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Anomalous absorption of spatial shear waves by a thin layer of viscous liquid
0020-7225/91 $3.00+ 0.00 Copyright @ 1991Pergamon Press plc
Int. J. Engng Sci. Vol. 29, No. 3, PP. 265-269, 1991 Printed in Great Britain. All rights reserved
ANOMALOUS ABSORPTION OF SPATIAL SHEAR WAVES BY A THIN LAYER OF VISCOUS LIQUID L. A. CHERNOZATONSKII, S. N. ERMOSHIN D. K. GRAMOTNEV
and
Center of Acoustic Microscopy, Institute of Chemical Physics, Academy of Sciences of the U.S.S.R., Kosygin Str. 4, 117334 Moscow, U.S.S.R. Abstract--It is theoretically shown that the phenomena of anomalously weak reflection and transition of shear acoustic waves may be observed in the studied structure (thin viscous liquid layer sandwiched between two solid halfspaces) when there exists an anomalously large wave absorption by a layer of optimal thickness.
In recent years much attention has been paid to studying the propagation of acoustic waves near solid-liquid interfaces. This is associated with the fact that the presence of the viscosity of liquid in such systems may lead to the appearance of physical effects which have no analogues in the acoustics of solid-state systems or in optics. Among such effects one may mention, for instance, the existence of surface shear waves with a specific law of dispersion [l], anomalous attenuation of Stoneley’s waves at the liquid-solid interface [2], etc. In the present work we shall consider one more such effect, that resides in anomalous absorption of spatial shear waves by a thin layer of viscous liquid, confined between two elastic half-spaces. We shall demonstrate that there exists an optimal frequency-independent value of the layer thickness, at which the absorption of the wave has a sharp maximum (up to complete absorption), the value of which, in its turn, is dependent neither on the frequency nor on the parameters of the liquid. In this case the ratio of the coefficient of the wave passage, corresponding to such optimal value of the layer thickness, to the coefficient of transmission through the interface of two half-spaces in the absence of the layer is a universal constant equal to 0.25. If, furthermore, the parameters of the elastic media are the same, the maximum of the coefficient of absorption (in terms of the layer thickness) and the energy coefficients of reflection and passage corresponding to it are also universal constants, equal to 0.5, 0.25, and 0.25, respectively. The quantitave interpretation of the effect of anomalous absorption is as follows. As is known, the coefficient of reflection of spatial shear waves with shifts perpendicular to the plane of incidence, from the interface of the elastic and liquid half-spaces depends on the dynamic viscosity of the liquid [3] and tends to unity as the frequency of the incident wave diminishes. The difference of this coefficient from unity is associated with the penetration and attenuation of the transverse wave in the liquid at distances on the order of --(~Y/o)~~ from the interface (here Y is the kinematic viscosity of the liquid and o is the frequency). In order to understand why there occurs a sharp increase of the wave absorption if we have a source of liquid, having a thickness h < (~Y/o)*‘~ between two elastic half-spaces, we shall consider the structure presented in Fig. 1. Let at first the angle of the wave incidence be 8 > f$, (0, being the angle of total internal reflection from the interface of the two half-spaces under consideration in the absence of the layer). Then shear vibrations in the half-space “2” (Fig. l), excited by the incident wave due to the coupling through the viscous liquid are improper for it. Therefore, the medium “2” “ resists” the origination of such vibrations in it, and this must lead to an increase of the liquid velocity gradients in the layer as its thickness decreases. This, in turn, brings about an increase in the dissipation in the liquid which is proportional to the square of the velocity gradient [4]. On the other hand, as the thickness of the layer diminishes, the volume, within which the absorption takes place, decreases, and the velocity gradient in the liquid when h+ 0 tends to a finite value which ensures stresses in the liquid, equal to the stresses at the interface of two solid half-spaces in the absence of the layer. This, in its turn, leads to the decrease of the dissipation and to its turning into zero when h-0. The interaction of the two opposite mechanisms discussed gives the optimal value h, to us, at which the dissipation has a maximum and the coefficient of reflection has a minimum. 265
266
et al.
