Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 4, pp. 761-778, Warsaw 2009 ANALYSIS OF THE PARAMETERS OF A SPHERICAL STRESS WAVE EXPANDING IN A LINEAR ISOTROPIC ELASTIC MEDIUM Edward Włodarczyk Faculty of Mechatronics, Military University of Technology, Warsaw, Poland e-mail: edward.wlodarczyk@wat.edu.pl Mariusz Zielenkiewicz Military Institute of Armament Technology, Zielonka, Poland e-mail: m.zielenkiewicz@chello.pl The vast qualitative and quantitative analysis of the characteristics of a spherical stress wave expanding in a linear-elastic medium was made. The wavewas generated by pressure p0 = const suddenly created in a spherical cavity of initial radius r0. From the analytical form of the solution to the problem it results that displacement and stresses decrease approximately in inverse proportion to the square and cube of the distance from cavity cen- ter. It was found that the cavity surface and successive spherical sections of the compressiblemediummove in the course of time with damped vibrating motion around their static positions. The remaining characteristics of the wave behave analogously.Material compressibility, represented by Poisson’s ratio ν in this paper, has the main influence on vibration damping. The increase of the parameter ν over 0.4 causes an intense decrease of the dam- ping, and in the limiting case ν = 0.5, i.e. in the incompressible material the damping vanishes completely. The incompressible medium vibrates like a conservativemechanical system of one degree of freedom. Key words: expanding spherical stresswave, isotropic elasticmedium, dyna- mic load 1. Introduction The problem of propagation of the spherical stress wave expanding in a com- pressible isotropic elasticmediumwas solved inWłodarczyk andZielenkiewicz (2009). Linear elasticity theory was used (Achenbach, 1975; Nowacki, 1970), i.e. 762 E. Włodarczyk, M. Zielenkiewicz εr = ∂u ∂r εϕ = εθ = u r v= ∂u ∂t σr =(2µ+λ) ∂u ∂r +2λ u r σϕ =2(µ+λ) u r +λ ∂u ∂r ρ0 ρ =1+ ∂u ∂r +2 u r (1.1) where λ= νE (1+ν)(1−2ν) µ= E 2(1+ν) u – radial displacement of elements of the medium v – radial velocity of the elements σr – radial stress σϕ =σθ – circumferential (tangential) stresses εr – radial strain εϕ = εθ – circumferential (tangential) strains ρ0,ρ – medium densities: initial and disturbed E – Young’s modulus ν – Poisson’s ratio r,t – Lagrangian coordinates. The wave was generated by pressure p(t) dynamically created inside the spherical cavity of initial radius r0 (Fig.1). Fig. 1. A scheme of the boundary value problem All wave parameters were determined with the use of a scalar potential ϕ(x) and its derivatives, namely Analysis of the parameters of a spherical stress wave... 763 u(r,t) = ϕ′(r−r0−at) r − ϕ(r−r0−at) r2 εr = ϕ′′ r −2 ϕ′ r2 +2 ϕ r3 εϕ = ϕ′ r2 − ϕ r3 ρ0 ρ =1+ ϕ′′ r σr =(2µ+λ) ϕ′′ r −4µ ϕ′ r2 +4µ ϕ r3 (1.2) σϕ =λ ϕ′′ r +2µ ϕ′ r2 −2µϕ r3 σϕ−σr =−2µ ϕ′′ r +6µ ϕ′ r2 −6µϕ r3 where a denotes the velocity of elastic stress wave propagation a= √ 1−ν (1+ν)(1−2ν) a0 a0 = √ E ρ0 (1.3) The apostrophes at the symbol ϕ denote derivatives of this function with respect to its argument. The potential ϕ for an arbitrary pressure p(t) has the following form ϕ(x) =−(1+ν) √ 1−2ν E r20 x ∫ 0 p (y−x0 a ) ehy sinωy dy (1.4) where h= 1−2ν 1−ν 1 r0 ­ 0 ω= √ 1−2ν (1−ν)r0 (1.5) For a constant pressure suddenly created in the spherical cavity, i.e. p(t)≡ p0 = const, from expression (1.4), we obtain ϕ(x)=−1+ν 2 r30 p0 E [ 1+ehx( √ 1−2ν sinωx− cosωx) ] ϕ′(x)=−(1+ν) √ 1−2νr20 p0 E ehx sinωx (1.6) ϕ′′(x)=−(1+ν)(1−2ν) 1−ν r0 p0 E ehx( √ 1−2ν sinωx+cosωx) where ωx= √ 1−2ν 1−ν ( r r0 −1− √ 1−ν (1+ν)(1−2ν) a0t r0 ) (1.7) hx= 1−2ν 1−ν ( r r0 −1− √ 1−ν (1+ν)(1−2ν) a0t r0 ) 764 E. Włodarczyk, M. Zielenkiewicz The static parameters of the problem, generated by the pressure p0 stati- cally created