a distance of 50 m nearly vanishes andthe curve climbs steeply at the cuf-off frequency.

The time histories of the pressures at the three points are givenin Fig. 6. Increase in observed distance leads to both a lag along the

0 0.01 0.02 0.03 0.04 0.05 0.06

-1

-0.5

0

0.5

1

x 104

t (s)

Pres

sure

(Pa)

M=4

M=6

M=8

M=10

Fig. 4. Pressure histories at the observing position for different numbers of sub-shells.

0 500 1000 1500 2000 2500 30000

20

40

60

80

100

Frequency (Hz)

Tran

sfer

func

tion

|Tp|

(dB

re μ

Pa N

-1)

distance = 8 m

distance = 25 m

distance = 50 mfc=46.8Hz

Fig. 5. Transfer functions for the observed points located 4 m below the sea surface.

0 0.02 0.04 0.06 0.08 0.1 0.12-1.5

-1

-0.5

0

0.5

1

1.5

x 104

t (s)

Pres

sure

(Pa)

distance = 8 mdistance = 25 mdistance = 50 m

Fig. 6. Time histories of pressure for the observed points located 4 m below the seasurface.

0 500 1000 1500 2000 2500 30000

20

40

60

80

100

Frequency (Hz)

Tran

sfer

func

tion

|Tp|

(dB

re μ

Pa N

-1)

depth = 0.5 m

depth = 4 m

depth = 7.5 m

Fig. 7. Transfer functions for the observed points located 8 m away from the pilesurface.

0 0.02 0.04 0.06 0.08 0.1 0.12

-3

-2

-1

0

1

2

x 104

t (s)

Pres

sure

(Pa)

depth = 0.5 mdepth = 4 mdepth = 7.5 m

Fig. 8. Time histories of pressure for the observed points located 8 m away from thepile surface.

92 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

time axis and an attentuation in amplitute. An estimated soundpropagation speed from the figure is 1504 m s�1 which matchesthe defined sound speed c ¼ 1500 m s�1 very precisely.

The force to pressure transfer functions for three observedpoints located 8 meters away from the pile surface are shown inFig. 7. The points are positioned at 0.5 m, 4 m and 7.5 m belowthe sea surface, respectively. It is obvious that the sound pressureis depth dependent. Below 1400 Hz, The point close to the seabedhas the highest peaks while the point close to the see surface hasthe lowest values. In the high frequency range, the trend no longerexists or even reverses. The pressure variation along depthdepends on the terms sin kxpðx� xesÞ in pressure modes, as givenin Eq. (10). For each mode, a pressure trough emerges on the pres-sure release boundary and a pressure crest emerges on the rigidseabed. The aforementioned phenomenon in the low frequencyrange occurs because the upper point is close to the trough whilethe lower point is close to the crest. As the frequency increases,fluid modes of higher value of kxp and smaller wavelengths becomedominant. In high frequency range, the phenomenon no longerexists because the dominant wavelengths become comparable toor even smaller than the distance from the observed points tothe adjacent boundaries.

Fig. 8 shows the time histories of the sound pressures at thethree observed points. The pressure curves see their first peaks atabout t ¼ 10 ms and then fluctuate with a rapid attenuation alongthe time axis. The pressure amplitudes decrease to less than5000 Pa after 70 ms. The point close to the seabed has the

maximum pressure peaks while the point near the sea surfacehas the minimum values. This is based on the fact that in thelow-frequency range the former observed point has the highestwhile the later one has the lowest pressure amplitude and mostenergy of the impact force is distributed in this frequency range.

Q. Deng et al. / Applied Acoustics 104 (2016) 85–100 93

The absence of strong peaks at the depth of 0.5 m is discussed fur-ther in the Section 5.3.

5.3. Pressure distributions in exterior fluid field

In Fig. 9, the pressure level distributions on a cross section ofthe fluid field are shown for 4 different frequencies. The valuesare presented in terms of sound pressure levels (SPL) calculatedby Eq. (24), with the reference value being ~p0 ¼ 1 lPa=Hz. The hor-izontal axis shows the radial distance from the pile surface and thevertical axis denotes the depth of the exterior fluid measured fromthe sea surface. As mentioned above, the pressure variation alongdepth depends on the terms sin kxpðx� xesÞ in pressure modes, withkxp given in Eq. (12). For each mode, the pressure vanishes on thesurface and reaches its wave crest on the seabed. From the Fig. 9(a), it is observed that the mode corresponding to p ¼ 0 seems tobe dominant over other modes. This can be explained by the dis-persion curves given in Fig. 3. For this frequencyx ¼ 2pf ¼ 314 rad s�1, only the mode corresponding to p ¼ 0 con-tributes as a propagating mode and all the other modes are evanes-cent. As for the Fig. 9 (b) whose radian frequency is 1256 rad s�1,there are two propagating modes contributing to the sound pres-sure, i.e. p ¼ 0 and p ¼ 1. A nodal line tends to emerge at the depthof 5.3 m, which indicates that the later mode is predominant overthe former one. When the frequency increases, there will be morepropagating modes involved and their relative contributions to thepressure field are dependent on the structural deformation of theshell. It can be seen from the other two figures that the dominantmodes for 1000 Hz and 2000 Hz are corresponding to p ¼ 2 andp ¼ 5, respectively. For all the four figures, the sound pressure

0

1

2

3

4

5

6

7

80 50 100 150 200

Radial distance from pile surface (m)

Dep

th o

f the

ext

erio

r flu

id d

omai

n (m

)

20

30

40

50

60

70

80

90

100

0

1

2

3

4

5

6

7

80 50 100 150 200

Radial distance from pile surface (m)

Dep

th o

f the

ext

erio

r flu

id d

omai

n (m

)

50

60

70

80

90

100

110

120

130

(a)

(c)

Fig. 9. Distributions of pressure level in dB re lPa/Hz for

levels attenuate as the radial distance increases. Due to the pres-ence of evanescent modes which are only significant near the pilesurface, the attenuations of the pressure levels are especiallynoticeable at the proximity of the shell surface.