L. A. CHERNOZATONSKII
Fig. 1
If 8 < &, the vibrations of the boundary of the half-space “2”, caused by the wave in the half-space “1” are proper waves, running away from the boundary y = h into the medium “2”. This leads to that the originated vibrations of the boundary under consideration become damped because of flow of the energy away from it, this leading to the reduction of the velocity of its travel and, hence, to an increase of the velocity gradients in the liquid. Therefore, when 8 < f&, there must also exist an optimal value of the layer thickness, at which the absorption is maximum. From the above stated it follows that as the angle of incidence approaches 8,, from the right and from the left the absorption must diminish markedly, since in the case of 8 = &, vibrations of the medium “2” are a proper wave, propagating along the layer. The outflow of the energy from the interface y = h into the depth of the medium “2” is absent and in the case of sufficiently small h : h << (2v/o)“* the vibrations of both interfaces occur in phase and with approximately the same amplitudes; this must lead to an appreciable decrease of the velocity gradient in the liquid and, consequently, of the dissipation as well. If we take into account that for the majority of liquids and of the sound wave frequencies usually employed 0 << &(piv
e)
sin
(1)
where pi and pi are the modulus of shear and the density of the medium “1” (Fig. l), the depth of penetration of transverse waves into the liquid and their length in the liquid are equal, respectively, to: h, = (2v/o)“* and A = 2~r(2~/0)~‘*. Consider the case when the layer thickness are small, when h/h,<<1
(2)
and the frequency is not too great and satisfies the inequality: w << (p2v)%
(3)
ICZI,
where p is the density of the liquid. Under the conditions set forth in (l)-(3) the expressions for the energy coefficients of reflection R and transmission T of the wave through the layer for the case of 8 =G8,~ are frequency-independent and have the form: R
=
GC2h c*
c2
+
v(C2 PY(G
4p2Y2c1c2
T 21
[cl&h
+
pV(c2
+
Cd Cd
’ ’
COS 8 + cl)]*COS
81’
(5)
Absorption of spatial shear waves
267
where C1 = G cos 8, C, = G cos 0,, e1 is the angle of refraction of the wave into the medium “2”, and p2 and p2 are the modulus of shear and the density of the latter. Here the coefficient of transmission is defined as the ratio of the mod&i of the Umov-Poynting vectors in the second and in the first half-spaces. Transition to the case of 8 > 8,-,in formula (4) occurs directly, if we put C2 = ip2y/o, where y2 = K: sin’ 8 - K& and K:,2 = ~~p~,~/p,,~. The spatial wave in the medium “2” becomes transformed into an exponentially decaying one, localized near the interface y = h, so that there is no sense in introducing the coefficient of transmission (5). Substituting the above relationships (4) and (5) into the expression for the coefficient of absorption M = 1 - R - T cos e,jc0s 8,
(6)
which we define as the ratio of the energy absorbed by the unity of the layer surface per unit of time to the y-component of the Umov-Poynting vector in the incident wave, we obtain that the wave dissipation has a maximum
(7)
M, = (1 + C,/C,)-‘, when the value of the layer thickness is h,
=
p(C2
+
(8)
G)IC$z
The coefficients of reflection and transmission in this case have the form R, = (1 + C&)-‘,
(9)
T,= c,c2cose[(c, + c2j2c0s
el]-1
(10)
Thus, in the maximum of the wave dissipation all the three coefficients obtained, namely, (7), (9), and (10) depend neither on the frequency nor the parameters of the liquid, while the coefficients of absorption (7) and reflection (9) depend only on the ratio of the impedances of the two solid half-spaces. At the same time, the thickness of the layer h,, at which the maximum of absorption is attained, is proportional to the dynamic viscosity of the liquid. Comparing expression (10) with the value of the coefficient of transmission To through the interface of two half-spaces in the absence of the layer, we obtain that the ratio T,/T, is a universal constant, not dependent on any material parameters and equal to 0.25. If the parameters of the elastic media are the same (pr = p2, pl = ,u2), the coefficients M,,, = 0.5, R, = T, = 0.25 are also universal constants, dependent neither on the angle of incidence, nor on the parameters of the resilient media or of the liquid, It should be noted that the maximum of the function M(h), which exists at any ratio of the parameters of two solids, in the case of 8 > & does not coincide with the minimum of the coefficient of reflection, which exists only at C, > C, and ho
=
PW,
-
Q/G
(11)
C2,
[in this case Rmin = 0 (see (4))]. But if the angle of incidence 8 > &, then the dissipation maximum apparently coincidences with the minimum of the coefficient of reflection Rmin = [(l + .)1’2 - l]/[(l
+ a)‘” + 11,
(12)
which is attained at the value
fi(J= H(1
+
.y2
where H = PYl(G
cos e),
u=
(~1/~2)2~~~2
e/(c0s2
eO - ~02
e).