on the inside of the cavity, can be determined by the following formulae us(r)= 1+ν 2 p0 E r0 (r0 r )2 εrs(r)=−(1+ν) p0 E (r0 r )3 εϕs(r)= 1+ν 2 p0 E (r0 r )3 σrs(r)=−p0 (r0 r )3 σϕs(r)= p0 2 (r0 r )3 (1.8) In order to simplify the quantitative analysis of the particular parameters of the expanding stresswave, the following dimensionless quantities have been introduced ξ= r r0 η= a0t r0 U = u r0 Us = us r0 V = v a0 P = p0 E R= ρ ρ0 Sr = σr p0 Srs = σrs p0 Sϕ = σϕ p0 Sϕs = σϕs p0 Sz = σϕ−σr p0 S0 = σ0 p0 Szs = σϕs−σrs p0 (1.9) The dimensionless independent variables ξ and η are containedwithin the following intervals 1¬ ξ¬∞ η­ √ (1+ν)(1−2ν) 1−ν (ξ−1) (1.10) Using expressions (1.2) and (1.6), the parameters of the expanding stress wave generated in the linear elastic mediumby pressure p0 = const suddenly created on the inside of the spherical cavity, can be found with the use of dimensionless quantities (1.9) bymeans of the following expressions Analysis of the parameters of a spherical stress wave... 765 U(ξ,η) = 1+ν 2 P ξ2 {1− [ √ 1−2ν(2ξ−1)sinωx+cosωx]ehx} V (ξ,η) = P ξ {[ (1−2ν) √ 1+ν 1−ν − √ 1−ν2 ξ ] sinωx+ + √ (1+ν)(1−2ν) 1−ν cosωx } ehx εr(ξ,η) =−(1+ν)P { 1 ξ3 + [√ 1−2ν (1−2ν 1−ν 1 ξ − 2 ξ2 + 1 ξ3 ) sinωx+ − ( 1 ξ3 − 1−2ν 1−ν 1 ξ ) cosωx ] ehx } (1.11) εϕ(ξ,η)= 1+ν 2 P ξ3 {1− [ √ 1−2ν(2ξ−1)sinωx+cosωx]ehx} R(ξ,η)= [1+εr(ξ,η)+2εϕ(ξ,η)] −1 Sr(ξ,η)=− 1 ξ3 {1+(ξ−1)[ √ 1−2ν(ξ−1)sinωx+(ξ+1)cosωx]ehx} Sϕ(ξ,η) = 1 2ξ3 { 1+ [√ 1−2ν ( − 2ν 1−ν ξ2−2ξ+1 ) sinωx+ − ( 2ν 1−ν ξ2+1 ) cosωx ] ehx } and Sz(ξ,η) =Sϕ(ξ,η)−Sr(ξ,η) = = 1 ξ3 {3 2 + [√ 1−2ν (1−2ν 1−ν ξ2−3ξ+ 3 2 ) sinωx+ (1.12) + (1−2ν 1−ν ξ2− 3 2 ) cosωx ] ehx } where ωx= √ 1−2ν 1−ν (ξ−1)− 1√ 1−ν2 η (1.13) hx= 1−2ν 1−ν (ξ−1)− √ 1−2ν 1−ν2 η According to (1.8) and (1.9), the dimensionless values of static parameters can be expressed with the following formulae 766 E. Włodarczyk, M. Zielenkiewicz Us(ξ)= 1+ν 2 Pξ−2 εrs(ξ)=−(1+ν)Pξ−3 εϕs(ξ)= 1+ν 2 Pξ−3 Srs(ξ)=−ξ−3 Sϕs(ξ)= 1 2 ξ−3 Sϕs(ξ)−Srs(ξ)= 3 2 ξ−3 (1.14) The closed analytical formulae presented above were derived in Włodar- czyk and Zielenkiewicz (2009). They are the basis of quantitative analysis of parameters of the spherical stresswave expanding in the linear isotropic elastic medium presented below. From the introductory analysis of the quoted formulae, it follows that the dynamic values of mechanical parameters of the expanding stress wave generated by constant pressure p0 suddenly created in the spherical cavity in the linear elastic medium, intensively decrease in space with an increase of distance from the system center. On the other hand, in particular spherical sections of themedium, the parameters oscillate around their static values. In a compressible medium, these oscillations decay in the course of time. This results from the spherical divergence of the expanding stress wave. The decay of oscillations is caused by the transport of mechanical energy by the stress wave propagating to successively disturbed regions of the medium. Nowwewill performthedetailed analysis ofwaveparameters.Themedium displacement will be considered first. 2. Analysis of the displacement for an infinite pressure impulse of the intensity p0 = const According to thegeneral remarkspresentedabove, themaximumdisplacement values represented in the dimensionless form by function U(ξ,η) occur on the cavity surface, i.e. for ξ = 1. In this case, expression (1.11)1 can be reduced to the form U(1,η) = 1+ν 2 P [ 1+ (√ 1−2ν sin η√ 1−ν2 −cos η√ 1−2ν ) exp ( − √ 1−2ν 1−ν2 η )] (2.1) From expression (2.1), it follows that function U(1,η) oscillates with dam- pedmotion versus dimensionless time η, and has relative extrema. The abso- lute maximum occurs at η = ηe determined by the first positive solution to Analysis of the parameters of a spherical stress wave... 767 the trigonometric