SPL ¼ 20log10

ffiffiffi2

p

2~pe x; r; h;xð Þj j

~p0

!ð24Þ

The evolution of the radiated exterior pressure with time for aradial distance up to 16 m is illustrated in Fig. 10. The time stepbetween two adjacent pictures is 2 ms. A compressional wave isexcited by the impact force at the pile top and propagates rapidlydownwards. When the structural wave goes through the fluid field,the fluid is compressed and generates a pressure peak which prop-agates outward along the radial direction. A minus pressure peak isgenerated immediately after the positive pressure peak due to therarefaction following after the compression. The structural wave inthe pile propagates faster than the sound velocity of the fluid,therefore a Mach cone [14,16] is formed in the fluid field with anangle awith respect to the pile axis. The angle depends on the rela-tion a ¼ tan�1ðc=csÞ in which cs is the propagation speed of thestructural wave in the pile. An estimated value of the angle fromthe figure is 16:5�. The approximate propagation speed of struc-tural wave is 5064 m s�1 which is slightly smaller than longitudi-nal wave velocity in the steel material cL ¼

ffiffiffiffiffiffiffiffiffiffiE=qs

p ¼ 5188 m s�1.This presence of Mach cones was initially mentioned in [14] andwas also observed in [16,17,23]. For a detailed description of thestructural wave velocity and its dispersion characteristics in thepile wall, the reader is referred to [21,22]. When the compressedstructural wave arrives at the pile bottom it is reflected backward

0

1

2

3

4

5

6

7

80 50 100 150 200

Radial distance from pile surface (m)

Dep

th o

f the

ext

erio

r flu

id d

omai

n (m

)

90

100

110

120

130

140

150

160

0

1

2

3

4

5

6

7

80 50 100 150 200

Radial distance from pile surface (m)

Dep

th o

f the

ext

erio

r flu

id d

omai

n (m

)

60

70

80

90

100

110

120

130

(b)

(d)

(a) 50 Hz, (b) 200 Hz, (c) 1000 Hz and (d) 2000 Hz.

-3

-2

-1

0

1

2

3

x 104

0 10

t=0.0125s

0 10

t=0.0105s

0 10

t=0.0085s

0 10

t=0.0065s

0 10

t=0.0045s0

1

2

3

4

5

6

7

80 10

t=0.0025s

Radial distance from the pile surface (m)

Dep

th o

f ext

erio

r flu

id fi

eld

(m)

Fig. 10. Evolution of the sound pressure in Pa for time steps starting from 2.5 ms to 12.5 ms.

94 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

due to the impedance difference on the pile end. The upwardreflected wave compresses the fluid and again an upward Machcone generates in the fluid. The upward Mach cone can beobserved in the last two pictures. It is worth noting that there isan area along the sea surface where the downward Mach conescannot be seen. Reinhall and Dahl [16] pointed out that the depthof the area is D� � R � tana, with R being the radial distance fromthe pile source and a being the angle of the Mach cone. Theabsence of downward Mach cones on this area leads to the lackof strong pressure peaks at points close to the sea surface, whichgives further explanation on the phenomenon observed in Fig. 8.Away from the pile interface and for the points 0.5 m below thesurface the powerful Mach cones are not present and strong pres-sure peaks cannot be found. However, if one moves very close tothe pile-fluid interface, the pressure difference in vertical directionwill no longer be so obvious due to the formation of the Machcones.

It takes about 0.01 s for the structural wave to travel twice thelength of the pile. We call this duration a quasi-periodicity. Inevery quasi-periodicity, the vibration energy of the pile decays sig-nificantly as a portion of it is transmited into water and comsumedon the pile-soil interface. As a result, the sound energy has a similardecaying trend. If we examine the curve in Fig. 4 by an interval ofabout 0.01 s, we can observe the quasi-periodicity and find thepressure decay in each quasi-periodicity is very obvious.

5.4. The effect of varied impact force on the sound radiation

Deeks and Randolph [31] revealed that the amplitude and theduration of the pile-driving force was strongly influenced by thecushion stiffness, the anvil mass and the cushion damping. Thevariation of the impact force can make a significant difference tothe pile driving noise. In this section, the effect of varied impactforce on the sound responses is investigated.

A set of half sinusoid forces FiðtÞ, as given in Eq. (25), are used as

the inputs of the model, with Fi0 and si being the peak distributed

force and the duration of the force FiðtÞ, respectively. They areassumed to impart equal impulse IF ¼

RFiðtÞdt to the pile head

and satisfy Eq. (26), with F0 ¼ 1� 106 Nm�1; s0 ¼ 2:5 ms. The eval-uated durations si increase from 1.25 ms to 5 ms, while the corre-

sponding peak distributed forces Fi0 decrease from 2F0 to 0:5F0. The

evaluated output quantities are the peak sound pressure levelðSPLpeakÞ and the single sound exposure level (SEL), derived fromthe sound pressure p at the observed position 4 m below the seasurface and 40 m away from the pile surface. The output quantitiesare defined as Eqs. (27) and (28), with the reference pressure and

time being p0 ¼ 1 lPa and t0 ¼ 1 s, respectively. The soundresponse of interest is within the interval between t1 and t2.

