- Email: [email protected]
On the transmission of thermoelastic plane waves across a plane boundary∗
J. Mech.Phys.Solids,1963, Vol. 11,pp. 35to 39. Pergamon PressLtd. Printedin GreatBritain.
ON
THE
TRANSMISSION
WAVES
OF
ACROSS
THERMOELASTIC
A PLANE
PLANE
BOUNDARY*
By H. DERESIEWICZ ColumbiaUniversity, New York
(Received alst July,
1062)
SurdnLutY
AMPLITUDE ratios are computed interface between two thermally
for the four reflected and transmitted conducting
waves generated on the
solids by an incident dilatational
E wave.
Several
interesting special cases are discussed.
1.
INTRODUCTION
DURING the past several years a number of papers have treated the effect of finite thermal conductivity on the propagation of plane elastic waves in the presence of boundaries. For example, reflexion from a plane, traction-free, thermally radiating boundary was studied by DERESIEMCZ (1960, 1962), and the propagation of waves of plane strain along the surface of a half-space, by LOCKETT (lg.!%), CHADWICK (1960) and DERESIEWICZ (1961). As to the problem of transmission across a plane boundary, LESSEN (195’7) set down the pertinent boundary conditions but confined himself to the observation that each of the two possible (normally) incident dilatational waves will generate reflected and transmitted waves of both types. While this is in general true, there are instances (e.g. special values of material constants or of certain combinations of the constants) for which degeneracy arises. The problem cannot, therefore, be said to be solved until quantitative expressions for the amplitude ratios of the displacements are exhibited.
2.
INCIDENT
E
WAVE
We have in mind dilatational waves .propagating in a thermally conducting elastic solid which are transmitted into a contiguous solid having, in general, different elastic and thermal properties. The boundary between the two solids is taken to be an infinite plane and, in order to keep the problem algebraically manageable, we limit our study to normally incident waves. As a result, we may expect generation at the boundary of dilatational waves but not of equivoluminal ones. Accordingly, with the motion parallel to t.he z-axis of a rectangular co-ordinate system, we employ the solution of the field equations of thermoelasticity (for the displacement and temperature deviation from equilibrium) of the fcrm *Support
by (UnitedStates)
National
Science Foundation
85
grant NSF-G21547
is grstefully
acknowledged.
H. DERESIEWICZ
J
where 71 == (L?c)* and 5’ \Ve may
n different That
notr
in passing
dilatational
that
inotlc~
of
q1
6)“.
( 1.1
each of the two potentials. nlotion.
lla\ ing
phase
+j, cwrcsponds
velocity
r,j,
to
muUJJ/fiflij.
is. “,2, g= 2’(‘,
Ccl, h (‘s. It is evident
that
S, and & are associated
with a predominantly
(5)
elastic
mode 01’
motion (E: wa\~c), while S2 and &pertain to a l~rcdominantly thermal mode (‘I’ wn\-~1. W’c shall distinguish the two media b!. assigning to them the indiwx 1 anti 2. The interface into medium from medium
being designated 1, the potential
by the equation
corresponding
z -- 0, with the positi\.v : dirwtetl to a normally incidcllt E: wa\-c ti-ai,clliilg
1 to L’ may be written d,]’ = A exp i h,(” 2.
((ia)
the subscript on the 6 denoting the type of waw and the superscript the medium. .4s noted above, reflected and transmitted motions of both types are to tw expected. Accordingly, w-c write the corresponding potentials in the form (reflected
E)
41’ = A, exp (-
i ?&(I’ Z),
(61,)
(reflected
T)
$2’ = A, exp (-
i S,(l) z),
(tic’)
(transmitted
E)
41’ L= 11, exp ({- i 81(Z)z),
(tit-i)
(transmitted
T)
&’ _- 11, exp ( -:- i 6,(2) 2).
(tic’)
On the transmission of thermoeiastic plane waves across a plane boundary 3.
~4MPLITUIIE
37
KATI0.S
the interface we require continuity of the displacement, traction, temperature and heat flux. With all the motions described by the expressions in (6) present, the displacement in each of t.hc two media will, in accordance with the first of (l), be At
,u (11 =
i [S,(l)
z
u,(2)
(r&i
::= i (glC2) $I’
+
417) 82’2) 4:)
82”
42’1 bw~,
(W
e’“‘.