equation, namely tan ( π− ηe√ 1−ν2 ) = √ 1−2ν ν (2.2) As can be directly seen from relations (2.1) and (2.2), for every value of Poisson’s ratio ν, the correspondingmaximumdisplacement of cavity surface Umax = U(1,ηe) can be found. Therefore, alike as in oscillating mechanical systems of one degree of freedom, the dynamic coefficient of loading characte- rising an expanding stress wave can be introduced, namely Ψ(ν)= U(1,ηe) Us (2.3) Values of the coefficient Ψ(ν) for selected Poisson’s ratios ν, are presented in Table 1 and their course versus ν is depicted in Fig.2. Table 1.Values of the dynamic coefficient of loading, Ψ(ν) ν 0.1 0.2 0.3 0.4 0.49 0.5 ηe 1.67 1.79 1.92 2.11 2.49 2.72 U(1,ηe) 0.672 0.746 0.832 0.950 1.242 1.500 Us 0.55 0.60 0.65 0.70 0.745 0.75 Ψ(ν) 1.222 1.244 1.280 1.357 1.667 2.000 As can be seen, the maximum effect of the dynamic load occurs in the in- compressiblemedium, i.e. for ν =0.5.Themediumcompressibility intensively cushions the effect of dynamic load, which can be seen in Fig.2. Fig. 2. Variation of coefficient Ψ(ν) versus Poisson’s ratio ν 768 E. Włodarczyk, M. Zielenkiewicz The exemplary variations of relative displacement (U/P) of the cavity surface (ξ = 1) versus η = a0t/r0 for several values of Poisson’s ratio ν, are depicted in Fig.3. As it turns out, the parameter ν, which is the measure of medium compressibility, has the significant influence on the course of quantity U(1,η)/P versus η. Fig. 3. Variation of relative displacement (U/P) of cavity surface (ξ=1) versus dimensionless time η for selected values of Poisson’s ratio ν We canmark out two ranges of ν in which the vibration of cavity surface is damped in a different manner. Thus, Poisson’s ratio in the range below about0.4 (media compressibility increase) causes an intensedecayof the cavity surface oscillations. For this values of ν, the displacement of cavity surface approaches its static value, i.e. (Us/P) = (1+ν)/2 already in the first cycle of vibration (Fig.3). On the other hand, in the range 0.4 < ν < 0.5, that is in quasi-compressible media, the damping of vibrations is very low. In the limiting case, ν =0.5, i.e. in the incompressiblematerial thedampingvanishes completely and the cavity surface oscillates harmonically around its static position with the constant amplitude (U/P)= 0.75 (Fig.3). Note the abnormal behaviour of the media in the range 0.4 < ν ¬ 0.5. In this case, insignificant increments ∆ν cause a considerable increase in the vibration amplitude of the cavity surface (Fig.4). For example, for the incre- ment ∆ν =0.5−0.4= 0.1, the maximum relative displacement increment is (∆U/P)= 1.5−0.7=0.8. The above-presented graphic analysis of the ratio U(1,η)/P concerns the movement of spherical cavity surface (ξ=1).Theparticular spherical sections for ξ > 1 oscillate analogously with adequately smaller values resulting from the spatial divergence of the stress wave (Fig.5). From this graph, it directly Analysis of the parameters of a spherical stress wave... 769 Fig. 4. Influence of parameter ν on variation of function U(1,η)/P Fig. 5. Spatial graph of function U(ξ,η)/P for ν=0.3 follows that the maximum displacements of particular sections for ν < 0.5 occur in the neighbourhood of the expanding wave front, i.e. near the line ξ−1= √ 1−ν (1+ν)(1−2ν) η 3. Analysis of the stress field for an infinite pressure impulse of the intensity p0 = const In real media (metals, rocks and the like), the limit of elasticity is always finite. The solution obtained for a studied problem is valid only in the elastic 770 E. Włodarczyk, M. Zielenkiewicz range. From this fact results the limitation of the maximum value of pressure created inside the cavity, i.e. p0 ¬ pmax. If p0 > pmax, then in the direct neighbourhood of cavity surface the elastoplastic strains