The SPLpeak and SEL curves for the forces are given in Fig. 11.Both curves experience a significant decline as the force durationincreases. Four times of decrease in force peak (from 2F0 to0:5F0) leads to a decline of SPLpeak and SEL as much as 20 dB, whichmeans the peak pressure decreases 10 times. The fact that thechange in sound pressure is more significant than the change inthe force peaks would be explained by evaluating the structuralwave excited by the hammer force. In addition to the force peak,the force duration make a significant impact on the structuralwave. At the very moment when the hammer bounce away fromthe pile top, the structural deformation covers a length of cssi onthe pile, which indicates that, for a short force duration si, thestructural deformation covers less length and the radial displace-ment varies more sharp in axial direction. It means that a morepowerful structural wave would generate, transmit on the pile,and radiate higher level of underwater sound. The results suggestthat it would be a potential way to reduce pile driving noise bychoosing appropriate anvil and cushion.

Fi tð Þ ¼ 2pRFi0 sin

ptsi

� �0 < t < s

0 t < 0 or t > s

(; ði ¼ 1;2; . . . ;21Þ ð25Þ

Z si

02pRFi

0 sinptsi

� �dt ¼

Z s0

02pRF0 sin

pts0

� �dt ð26Þ

SPLpeak ¼ 20log10max½abs pð Þ�

p0

� �ð27Þ

SEL ¼ 10log101t0

Z t2

t1

p2

p20

dt� �

ð28Þ

6. The case with a non-axisymmetric impact force

In the previous case, the impact force is an ideal one which onlyexcites axisymmetric vibration and sound radiation. In practice, itis difficult to keep the impact force absolutely parallel with pileaxis and uniformly distributed on the circumference. In this sec-tion, a non-axisymmetric case whose impact force is dependenton circumferential angle is studied. The material and geometricalparameters are summarized in Table 1 and the impact force ofthe hammer is an assumed distributed force given in Fig. 12. Theforce can be considered as the summation of a uniformly dis-tributed force F1 ¼ 1� 106 N m�1 and an angle-dependent dis-tributed force. The time function HðtÞ is the same with the

1 1.5 2 2.5 3 3.5 4 4.5 5160

165

170

175

180

185

190

195

200

205

210

Force duration (ms)

SEL

/ SPL

peak

(dB)

SEL

SPLpeak

Fig. 11. SPLpeak (re 1 lPa) and SEL (re 1 lPa2=s) values for varied impact force.Fig. 13. Pressure levels of four observed positions distributed 4 m below the seasurface.

Q. Deng et al. / Applied Acoustics 104 (2016) 85–100 95

previous case and is given in Eq. (21). Other than the axisymmetriccase in which only the axisymmetric modes (corresponding ton ¼ 0) are included in the program, a non-axisymmetric case musttake the non-axisymmetric modes (corresponding to n > 1) intoconsideration. For the specified force given in Fig. 11, one onlyneeds to include the modes with n ¼ 0 and n ¼ 1 into the programbecause the force doesn’t excite pressure modes with higher cir-cumferential orders.

In Fig. 13, the sound responses in frequency domain of fourobserved positions located 40 m away from the pile surface and4 m below the sea surface are shown. The first 3 curves are spec-trums of the points positioned at h ¼ 0; p=2 rad and p rad, respec-tively. The last curve is the pressure response of the axisymmetriccase in which only the axisymmetric distributed force F1 is applied.In this non-axisymmetric case, the impact force as well as theacoustic response can be regarded as the summation of two com-ponents, i.e. the axisymmetric component and the non-axisymmetric component. Therefore, the difference in lPa sbetween each curve and the fourth curve equal the pressureresponse caused by the non-axisymmetric component. Theaxisymmetric response is completely overlapped with the pressurespectrum of the position located at h ¼ p=2 rad, which means thatthe sound pressure caused by the non-axisymmetric componentvanishes at this point. Theoretically, there exists a cross sectionplane (h ¼ p=2 rad) in the exterior sound field on which the soundpressure caused by the non-axisymmetric component vanishes.This is based on the fact that the pressure modes correspondingto n ¼ 1 contribute to the non-axisymmetric sound radiation andthis section plane is on the nodal plane of the pressure modes.