>
only non-zero component of traction is the normal one,
‘I’&
I
3u
az -%
crz=2p
-+- i j=I
($2
-
w2/2c2%) $j
e(*l , I
where cp2 = p/p. which, for normal incidence, reduces, by virtue of the first of (1) and (2), to
Thus,
The temr~erature deviation, obtained from the second of (l), is
Finally, the heat flux, Q = k aT/&z, where k is the thermal conductivity,
-
S,“‘2
&,a’
-
-
d
is given by
eiwt,
Q.(l)2 (7d)
$1”
We now set P
=
p’2’/p’l’
f
.
E=
,(a,
Ic
z (1)
9
f(2)p,
1~ =
k(2),%(‘)
f
il) - cp)/tp,
(8)
and “j = A&4, and TV being amplitude respectively.
rj
tj = B&4,
j = 1, 2,
ratios of the reflected and transmitted
P-4
potentials,
H. DERESIEWICZ
38
Insertion of (7), (8) and (9) into the boundary the set of algebraic equations ?.i +a,r,
+a,t,
T1 +
r; -
pt, -
r1 +
b, r2 -
conditions
+a,t,= pt, =
(on z = 0) results in
1.
-
1)
(10)
‘DJ-t, - O,fi, =: - 1, 71 + a, 6, y2 + a2 0, kj-ttl + a3 6, kj-tz = 1. in which,
to the approximation
consistent
with Sz (and
1 J
12) < 1, the coefficients
ai, 6, are given by
S2(1)2 -
(W/C~(l))2
=
g1m2
(w/4l))2
2=
glm2
4
b
-
A -
(1 -
i t1-2
Sl(2)2 - (W/CT(2))2 A 621-z[l -
(w/cT(l))2
81(1)2
-
-
522/d2’)], (lib)
*
iev-2
52-2
[l
_
i
(512/&’
-
(rather
medium. The amplitude
than the superscripts.
ratios of the displacements
as elsewhere)
may be obtained
on 5 refer to the from those of t,he
by means of the relations* R, = (Sj(i)/8,‘\‘)
the rj and tj being, final results
of course,
:
1‘1, Z’j =- (Sj’2’/Si’i’) roots of the system
tj,
*P27J2_1’E
P”+l
(1
-+ i) ?‘l’ 8’2’
ZEfV
p 02 -
c.f
. pv +
We record
1
(12) herewith
the
’
0 (max. [Q1112,R21’2]),
>
-_
j = 1. 2,
(10).
pv--1 1 _ (1+ i)/~ +' 1'2'p v2- cf F ___
R, =
522/ew)],
(w/c*(‘))2
and the subscripts
potentials
i (<,“/dl’
-
b, = h?t2)2- (4cT(2))2
2i &2/d”),
F
1 ‘2’
Q 0 (max. [Q11’2, Q2’2]), [ 0, R
=
(1 + i) k ~,(l’ 77(Z) v j- c.f’
_
p.p.-)
2
T,
V
zzz
pv+1 T2 =
PV+1
Cl2 E
1 _ (1 + i)7p’ ?y _pv2 2 cfv pv+l
_.?-
(1 + 4 q”’
cf :
confusion, the subscript
aseociated with T waves.
Ef F ‘72 I ’
T’ + c.f L2 . -__ . --, PV+~
*To obviate posible necessarily T waves
P’
(13)
p’u -- 1 E pz’2 -
for
E
we reiterate that the l’j correspond 1 in (12) pertains to quantities
to transmitted
waves of both types and are not
associated with E waves, the subscript
2 to those
On
the taammieeion of thermoelaeticpI8ne wavea aerosf~a plane boundary.
39
where (14)
The expressions in (13) and (14) yield some interesting information. For example, we note that, in the first of the expressions for R, (the reflected E wave) and in the expression for !!ll (the transmitted E wave), the factors preceding the bracket represent, in each case, the amplitude ratio appropriate to the classicai elastic solution ; the second portion of the bracketed expressions represents the correct,ion terms, these being of 0 (5), or, by virtue of (P), of 0 (Q*). In the classical solution, when the two media in contact have matched impedances (i.e. p,J1)c(l) = pt2)cS(~)), no reflected wave is generated, all of the energy being transmitted into the second medium. Such is not the case here, for, when the impedances are matched (or very nearly matched), the amplitude ratio of the refected wave is non-zero; it is exhibited as the second of the expressions for R,. 1n addition, of course, energy is reflected back in the form of a T wave. From a practical point of view, these quantities will assume importance only at high frequencies, being, again, of O(e). Under certain circumstances, however, the reflected E wave may be totally extinguished. One such case is given in the third of the expressions for RI. It requires, in effect, that the quantities* pcs and /? (1 $- E)/Ebe the same for both media. A second such case can be extracted from the second of the expressions for R,. Again we require that the mechanical impedances match, i.e. that pcs be the same for the two solids ; in addition, F, given by the second of (lb), must vanish, which occurs when EW//~T,, (1 + 6) is the same for both solids. However, since w is the same by virtue of the boundary conditions and T,,, the equilibrium temperature, has been assumed (in the boundary conditions) the same, this condition is seen to be identical with the one discussed above. Several other special cases are of interest. Thus, if the coefficient of thermal expansion, G(,is zero for the first medium (i.e. the one in which the incident E wave originates), a T wave is not reflected, while if u = 0 for the second medium, a T wave is not transmitted. However, in either of these cases, the corrections on both the reflected and the transmitted E waves vanish (to the present order of approximation). Finally, it can be shown that the expressions in (13) reduce to the classical results when either solid is thermally nonconducting. More precisely, if one of the media is nonconducting, the amplitude of the corresponding T wave vanishes identically, but the amplitude of the T wave in the contiguous medium can be shcwn to be of 0 ([?), or 0 (Q), this being also the order of magnitude of the correction on the amplitudes of the two E waves. *These donot,here,representratios ; the subscriptshave merely beenomittedfor convenience. REE.EREN CES hADWICE,P.