occur in elastoplastic metals or cracks in brittle media (cast iron, rocks). In this range of pressure, the solution quoted in this paper loses its physical sense. Bearing inmind this limitation, wewill performa thorough analysis of the stress field in the studied medium. As it is known, plastic strains in metals are caused by the components of stress deviator. Therefore, it can be assumed that the condition of material plastic flow depends only on the difference of stresses σϕ − σr. Indeed, the expression (σϕ −σr)/2 determines the maximum value of tangential stress. So, according to Tresca’s plasticity condition, and in the case of spherical symmetry – also Huber-Mises-Hencky’s condition, we have σz =σϕ−σr =σ0 (3.1) where σ0 is the dynamic yield point obtained in the tension test of a given material. According to the above-mentioned remarks, in further considerations we will concentrate the main effort on the analysis of relative reduced stress, i.e. Sz =σz/p0 =(σϕ−σr)/p0. As was mentioned, function Sz(ξ,η) reaches its maximum on the cavity surface, i.e. for ξ = 1. According to expressions (1.12) and (1.13) for ξ = 1, function Sz(ξ,η) can be reduced to the form Sz(1,η) = 3 2 + 1+ν 2(1−ν) [√ 1−2ν sin η√ 1−ν2 −cos η√ 1−ν2 ] exp ( − √ 1−2ν 1−ν2 η ) (3.2) From comparison of expressions (2.1) and (3.2), it follows that functions U(1,η) and Sz(1,η) have analogous forms, so Sz(1,η) reaches its absolute maximum for η = ηe, too. The value of ηe is determined by trigonometric equation (2.2). This means that despite rapid pressure rise in the spherical cavity, the reduced stress Sz(1,η) monotonically increases to its maximum value and reaches it after the finite time te = (r0/a0)ηe 6=0. As can be seen, some ”inertia” occurs in the increasing of stress Sz(1,η) to itsmaximumvalue in comparison with the pressure p0 (Fig.6). The variation of quantity Sz(1,η) versus η for selected values of Pois- son’s ratio ν is shown in Fig.6. From comparison of graphs depicted in Fig.3 and Fig.6, it follows that the courses of functions U(1,η) and Sz(1,η) are similar. Analogously to U(1,η), the significant influence of medium compres- Analysis of the parameters of a spherical stress wave... 771 Fig. 6. Variation of relative reduced stress Sz(1,η) versus dimensionless time η for selected values of Poisson’s ratio ν; (1) Sz =1.64, ηe =1.67, (2) Sz =1.68, ηe =1.79, (3) Sz =1.76, ηe =1.92, (4) Sz =1.92, ηe =2.11, (5) Sz =2.47, ηe =2.49, (6) Sz =3.00, ηe =2.72, Szs – static value sibility (ν parameter) on the pulsating variation of Sz(1,η) versus η is no- teworthy. In the compressible media for ν < 0.4, the oscillation of function Sz(1,η) is intensively damped to its static value Szs = 1.5. For example, for ν < 0.3 already after time t = 4r0/a0 we have Sz(1.4) ≈ Szs = 1.5. On the contrary, for ν =0.5, i.e. in the incompressiblemedium, the damping vanishes completely and stress Sz(1,η) harmonically pulsates around its static value Szs = 1.5 with a constant amplitude equal to 1.5. The abnormal influence of ν on the course of function Sz(1,η) is depicted spatially in Fig.7. Fig. 7. Influence of parameter ν on variation of function Sz(1,η) 772 E. Włodarczyk, M. Zielenkiewicz In turn, the variation of stress Sz(ξ,ηi) versus ξ for three selected values of η and ν =0.3 is shown in Fig.8. The intense decrease of stress Sz with an increase in the distance from cavity center is apparent. Fig. 8. Variation of relative reduced stress Sz(ξ,η) versus dimensionless radius ξ for three selected values of dimensionless time η and for ν=0.3; Szs – static value Full spatial graphs of function Sz(ξ,η) for ν = 0.3 and ν = 0.49 are depicted in Fig.9 and Fig.10. The significant influence of ν on the values of stress Sz(ξ,η) is noticeable there. Fig. 9. Spatial graph of function Sz(ξ,η) for ν=0.3 From the