The curves in Fig. 13 experience several evident troughs as thefrequency increases. An analysis of the frequency spectrum of theimpact force finds that its amplitude decreses as the frequency

Fig. 12. Modeling of the non-a

increases and vanishes at the frequencies f q ¼ ð1=2þ qÞ=s, withthe index being q ¼ 1;2;3; . . .. Therefore, the pressure spectrumsof the observed points see a downward trend as the frequencyarises and experience several sharp troughs. The first troughemerges at f 1 ¼ 600 Hz and troughs appear again every 400 Hzalong the frequency axis. For a specific shape of the impact force,a small duration of the force will lead to a result that the mainenergy of the force distributes on a wide range. In other words,the faster the force attenuation is in time domain, the slower thespectrum attenuation is in frequency domain. The impact forceused in the model determines how the energy is distributed inthe frequency domain. Actually, the frequency spectrum of thesound pressure in the fluid field is atrongly influenced by the waveshape, amplitude and duration of the hammer force. Generally, theimpact force of pile driving has a duration within several millisec-onds, and the sound energy is mainly distributed on the low fre-quency range. As can be seen from Fig. 13, the SPL peaks over2000 Hz are approximately 40 dB less than the peaks below500 Hz. The impact forces in these two cases are hypothetical oneswhich create great convenience for the computation. However, asmentioned in Section 3.5, systematic studies and further discus-sions are necessary for more accurate and practical evaluationsof the impact force with a variery of factors consided.

Fig. 13 shows that the axisymmetric response has an obvioustrough around the ring frequency (f ring ¼ 860 Hz), which is consis-tent with the phenomenon observed in Fig. 5. However, the troughcannot be found from the non-axisymmetric responses at theobserved points with h ¼ 0 and h ¼ p=2 rad. The comparisonbetween the curves shows that the radiated sound is not uniformlydistributed in circumferential direction for the entire frequency

xisymmetric impact force.

F

0u

cmanvil

Fig. 14. The impact model of the anvil.

0 500 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

Frequency (Hz)

SPL

(dB

re 1

μPa/

Hz)

without anvilwith anvil

Fig. 15. Pressure levels at the point located 4 m below the sea surface and 10 maway from the pile surface.

96 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

range and the circumferential non-uniformity is especially obviousaround the ring frequency of the pile. It is clear that around the ringfrequency only the responses of non-axisymmetric modes are sig-nificant. The difference between the curves indicates that experi-mental signal acquisition of underwater pile-driving noise in justone direction would not be enough, especially when the impactforce has a significant non-axisymmetric component. The discoveryof circumferential non-uniformity of the sound field is of guidingsignificance for the experimental tests of the underwater pile driv-ing noise.

7. The case including the pile cap

On the one hand, the interaction between the ram, the cushion,the anvil and the pile influences the impact force [31,35], and onthe other hand their presence would influence the mechanicalresponses of the pile. In this section, a pile anvil seated on the pilehead is included into the model, as illustrated in Fig. 14. Followingassumptions are used to this model: (a) the impact force is a ver-tical force coaxial with the pile axis; (b) the anvil is assumed as alumped mass simply supported on the pile head, i.e., the anvilmoves vertically together with the pile head without separationand the interface between the anvil and the pile head doesn’t sus-tain shear forces and bending moments. The impact force is thesame with the one used in Sections. 5.1, 5.2, 5.3.

The second assumption means that an additional axial inertiaforce is applied to the pile head as the anvil moves along with it.The effect of the anvil is taken into account by incorporating thekinetic energy of the anvil Tc into the variational Eq. (1). Thenthe variational equation turns to be Eq. (29), with Tc given in Eq.(30). u0 is the displacement of the pile head in axial direction. Bythis approach, one only needs to change the mass matrix of thefirst sub-shell to get the new governing equations. The mass sub-matrix of this case is also given in Appendix B.Z t1

t0

dTc þXI

i¼1

ðdTi � dUi þ dWiÞ" #

dt þZ t1

t0

Xi;iþ1

dPkjdt ¼ 0 ð29Þ

Tc ¼ 12mcu2

0 ð30Þ

In Fig. 15, the SPL curve of the case including an anvil(mc ¼ 600 kg) is compared with the case without the anvil at anobserved position. It is shown that the anvil tends to reduce thecorresponding frequencies of the peaks, which can be explainedby the fact that the anvil introduces additional mass to the cou-pling system. Furthermore, the corresponding amplitudes of thepeaks tend to decrease due to the presence of the anvil. Both thetrends become more evident in the high-frequency range. Theincorporation of the anvil changes the geometrical boundary con-dition on the pile top, and changes the resonance frequenciesand the response amplitudes of the model. The change becomesincreasingly sensitive to the variation of boundary condition asthe frequency goes up. In this case, the incorporation of the anvildoesn’t result in obvious losses in SPLpeak and SEL at the position,which are just 0.5 dB and 0.3 dB, respectively. It can be concludedthat the losses in SPLpeak and SEL will be more significant if a shar-per force impacts on the pile, since a sharper force carries moreenergy on the high frequency range.

The variational formulationmakes it quite tractable to deal withpiles with complex boundary conditions in which some attach-ments are connected to the pile ends, such as the pile cap (anvil),the pile cushion and the pile footing. Compared to themethod usingdifferential equations to solve the shell displacements, the effort toseek in-vacuo shell modes for different geometrical boundary con-ditions is avoided. This advantage makes it possible to establish a

more complete model for influence evaluation of the attachmentsand to explore potential noise-reducing measures, which wouldbe a potential work for the authors and the readers.

8. Experimental validation

To validate the modeling approach described in Section 3, acomparison between measured data from the sound test of a minipile and computed values from the mechanical model is given inthis section. The experiment was conducted in a lake whose waterdepth was much less than that of an offshore site and the size ofthe pile was also far lower than that of a real pile used in offshorepile driving. The data comparison presented in this work offersexperiment support for the modeling approach, but both the mea-sured data and the computed values cannot reveal the real charac-teristics of the offshore pile driving noise due to the huge sizedifference.