Thus, the value Rmin in the case under consideration is independent of the frequency or on the kind of the liquid, and & is proportional to the dynamic viscosity of the liquid. Presented in Fig. 2 are the lines of the coefficient of reflection level vs. the thickness of the layer and the angle of incidence for the fused SiO,-glycerol-Si system (pr = 3.11 - 10” dyne/cm*, p1 = 2.2 g/cm, p = 1.25 g/cm3, Y = 6.8 cm2/s, p2 = 7.9 - 101’ dyne/cm2, p2 = 2.3 g/cm3, 8, = 39.9”). Hatching lines indicate the relationships ho and ho the characteristic
L. A. CHERNOZATONSKII
268
et al.
4
3
2 3
2
.c
1
0
10
50
30
8-8,
(grad)
Fig. 2
values of these thickness being: h,( 8 = O”) = 400 A, A,( 8 = 50”) = 2000 A, h,( 8 = 0”) = 1760 A. Figure 3 illustrates the relationships R(h)/8 = const and T(h)/8 = const for four different values of the angle of incidence. Figure 4 illustrates the relationships R(B)/h = const and T(8)/h = const for different values of the layer thickness. All these relationships are in fair agreement with the physical interpretation of the effect of anomalous absorption, presented above. All the formulas given above can be derived from a consideration of contact of two bodies with liquid friction, the force of which is proportional to the relative velocity of motion of the
0.6
1.0 h
(,urn)
Fig. 3(a)
Fig. 3(b)
269
Absorption of spatial shear waves
8 (grad1
Fig. 4(a)
0
--Y
-
I
c
20
I
40 e
I
60
I
80
igrodl
Fig. 4(b)
surfaces of these solid bodies. Thus, the conditions (l)-(3) are the conditions of applicability of the approximation of contact with friction (with coefficient W = prlfi) in the consideration of passage of acoustic waves through a thin layer of liquid. At the same time, the effect of anomalous absorption takes place also when these conditions are not fulfilled. In conclusion, we would like to point out that the effect of anomalous absorption must take place also for transverse acoustic waves falling perpendicular to the plane of the layer, and for shear waves with shifts, lying in the plane of the wave incidence. It may be used for studying the mechanical properties of small (%1V5 cm3) volumes of liquids, for studying the solid-liquid interface, for example, the effects associated with the formation of a double electric layer or adoption-devotion in liquid, for e%cient transmi~ion of energy and wading-up of thin liquid layers, and also for finding the thicknesses of such layers and, as a ~nsequence, for determining the microrelief and heterogeneities on the surface of a solid.
REFERENCES [l] V. P. PLESSKYand YU. A. TEN, Pis’mu IJZhTF10(5), 296-300 (in Russian) (1984). [2] M. M. VOLKENSmIN, and V. M. LEVIN, Pis’ma v ZKFF l2(24), 1498-1503 (in Russian) (1986). [3] R. MOORE and G. MCSKIMIN, in Phy&al Acowtics (Edited by W. MASON and R. THURSTONE), Vol. 6, p. 203 (Russian transl.). Mir, Moscow (1973). 141 L. D. LANDAU and E. M. LIFSHiTS, Hydr~dynumicr Nauka, Moscow (in Russian) (1986).