analysis of the variations of stress Sz, it results that in the direct neighbourhood of cavity surface, with a pressure p0 high enough, themedium yield pointwill be exceeded. In this range, the solution described in this paper Analysis of the parameters of a spherical stress wave... 773 Fig. 10. Spatial graph of function Sz(ξ,η) for ν=0.49 loses its physical sense. The maximum pressure by which the dynamic yield point is not exceeded in themedium layer directly surrounding the cavity can be determined from the following expression pmax = σ0 Sz(1,ηe) or pmax σ0 = 1 Sz(1,ηe) (3.3) The values of pmax/σ0 ratio for selected values of parameter ν, are pre- sented in Table 2. Table 2.Values of pmax/σ0 ratio for selected ν ν 0.1 0.2 0.3 0.4 0.49 0.5 ηe 1.67 1.79 1.92 2.11 2.49 2.72 Sz(1,ηe) 1.64 1.68 1.76 1.92 2.47 3.00 pmax/σ0 0.61 0.60 0.57 0.52 0.40 0.33 The variation of relative circumferential stress Sϕ(1,η) = σϕ/p0 on the cavity surface versus η for selected values of the parameter ν is shown in Fig.11. The courses are analogous to function Sz(1,η) (Fig.6). Note the fact that in the initial period of cavity expansion in the compressible medium (ν < 0.5) the stress Sϕ(1,η) is negative. The negative value of stress Sϕ propagates at the front of the wave and in its direct neighbourhood, which can be seen in Fig.12. It is the dynamic effect of inertial action of themedium in the direct neighbourhood of strong discontinuity wave front. The course of relative radial stress Sr(ξ,η) versus η for selected values of ξ and ν =0.3 is presented in Fig.13. Stress Sr(ξ,η) reaches its maximum 774 E. Włodarczyk, M. Zielenkiewicz Fig. 11. Variation of relative circumferential stress Sϕ(1,η) versus dimensionless time η for selected values of Poisson’s ratio ν Fig. 12. Variation of relative circumferential stress Sϕ(ξ,η) versus dimensionless time η for selected values of dimensionless radius for ξ= const and ν =0.3 absolute values on the wave front. This value intensively decreases with the increase of the distance from cavity center. Note the fact that in the medium sections sufficiently distant from the cavity surface, behind the front of wave, regions occur in which the medium is radially stretched (Sr(ξ,η) > 0). It is the result of vibrating movement of the compressible medium. At the end of the presented analysis, it can be stated that for the entire range of the parameter ν the material density varies insignificantly (Fig.14). Themaximum increments ∆R do not exceed a few tens per cent. Analysis of the parameters of a spherical stress wave... 775 Fig. 13. Variation of relative radial stress Sr(ξ,η) versus dimensionless time η for selected values of dimensionless radius for ξ= const and ν=0.3 Fig. 14. Variation of relativemedium density R(1,η) versus dimensionless time η for ξ=1 and selected values of Poisson’s ratio ν 4. Final conclusions From the analysis of the studied problem, the following conclusions can be drawn: • From the form of the obtained analytical solution, it results that displa- cements and stresses decrease approximately in inverse proportion to the square and cube of the distance from the cavity center. Therefore, the maximum absolute values of stress wave parameters occur on the cavity surface. It is the result of the spherical divergence of the expandingwave. 