A pile with a length of 4.6 m, a diameter of 139 mm and a thick-ness of 2 mm was driven into the lake bottom. The length of thepile below the water–soil interface was about 0.4 m. The materialparameters of the pile are listed in Tab. 1. The average water depthof the site was about 1.7 m. A disc-shaped pile cap with a mass of2.5 kg was fixed on the pile top by a flange connection. A hammer,equipped with a force sensor, impacted on the geometrical center

Q. Deng et al. / Applied Acoustics 104 (2016) 85–100 97

of the upper surface of the cap and the sound signal was collectedby a hydrophone located at the position 3 m away from the pilesurface and 0.65 m below the air–water surface. The impacts wererepeated 5 times for average and the time interval between anytwo adjacent impacts was large enough for the sound signalexcited by the former impact to vanish. The average peak forcewas 1817 N and the average duration of the force was 1.95 ms.The data were collected with a sampling frequency of 8192 Hz.The test of the ambient noise found it was negligible comparedwith collected pile driving noise. At the observed position, no sig-nificant reflected sound wave from the lake shore was detectedduring the time interval of interest.

The bottom of the site was characterized by a thin layer of softclay with a thickness of approximately 0.8 m, overlying a relativelyhard clay bottom. In the theoretical computation, the soil parame-ters were chose according to the upper layer of soil. A sampleanalysis of the upper layer suggested its density was aboutqsoil ¼ 1720 kg=m3. The soil’s elastic modulus and the Poisson’sratio used for theoretical computation were respectivelyEsoil ¼ 3:5 MPa and lsoil ¼ 0:4, which were chose according to thesuggestion of the reference [36] for soft clay. In the theoreticalmodel, the water–soil interface is assumed to be a rigid acousticboundary, and the radial displacement w at the pile top is confinedto be zero in consideration of the effect of the flange connection.

A comparison between the computed sound pressure and themeasured sound pressure is given in Fig. 16, in which the mea-sured data are the average output of the five impacts. The curvesshow that the model can reproduce the main characteristics ofthe response in both time domain (Fig. 16a) and frequency domain(Fig. 16b), albeit with some deviations on the pressure peaks andresonance frequencies. In the frequency range of interest, the res-onance frequencies are not as dense as those presented in Figs. 5and 7, which can be explained by the fact that the modal densityof the mini pile is far lower than that of the previous one withinthis frequency range. The computed SPLpeak and SEL values arerespectively 3.4 dB and 1.4 dB larger than their correspondingmeasured data. Both the soil simplification and the rigidassumption of acoustic bottom introduce accuracy loss to themodel, and these errors are inevitable on this topic due to the

0.05 0.1 0.15-40

-20

0

20

40

t (s)

Soun

d pr

essu

re (P

a) computed valuemeasured value

200 400 600 800 1000 1200 1400 16000

0.1

0.2

0.3

0.4

f (Hz)

Pres

sure

am

plitu

de (P

a/H

z)

computed valuemeasured value

(a)

(b)

Fig. 16. Comparison of computed and measured sound pressures at the positionlocated 3 m away from the pile surface and 0.65 m below the air–water interface in(a) time domain and (b) frequency domain.

complex composition of the model. Due to a number of uncertain-ties, prediction errors of underwater pile driving noise at thesescales are common [19,23], such as the parameter uncertaintiesof soil, the contact condition uncertainties at the pile-soil andanvil-pile interfaces, and the variations of the pile geometry. Inaddition to the overestimate of pressure peak, the theoreticalmodel gives a higher evaluation to the pressure decay over time,which suggests that the spring and dashpot model would overratethe damping of the soil. The comparison to the experiment dataindicates that the presented model is capable of predicting theunderwater pile driving noise quantitatively in both time domainand frequency domain, albeit with some errors.

9. Conclusions

In this work, a new semi-analytical methodology is proposed toestablish the mechanical model for predicting the underwaternoise for a pile-water–soil interaction system subject to pilinghammer impacts. Based on the study, following conclusions canbe obtained:

The dispersion relation curves show that for perfectly rigidseabed there is a cut-off frequency for sound propagation in theexterior fluid field. Underwater noise below the cut-off frequencycannot propagate effectively away from the pile. The value of thecut-off frequency is inversely proportional to the depth of the exte-rior fluid, which indicates that low-frequency pile-driving noise inshallow waters is harder to propagate away than that in deepwaters. The sound outputs of a set of impact forces with equalimpulse shows that both the SEL and the SPLpeak decreases signifi-cantly as the force duration increases. The non-axisymmetric caseindicates that the radiated sound pressure has dependence on cir-cumferential angle, and around the ring frequency of the pile thesound pressure is mainly contributed by the non-axisymmetricpressure modes. The presence of the anvil tends to reduce the cor-responding frequencies and amplitudes of the peaks of the soundpressure level, and this tendency becomes more obvious as the fre-quency increases. However, in the presented case the anvil doesn’tinfluence the underwater SEL and the SPLpeak significantly.

A mini pile experiment has demonstrated that the developedsemi-analytical method is reliable to predict the pile driving noisein both time and frequency domains, albeit with some accuracyloss.

Acknowledgements

This work is supported by National Natural Science Foundationof China (Grant No. 11272208). The authors gratefully acknowl-edge Dr. Haijun Wu from Shanghai Jiao Tong University for hisvaluable discussions and suggestions.