1960
DERESIEWICZ, H.
1960 1961 1962 1957 1958
LESSEN, M.
LOCKEXT,F. J.
in Solid Mechnniq Vol. I, pp. 307-312 (North-Holland Publishing Co., Amsterdam). J. Mech. Phys. Solids 8, 164. J. Mech. Phys. Solids 9, 191. J. Mech. Phys. Solids 10, 179. Proc. 9th kt. Congr. Appl. Mech. (Brussels), 7, 154. J. Mech. Phys. Solids 7, 71. Progress
ON
THE
TRANSMISSION
WAVES
OF
ACROSS
THERMOELASTIC
A PLANE
PLANE
BOUNDARY*
By H. DERESIEWICZ ColumbiaUniversity, New York
(Received alst July,
1062)
SurdnLutY
AMPLITUDE ratios are computed interface between two thermally
for the four reflected and transmitted conducting
waves generated on the
solids by an incident dilatational
E wave.
Several
interesting special cases are discussed.
1.
INTRODUCTION
DURING the past several years a number of papers have treated the effect of finite thermal conductivity on the propagation of plane elastic waves in the presence of boundaries. For example, reflexion from a plane, traction-free, thermally radiating boundary was studied by DERESIEMCZ (1960, 1962), and the propagation of waves of plane strain along the surface of a half-space, by LOCKETT (lg.!%), CHADWICK (1960) and DERESIEWICZ (1961). As to the problem of transmission across a plane boundary, LESSEN (195’7) set down the pertinent boundary conditions but confined himself to the observation that each of the two possible (normally) incident dilatational waves will generate reflected and transmitted waves of both types. While this is in general true, there are instances (e.g. special values of material constants or of certain combinations of the constants) for which degeneracy arises. The problem cannot, therefore, be said to be solved until quantitative expressions for the amplitude ratios of the displacements are exhibited.
2.
INCIDENT
E
WAVE
We have in mind dilatational waves .propagating in a thermally conducting elastic solid which are transmitted into a contiguous solid having, in general, different elastic and thermal properties. The boundary between the two solids is taken to be an infinite plane and, in order to keep the problem algebraically manageable, we limit our study to normally incident waves. As a result, we may expect generation at the boundary of dilatational waves but not of equivoluminal ones. Accordingly, with the motion parallel to t.he z-axis of a rectangular co-ordinate system, we employ the solution of the field equations of thermoelasticity (for the displacement and temperature deviation from equilibrium) of the fcrm *Support
by (UnitedStates)
National
Science Foundation
85
grant NSF-G21547
is grstefully
acknowledged.
H. DERESIEWICZ
J
where 71 == (L?c)* and 5’ \Ve may
n different That
notr
in passing
dilatational
that
inotlc~
of
q1
6)“.
( 1.1
each of the two potentials. nlotion.
lla\ ing
phase
+j, cwrcsponds
velocity
r,j,
to
muUJJ/fiflij.
is. “,2, g= 2’(‘,
Ccl, h (‘s. It is evident
that
S, and & are associated
with a predominantly
(5)
elastic
mode 01’
motion (E: wa\~c), while S2 and &pertain to a l~rcdominantly thermal mode (‘I’ wn\-~1. W’c shall distinguish the two media b!. assigning to them the indiwx 1 anti 2. The interface into medium from medium
being designated 1, the potential
by the equation
corresponding
z -- 0, with the positi\.v : dirwtetl to a normally incidcllt E: wa\-c ti-ai,clliilg
1 to L’ may be written d,]’ = A exp i h,(” 2.