&S
29r3-B
Int. J. Engng Sci. Vol. 29, No. 3, PP. 265-269, 1991 Printed in Great Britain. All rights reserved
ANOMALOUS ABSORPTION OF SPATIAL SHEAR WAVES BY A THIN LAYER OF VISCOUS LIQUID L. A. CHERNOZATONSKII, S. N. ERMOSHIN D. K. GRAMOTNEV
and
Center of Acoustic Microscopy, Institute of Chemical Physics, Academy of Sciences of the U.S.S.R., Kosygin Str. 4, 117334 Moscow, U.S.S.R. Abstract--It is theoretically shown that the phenomena of anomalously weak reflection and transition of shear acoustic waves may be observed in the studied structure (thin viscous liquid layer sandwiched between two solid halfspaces) when there exists an anomalously large wave absorption by a layer of optimal thickness.
In recent years much attention has been paid to studying the propagation of acoustic waves near solid-liquid interfaces. This is associated with the fact that the presence of the viscosity of liquid in such systems may lead to the appearance of physical effects which have no analogues in the acoustics of solid-state systems or in optics. Among such effects one may mention, for instance, the existence of surface shear waves with a specific law of dispersion [l], anomalous attenuation of Stoneley’s waves at the liquid-solid interface [2], etc. In the present work we shall consider one more such effect, that resides in anomalous absorption of spatial shear waves by a thin layer of viscous liquid, confined between two elastic half-spaces. We shall demonstrate that there exists an optimal frequency-independent value of the layer thickness, at which the absorption of the wave has a sharp maximum (up to complete absorption), the value of which, in its turn, is dependent neither on the frequency nor on the parameters of the liquid. In this case the ratio of the coefficient of the wave passage, corresponding to such optimal value of the layer thickness, to the coefficient of transmission through the interface of two half-spaces in the absence of the layer is a universal constant equal to 0.25. If, furthermore, the parameters of the elastic media are the same, the maximum of the coefficient of absorption (in terms of the layer thickness) and the energy coefficients of reflection and passage corresponding to it are also universal constants, equal to 0.5, 0.25, and 0.25, respectively. The quantitave interpretation of the effect of anomalous absorption is as follows. As is known, the coefficient of reflection of spatial shear waves with shifts perpendicular to the plane of incidence, from the interface of the elastic and liquid half-spaces depends on the dynamic viscosity of the liquid [3] and tends to unity as the frequency of the incident wave diminishes. The difference of this coefficient from unity is associated with the penetration and attenuation of the transverse wave in the liquid at distances on the order of --(~Y/o)~~ from the interface (here Y is the kinematic viscosity of the liquid and o is the frequency). In order to understand why there occurs a sharp increase of the wave absorption if we have a source of liquid, having a thickness h < (~Y/o)*‘~ between two elastic half-spaces, we shall consider the structure presented in Fig. 1. Let at first the angle of the wave incidence be 8 > f$, (0, being the angle of total internal reflection from the interface of the two half-spaces under consideration in the absence of the layer). Then shear vibrations in the half-space “2” (Fig. l), excited by the incident wave due to the coupling through the viscous liquid are improper for it. Therefore, the medium “2” “ resists” the origination of such vibrations in it, and this must lead to an increase of the liquid velocity gradients in the layer as its thickness decreases. This, in turn, brings about an increase in the dissipation in the liquid which is proportional to the square of the velocity gradient [4]. On the other hand, as the thickness of the layer diminishes, the volume, within which the absorption takes place, decreases, and the velocity gradient in the liquid when h+ 0 tends to a finite value which ensures stresses in the liquid, equal to the stresses at the interface of two solid half-spaces in the absence of the layer. This, in its turn, leads to the decrease of the dissipation and to its turning into zero when h-0. The interaction of the two opposite mechanisms discussed gives the optimal value h, to us, at which the dissipation has a maximum and the coefficient of reflection has a minimum. 265
266
et al.