Appendix A. Expressions of the spring and damping coefficients

The equivalent spring coefficients for the soil model in threedirections are given as

kx ¼ qVsx � ReHð2Þ

1p4xRVs

� �Hð2Þ

0p4xRVs

� �24 35;

kr ¼ 12qx � Re Vs

Hð2Þ1

p4xRVs

� �Hð2Þ

0p4xRVs

� �þ VLa

Hð2Þ1

p4

xRVLa

� �Hð2Þ

0p4

xRVLa

� �24 35;

kh ¼ 12qVsx � Re

Hð2Þ1

p4xRVs

� �Hð2Þ

0p4xRVs

� �24 35; ðA:1a-cÞ

98 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

with Vs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEsoil=½2 1þ lsoil

� �qsoil�

qbeing the shear wave velocity of

the soil and VLa ¼ 3:4Vs=½pð1� lsoilÞ� denoting the Lysmer’s analog-wave velocity [29].

The equivalent damping coefficients in three directions aregiven as

cx ¼ qVs � Re �iHð2Þ

1p4xRVs

� �Hð2Þ

0p4xRVs

� �24 35þ 2kx

bsoil

x;

cr ¼ 12q � Re �i � Vs

Hð2Þ1

p4xRVs

� �Hð2Þ

0p4xRVs

� �� i � VLa

Hð2Þ1

p4

xRVLa

� �Hð2Þ

0p4

xRVLa

� �24 35þ 2kr

bsoil

x;

ch ¼ 12qVs � Re �i

Hð2Þ1

p4xRVs

� �Hð2Þ

0p4xRVs

� �24 35þ 2kh

bsoil

x:

ðA:2a-cÞ

In each equation,the first part in the right side is the radiationdamping coefficient in corresponding direction [29], while the sec-ond part denotes corresponding material damping coefficient, withbsoil being the equivalent damping ratio depending on the level ofsoil strain [30], which can be evaluated roughly by examining thevibration responses of the soil-free model.

Appendix B. Expressions of M;K ;Kk;Kj and FðxÞ

The disjoint mass and stiffness matrices M;K are respectivelygiven as

M ¼ diag½M1;M2; . . . ;Mi; . . . ;MI�; K ¼ diag½K1;K2; . . . ;K i; . . . ;K I�ðB:1a-bÞ

in which sub-matrices Mi and K i are mass and stiffness matrices ofthe ith sub-shell given as

Mi ¼Mi

uu

Mivv

Miww

26643775; K i ¼

K iuu K i

uv K iuw

K i;Tuv K i

vv K ivw

K i;Tuw K i;T

vw K iww

26643775 ðB:2a-bÞ

According to Reissner–Naghdi’s thin shell theory, the elementsof the sub-matrices Mi and K i can be given as Eqs. (B.3a-c) and

(B.4a-f), where K ¼ Eh=ð1� l2Þ and D ¼ Eh3=½12 1� l2

� �� are themembrane stiffness and bending stiffness of the shell, respectively.

For the case with the anvil included, the additional mass sub-matrix mcU

Tx¼0Ux¼0 is added to the right side of the expression of

M1uu which is the mass sub-matrix corresponding to the sub-shell

connected to the anvil. Ux¼0 is the polynomial vector of the axialshell displacement valued at x ¼ 0.

Miuu ¼

Z ZSi

qhUTURdxdh;

Mivv ¼

Z ZSi

qhVTVRdxdh;

Miww ¼

Z ZSi

qhWTWRdxdh:

ðB:3a-cÞ

K iuu ¼

Z ZSi

K@UT

@x@U@x

þ 1� l2R2

@UT

@h@U@h

!Rdxdh;

K iuv ¼

Z ZSi

KlR@UT

@x@V@h

þ 1� l2R

@UT

@h@V@x

!Rdxdh;

K iuw ¼

Z ZSi

KlR@UT

@xWRdxdh;

K ivv ¼

Z ZSi

1�l2

Kþ D

R2

� �@VT

@x@V@x

þ 1R2 Kþ D

R2

� �@VT

@h@V@h

" #Rdxdh;

K ivw ¼

Z ZSi

1R2

@VT

@hKW�lD

@2W@x2

� D

R2

@2W@h2

!"

�1�lR2 D

@VT

@x@2W@x@h

#Rdxdh;

K iww ¼

Z ZSi

K

R2WTWþD

@2WT

@x2@2W@x2

"(

þ lR2

@2WT

@x2@2W@h2

þ@2WT

@h2@2W@x2

!

þ 1R4

@2WT

@h2@2W@h2

þ2 1�lð ÞR2

@2WT

@x@h@2W@x@h

#)Rdxdh: ðB:4a-fÞ

The stiffness matrices Kk and Kj can be obtained respectively

by assembling the sub-matrices K i;iþ1k and K i;iþ1

j which are theinterface matrices between the ith subshell and the ðiþ 1Þth sub-

shell. The interface matrix K i;iþ1k is given as

K i;iþ1k ¼

Z 2p

0

Kuiui Kuiv iKuiwi

Kuiuiþ1Kuiv iþ1

0

KTuiv i

Kv iv iKv iwi

Kv iuiþ1Kv iv iþ1

Kv iwiþ1

KTuiwi

KTv iwi

KwiwiKwiuiþ1

Kwiv iþ1Kwiwiþ1

KTuiuiþ1

KTv iuiþ1

KTwiuiþ1

0 0 0

KTuiv iþ1

KTv iv iþ1

KTwiv iþ1

0 0 0

0 KTv iwiþ1

KTwiwiþ1

0 0 0

2666666666664

3777777777775ðB:5Þ

The elements in the matrix K i;iþ1k are given by Eqs. (B.6)–(B.19),

with K and D defined as K ¼ Kð1� lÞ=2 and D ¼ Dð1� lÞ=2.