776 E. Włodarczyk, M. Zielenkiewicz • The cavity surface and successive spherical sections of the compressi- ble medium move in the course of time with damped vibrating motion around their static positions caused by the pressure p0 created statical- ly inside the cavity. The remaining characteristics of the wave behave analogously. • Material compressibility, represented in this paper by Poisson’s ratio ν, has the main influence on the vibration damping. Poisson’s ratio in the range belowabout 0.4 (media compressibility increase) causes an intense decay of the wave oscillations. For such values of ν, the wave parame- ters approach their static values already in the first cycle of vibration (Fig.3 and Fig.6). On the other hand, in the range 0.4 < ν < 0.5 (quasi-compressiblemedia) the damping of vibrations is very low. In the limiting case ν = 0.5, i.e. in the incompressible material, the damping vanishes completely and the cavity surface pulsates harmonically around its static position with the constant amplitude (U/P)= 0.75 (Fig.3). • The velocity of propagating spherical stress wave in a compressible line- arly elastic medium is an increasing function of Poisson’s ratio ν a= √ (1−ν)E (1+ν)(1−2ν)ρ0 This velocity determines the rate of energy transfer to successive layers of the medium from the loaded cavity surface. In the limiting case, the mediumbecomes incompressible and vibrates like a conservativemecha- nical system with one degree of freedom and the natural frequency ω0 = 2 r0 √ E 3ρ0 Note that the frequency varies in inverse proportion to the radius of cavity. • In the incompressiblemedium (ν =0.5), the parameters of the problem are determined by the following formulae U(ξ,η) = 3 4 P 1 ξ2 ( 1−cos 2√ 3 η ) V (ξ,η) = √ 3 2 P 1 ξ2 sin 2√ 3 η εr(ξ,η) =− 3 2 P 1 ξ3 ( 1−cos 2√ 3 η ) Analysis of the parameters of a spherical stress wave... 777 εϕ(ξ,η) = 3 4 P 1 ξ3 ( 1− cos 2√ 3 η ) R(ξ,η)= 1 Sr(ξ,η) =− 1 ξ3 [ 1+(ξ2−1)cos 2√ 3 η ] Sϕ(ξ,η)= 1 2ξ3 [ 1− (2ξ2+1)cos 2√ 3 η ] Sz(ξ,η) =Sϕ(ξ,η)−Sr(ξ,η) = 3 2ξ3 ( 1− cos 2√ 3 η ) • Thecircumferential stress Sϕ(1,η) on the cavity surfaceduring itsmove- ment increases from the initial negative value. From this it follows that the maximum value of reduced stress Szmax depends on the pressure pulse duration.We will consider this problem in a separate paper. • The results of analyses presented in this paper canbeused, amongothers to investigate spherical ballistic casings. In addition, from our point of view, the results of this analysis are amodest contribution of knowledge to the theory of stress waves propagation in elastic media. References 1. Achenbach J.D., 1975, Wave Propagation in Elastic Solids, North-Holland Publ. Co., American Elsevier, Amsterdam-NewYork 2. Nowacki W., 1970,Theory of Elasticity, PWN,Warszawa [in Polish] 3. WłodarczykE., ZielenkiewiczM., 2009. Influence of elasticmaterial com- pressibility on parameters of the expanding spherical stress wave. I Analytical solution to the problem, Journal of Theoretical and Applied Mechanics, 47, 1, 127-141 Analiza parametrów kulistej fali naprężenia ekspandującej w liniowym sprężystym ośrodku izotropowym Streszczenie Dokonano obszernej jakościowej i ilościowej analizy charakterystyk ekspandującej kulistej fali naprężenia w liniowym sprężystym ośrodku izotropowym. Falę wygene- rowano nagle wytworzonym w kulistej kawernie o początkowym promieniu r0 sta- łym ciśnieniem p0 = const. Z postaci analitycznego rozwiązania problemu wynika, 778 E. Włodarczyk, M. Zielenkiewicz że przemieszczenie i naprężenia maleją w przybliżeniu odwrotnie proporcjonalnie do kwadratu i sześcianuodległości od centrumkawerny.Stwierdzono, że powierzchniaka- werny i kolejne przekroje sferyczne ośrodka ściśliwego przemieszczają się z upływem czasu tłumionymruchemdrgającymwokółprzemieszczenia statycznego.Podobnie za- chowują się pozostałe charakterystyki fali. Na tłumienie drgań decydującywpływma ściśliwość ośrodka, reprezentowanaw pracy przez liczbę Poissona ν.Wzrost parame- tru ν ponad0.4powodujegwałtownyspadek intensywności tłumienia, awgranicznym przypadkudla ν=0.5, tj. w ośrodkunieściśliwym, tłumienie całkowicie zanika.Ośro- dek nieściśliwy drga, jak zachowawczyukładmechaniczny o jednym stopniu swobody. Manuscript received December 23, 2008; accepted for print March 2, 2009