L. A. CHERNOZATONSKII
Fig. 1
If 8 < &, the vibrations of the boundary of the half-space “2”, caused by the wave in the half-space “1” are proper waves, running away from the boundary y = h into the medium “2”. This leads to that the originated vibrations of the boundary under consideration become damped because of flow of the energy away from it, this leading to the reduction of the velocity of its travel and, hence, to an increase of the velocity gradients in the liquid. Therefore, when 8 < f&, there must also exist an optimal value of the layer thickness, at which the absorption is maximum. From the above stated it follows that as the angle of incidence approaches 8,, from the right and from the left the absorption must diminish markedly, since in the case of 8 = &, vibrations of the medium “2” are a proper wave, propagating along the layer. The outflow of the energy from the interface y = h into the depth of the medium “2” is absent and in the case of sufficiently small h : h << (2v/o)“* the vibrations of both interfaces occur in phase and with approximately the same amplitudes; this must lead to an appreciable decrease of the velocity gradient in the liquid and, consequently, of the dissipation as well. If we take into account that for the majority of liquids and of the sound wave frequencies usually employed 0 << &(piv
e)
sin
(1)
where pi and pi are the modulus of shear and the density of the medium “1” (Fig. l), the depth of penetration of transverse waves into the liquid and their length in the liquid are equal, respectively, to: h, = (2v/o)“* and A = 2~r(2~/0)~‘*. Consider the case when the layer thickness are small, when h/h,<<1
(2)
and the frequency is not too great and satisfies the inequality: w << (p2v)%
(3)
ICZI,
where p is the density of the liquid. Under the conditions set forth in (l)-(3) the expressions for the energy coefficients of reflection R and transmission T of the wave through the layer for the case of 8 =G8,~ are frequency-independent and have the form: R
=
GC2h c*
c2
+
v(C2 PY(G
4p2Y2c1c2
T 21
[cl&h
+
pV(c2
+
Cd Cd
’ ’
COS 8 + cl)]*COS
81’
(5)
Absorption of spatial shear waves
267
where C1 = G cos 8, C, = G cos 0,, e1 is the angle of refraction of the wave into the medium “2”, and p2 and p2 are the modulus of shear and the density of the latter. Here the coefficient of transmission is defined as the ratio of the mod&i of the Umov-Poynting vectors in the second and in the first half-spaces. Transition to the case of 8 > 8,-,in formula (4) occurs directly, if we put C2 = ip2y/o, where y2 = K: sin’ 8 - K& and K:,2 = ~~p~,~/p,,~. The spatial wave in the medium “2” becomes transformed into an exponentially decaying one, localized near the interface y = h, so that there is no sense in introducing the coefficient of transmission (5). Substituting the above relationships (4) and (5) into the expression for the coefficient of absorption M = 1 - R - T cos e,jc0s 8,
(6)
which we define as the ratio of the energy absorbed by the unity of the layer surface per unit of time to the y-component of the Umov-Poynting vector in the incident wave, we obtain that the wave dissipation has a maximum
(7)
M, = (1 + C,/C,)-‘, when the value of the layer thickness is h,
=
p(C2
+
(8)
G)IC$z
The coefficients of reflection and transmission in this case have the form R, = (1 + C&)-‘,
(9)
T,= c,c2cose[(c, + c2j2c0s
el]-1
(10)
Thus, in the maximum of the wave dissipation all the three coefficients obtained, namely, (7), (9), and (10) depend neither on the frequency nor the parameters of the liquid, while the coefficients of absorption (7) and reflection (9) depend only on the ratio of the impedances of the two solid half-spaces. At the same time, the thickness of the layer h,, at which the maximum of absorption is attained, is proportional to the dynamic viscosity of the liquid. Comparing expression (10) with the value of the coefficient of transmission To through the interface of two half-spaces in the absence of the layer, we obtain that the ratio T,/T, is a universal constant, not dependent on any material parameters and equal to 0.25. If the parameters of the elastic media are the same (pr = p2, pl = ,u2), the coefficients M,,, = 0.5, R, = T, = 0.25 are also universal constants, dependent neither on the angle of incidence, nor on the parameters of the resilient media or of the liquid, It should be noted that the maximum of the function M(h), which exists at any ratio of the parameters of two solids, in the case of 8 > & does not coincide with the minimum of the coefficient of reflection, which exists only at C, > C, and ho
=
PW,
-
Q/G
(11)
C2,
[in this case Rmin = 0 (see (4))]. But if the angle of incidence 8 > &, then the dissipation maximum apparently coincidences with the minimum of the coefficient of reflection Rmin = [(l + .)1’2 - l]/[(l
+ a)‘” + 11,
(12)
which is attained at the value
fi(J= H(1
+
.y2
where H = PYl(G
cos e),
u=
(~1/~2)2~~~2
e/(c0s2
eO - ~02
e).