Kuiui ¼ K@UT

i

@xU i þ KUT

i@U i

@xðB:6Þ

Kuiv i¼ Kl

RUT

i@V i

@hþ K

R@UT

i

@hV i ðB:7Þ

Kuiwi¼ Kl

RUT

i W i ðB:8Þ

Kuiuiþ1¼ �K

@UTi

@xU iþ1 ðB:9Þ

Kuiv iþ1¼ �K

R@UT

i

@hV iþ1 ðB:10Þ

Kv iv i¼ K þ D

R2

!@VT

i

@xV i þ VT

i@V i

@x

!ðB:11Þ

Kv iwi¼ D

R2

@2VTi

@x@hW i � lD

R2

@VTi

@h@W i

@x� 2D

R2 VTi@2W i

@x@hðB:12Þ

Kv iuiþ1¼ �lK

R@VT

i

@hU iþ1 ðB:13Þ

Kv iv iþ1¼ � K þ D

R2

!@VT

i

@xV iþ1 ðB:14Þ

Kv iwiþ1¼ � D

R2

@2VTi

@x@hW iþ1 þ lD

R2

@VTi

@h@W iþ1

@xðB:15Þ

Q. Deng et al. / Applied Acoustics 104 (2016) 85–100 99

Kwiwi¼ D

� @3WTi

@x3 � 2�lR2

@3WTi

@x@h2

� �W i þ @2WT

i@x2 þ l

R2@2WT

i

@h2

� �@W i@x

þWTi � @3W i

@x3 � 2�lR2

@3W i

@x@h2

� �þ @WT

i@x ð@2W i

@x2 þ lR2

@2W i

@h2Þ

264375 ðB:16Þ

Kwiuiþ1¼ �lK

RWT

i U iþ1 ðB:17Þ

Kwiv iþ1¼ 2D

R2

@2WTi

@x@hV iþ1 ðB:18Þ

Kwiwiþ1¼D

@3WTi

@x3þ2�l

R2

@3WTi

@x@h2

!W iþ1� @2WT

i

@x2þ lR2

@2WTi

@h2

!@W iþ1

@x

ðB:19ÞThe interface matrix K i;iþ1

j introduced by the least-squaresweighted residual terms between the ith subshell and theðiþ 1Þth subshell is given as

K i;iþ1j ¼

Z 2p

0

eK uiui 0 0 eK uiuiþ10 0

0 eK v iv i0 0 eK v iv iþ1

0

0 0 eKwiwi0 0 eKwiwiþ1eK T

uiuiþ10 0 0 0 0

0 eK Tv iv iþ1

0 0 0 0

0 0 eK Twiwiþ1

0 0 0

26666666666664

37777777777775Rdh

ðB:20ÞThe elements in the interface matrix K i;iþ1

j are given by Eqs.(B.21)–(B.29), with au;av ;aw and ar being the least-squaresweighted residual parameters [25].eK uiui ¼ juU

Ti U i ðB:21Þ

eK uiuiþ1¼ �juU

Ti U iþ1 ðB:22Þ

eK v iv i¼ jvVT

i V i ðB:23Þ

eK v iv iþ1¼ �jvVT

i V iþ1 ðB:24Þ

eKwiwi¼ jwW

Ti W i þ jr

@WTi

@x@W i

@xðB:25Þ

eKwiwiþ1¼ �jwW

Ti W iþ1 � jr

@WTi

@x@W iþ1

@xðB:26Þ

eK uiþ1uiþ1¼ juU

Tiþ1U iþ1 ðB:27Þ

eK v iþ1v iþ1¼ jvVT

iþ1V iþ1 ðB:28Þ

eKwiþ1wiþ1¼ jwW

Tiþ1W iþ1 þ jr

@WTiþ1

@x@W iþ1

@xðB:29Þ

dWi ¼ �d½ReðwTi Þejxt � � Re

Z 2p

0

Z xiþ1

xi

WT � Reð~peejxtÞRdxdh !

¼ �d ReðwTi Þejxt

� Re R 2p0

R xiþ1xi

WT �Xa;n;p

ePe;1a1np;Xa;n;p

ePe;1a2np; .

"Xa;n;p

ePe;2a1np; . . . ;Xa;n;p

ePe;2aMnp; . . . ;Xa

0BBBBB@� w1a1n;w1a2n; . . . ;w1aMn;w2a1n; . . . ;w2aMn; .½

8>>>>>>>><>>>>>>>>:¼ �d½ReðwT

i Þejxt � � Re eC i1; eC i2; . . . ; eC iIe

h i� wT

1;wT2; . . . ;w

TIe

Tejxt

n o;

Assuming that the exciting forces acting on the circumferenceof the pile top are, respectively, Fx in vertical direction, Fh incircumferential direction and Fr in radial direction, the generalizedforce vector FðxÞ can be given as

F xð Þ ¼ f x1 ; f h1 ; f r1 ;0;0;0; . . . ;0;0;0 T

; ðB:30Þwhere f x1 ; f h1 , f r1 are given by Eqs. (B.31)–(B.33). For verticalaxisymmetric impact forces, Fh ¼ Fr ¼ 0 and the vectorsf h1 ¼ f r1 ¼ 0.

f x1 ¼Z 2p

0UTFxRdh ðB:31Þ

f h1 ¼Z 2p

0VTFhRdh ðB:32Þ

f r1 ¼Z 2p

0WTFrRdh ðB:33Þ

Appendix C. Coupling matrices Ce;Ci and Csoil

The sound pressure in exterior surface of the shell can beexpressed in matrix form as Eq. (C.1).