Thus, the value Rmin in the case under consideration is independent of the frequency or on the kind of the liquid, and & is proportional to the dynamic viscosity of the liquid. Presented in Fig. 2 are the lines of the coefficient of reflection level vs. the thickness of the layer and the angle of incidence for the fused SiO,-glycerol-Si system (pr = 3.11 - 10” dyne/cm*, p1 = 2.2 g/cm, p = 1.25 g/cm3, Y = 6.8 cm2/s, p2 = 7.9 - 101’ dyne/cm2, p2 = 2.3 g/cm3, 8, = 39.9”). Hatching lines indicate the relationships ho and ho the characteristic
L. A. CHERNOZATONSKII
268
et al.
4
3
2 3
2
.c
1
0
10
50
30
8-8,
(grad)
Fig. 2
values of these thickness being: h,( 8 = O”) = 400 A, A,( 8 = 50”) = 2000 A, h,( 8 = 0”) = 1760 A. Figure 3 illustrates the relationships R(h)/8 = const and T(h)/8 = const for four different values of the angle of incidence. Figure 4 illustrates the relationships R(B)/h = const and T(8)/h = const for different values of the layer thickness. All these relationships are in fair agreement with the physical interpretation of the effect of anomalous absorption, presented above. All the formulas given above can be derived from a consideration of contact of two bodies with liquid friction, the force of which is proportional to the relative velocity of motion of the
0.6
1.0 h
(,urn)
Fig. 3(a)
Fig. 3(b)
269
Absorption of spatial shear waves
8 (grad1
Fig. 4(a)
0
--Y
-
I
c
20
I
40 e
I
60
I
80
igrodl
Fig. 4(b)
surfaces of these solid bodies. Thus, the conditions (l)-(3) are the conditions of applicability of the approximation of contact with friction (with coefficient W = prlfi) in the consideration of passage of acoustic waves through a thin layer of liquid. At the same time, the effect of anomalous absorption takes place also when these conditions are not fulfilled. In conclusion, we would like to point out that the effect of anomalous absorption must take place also for transverse acoustic waves falling perpendicular to the plane of the layer, and for shear waves with shifts, lying in the plane of the wave incidence. It may be used for studying the mechanical properties of small (%1V5 cm3) volumes of liquids, for studying the solid-liquid interface, for example, the effects associated with the formation of a double electric layer or adoption-devotion in liquid, for e%cient transmi~ion of energy and wading-up of thin liquid layers, and also for finding the thicknesses of such layers and, as a ~nsequence, for determining the microrelief and heterogeneities on the surface of a solid.
REFERENCES [l] V. P. PLESSKYand YU. A. TEN, Pis’mu IJZhTF10(5), 296-300 (in Russian) (1984). [2] M. M. VOLKENSmIN, and V. M. LEVIN, Pis’ma v ZKFF l2(24), 1498-1503 (in Russian) (1986). [3] R. MOORE and G. MCSKIMIN, in Phy&al Acowtics (Edited by W. MASON and R. THURSTONE), Vol. 6, p. 203 (Russian transl.). Mir, Moscow (1973). 141 L. D. LANDAU and E. M. LIFSHiTS, Hydr~dynumicr Nauka, Moscow (in Russian) (1986).
&S
29r3-B