~pe x;R;h;xð Þ ¼XIei¼1

X1a¼0

XMm¼0

X1n¼0

X1p¼0

wiamn � ePe;iamnp

¼Xa;n;p

ePe;1a1np;Xa;n;p

ePe;1a2np; . . . ;Xa;n;p

ePe;1aMnp;

"

�Xa;n;p

ePe;2a1np; . . . ;Xa;n;p

ePe;2aMnp; . . . ;Xa;n;p

ePe;IeaMnp

#� ½w1a1n;w1a2n; . . . ;w1aMn;w2a1n; . . . ;w2aMn; . . . ;wIeaMn�T

ðC:1ÞBy substituting the displacement functions of the sub-shells

and Eq. (C.1) into Eq. (14), the total virtual work of sound pressureon the ith submerged sub-shell can be obtained as Eq. (C.2),

with the sub-matrix eC ik given in Eq. (C.3). The component term

�d½ReðwTi e

jxt �ReðeC ikwkejxtÞ presents the virtual work done by thesoundpressure contributedby the kth sub-shell.Due to thepresence

of matrixes eC ik and eC ki, an additional coupling relation is appliedbetween the vibrations of the ith submerged sub-shell and the kthsubmerged sub-shell. The terms d½ReðwT

i ejxt � can be removed from

the variational equation due to the randomicity of the variation. Atlast, the variational equation Eq. (1) is simplified to a linear equationwithout 2th order terms of complex variables. The global coupling

matrix Ce can be obtained by assembling the sub-matrixes eC ik

together based on the arrangement of the generalized coordinatevectors as shown in Eq. (19). The derivation of the interior couplingmatrix C i is similar with that of the exterior coupling matrix.

. . ;Xa;n;p

ePe;1aMnp;

;n;p

ePe;IeaMnp

#� Rdx

1CCCCCAdh

. . ;wIeaMn�Tejxt

9>>>>>>>>=>>>>>>>>;ð1 6 i 6 IeÞ

ðC:2Þ

100 Q. Deng et al. / Applied Acoustics 104 (2016) 85–100

eC ik ¼Z 2p

0

Z xiþ1

xi

WT �Xa;n;p

ePe;ka1np;Xa;n;p

ePe;ka2np;

"

. . . ;Xa;n;p

ePe;kaMnp

#Rdx

0BBBBB@

1CCCCCAdh; ð1 6 k 6 IeÞ

ðC:3ÞProvided that the minimum and maximum vertical coordinates

of the pile-soil contact surface on a sub-shell are xs1 and xs2,respectively. By substituting displacement functions (u ¼U � uejxt ;v ¼ V � vejxt ;w ¼ W �wejxt) into Eqs. (15) and (16), thevirtual works done by the springs and dashpots can be given asEqs. (C.4) and (C.5), respectively. Therefore, the additional termincorporated into the governing equation can be expressed asEq. (C.6), with its sub-matrices given in Eqs. (C.7a-c) and (C.8a-c).The global coupling matrix Csoil can be obtained by assemblingthe matrixes Cs0 together.

dWspring ¼ �d½ReðuTejxtÞ�

� ReZ 2p

0

Z xs2

xs1

kxUTURdxdh � uejxt

� �� d½ReðvTejxtÞ�Re

Z 2p

0

Z xs2

xs1

khVTVRdxdh � vejxt

� �� d½ReðwTejxtÞ�Re

Z 2p

0

Z xs2

xs1

krWTWRdxdh �wejxt

� �;

ðC:4Þ

dWdashpot ¼ �d Re uTejxt� �

ReZ 2p

0

Z xs2

xs1

jxcxUTURdxdh � uejxt

� �� d½ReðvTejxtÞ�Re

Z 2p

0

Z xs2

xs1

jxchVTVRdxdh � vejxt

� �� d½ReðwTejxtÞ�Re

Z 2p

0

Z xs2

xs1

jxcrWTWRdxdh �wejxt

� �;

ðC:5Þ

jxCs0

uvw

264375 ¼ jx

Kuujx þ Cuu 0 0

0 Kvvjx þ Cvv 0

0 0 Kwwjx þ Cww

26643775

uvw

264375 ðC:6Þ

Kuu ¼ kx

Z 2p

0

Z xs2

xs1

UTURdxdh

Kvv ¼ kh

Z 2p

0

Z xs2

xs1

VTVRdxdh

Kww ¼ kr

Z 2p

0

Z xs2

xs1

WTWRdxdh

ðC:7a-cÞ

Cuu ¼ cx

Z 2p

0

Z xs2

xs1

UTURdxdh

Cvv ¼ ch

Z 2p

0

Z xs2

x1

VTVRdxdh

Cww ¼ cr

Z 2p

0

Z xs2

xs1

WTWRdxdh

ðC:8a-cÞ

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