((ia)
the subscript on the 6 denoting the type of waw and the superscript the medium. .4s noted above, reflected and transmitted motions of both types are to tw expected. Accordingly, w-c write the corresponding potentials in the form (reflected
E)
41’ = A, exp (-
i ?&(I’ Z),
(61,)
(reflected
T)
$2’ = A, exp (-
i S,(l) z),
(tic’)
(transmitted
E)
41’ L= 11, exp ({- i 81(Z)z),
(tit-i)
(transmitted
T)
&’ _- 11, exp ( -:- i 6,(2) 2).
(tic’)
On the transmission of thermoeiastic plane waves across a plane boundary 3.
~4MPLITUIIE
37
KATI0.S
the interface we require continuity of the displacement, traction, temperature and heat flux. With all the motions described by the expressions in (6) present, the displacement in each of t.hc two media will, in accordance with the first of (l), be At
,u (11 =
i [S,(l)
z
u,(2)
(r&i
::= i (glC2) $I’
+
417) 82’2) 4:)
82”
42’1 bw~,
(W
e’“‘.
>
only non-zero component of traction is the normal one,
‘I’&
I
3u
az -%
crz=2p
-+- i j=I
($2
-
w2/2c2%) $j
e(*l , I
where cp2 = p/p. which, for normal incidence, reduces, by virtue of the first of (1) and (2), to
Thus,
The temr~erature deviation, obtained from the second of (l), is
Finally, the heat flux, Q = k aT/&z, where k is the thermal conductivity,
-
S,“‘2
&,a’
-
-
d
is given by
eiwt,
Q.(l)2 (7d)
$1”
We now set P
=
p’2’/p’l’
f
.
E=
,(a,
Ic
z (1)
9
f(2)p,
1~ =
k(2),%(‘)
f
il) - cp)/tp,
(8)
and “j = A&4, and TV being amplitude respectively.
rj
tj = B&4,
j = 1, 2,
ratios of the reflected and transmitted
P-4
potentials,
H. DERESIEWICZ
38
Insertion of (7), (8) and (9) into the boundary the set of algebraic equations ?.i +a,r,
+a,t,
T1 +
r; -
pt, -
r1 +
b, r2 -
conditions
+a,t,= pt, =
(on z = 0) results in
1.
-
1)
(10)
‘DJ-t, - O,fi, =: - 1, 71 + a, 6, y2 + a2 0, kj-ttl + a3 6, kj-tz = 1. in which,
to the approximation
consistent
with Sz (and
1 J
12) < 1, the coefficients
ai, 6, are given by
S2(1)2 -
(W/C~(l))2
=
g1m2
(w/4l))2
2=
glm2
4
b
-
A -
(1 -
i t1-2
Sl(2)2 - (W/CT(2))2 A 621-z[l -
(w/cT(l))2
81(1)2
-
-
522/d2’)], (lib)
*
iev-2
52-2
[l
_
i
(512/&’
-
(rather
medium. The amplitude
than the superscripts.
ratios of the displacements
as elsewhere)
may be obtained
on 5 refer to the from those of t,he
by means of the relations* R, = (Sj(i)/8,‘\‘)
the rj and tj being, final results
of course,
:
1‘1, Z’j =- (Sj’2’/Si’i’) roots of the system
tj,
*P27J2_1’E
P”+l
(1
-+ i) ?‘l’ 8’2’
ZEfV
p 02 -
c.f
. pv +
We record
1
(12) herewith
the
’
0 (max. [Q1112,R21’2]),
>
-_
j = 1. 2,
(10).
pv--1 1 _ (1+ i)/~ +' 1'2'p v2- cf F ___
R, =
522/ew)],
(w/c*(‘))2
and the subscripts
potentials
i (<,“/dl’
-
b, = h?t2)2- (4cT(2))2
2i &2/d”),
F
1 ‘2’
Q 0 (max. [Q11’2, Q2’2]), [ 0, R
=
(1 + i) k ~,(l’ 77(Z) v j- c.f’
_
p.p.-)
2
T,
V
zzz
pv+1 T2 =
PV+1
Cl2 E
1 _ (1 + i)7p’ ?y _pv2 2 cfv pv+l
_.?-
(1 + 4 q”’
cf :
confusion, the subscript
aseociated with T waves.
Ef F ‘72 I ’
T’ + c.f L2 . -__ . --, PV+~
*To obviate posible necessarily T waves
P’
(13)
p’u -- 1 E pz’2 -
for
E
we reiterate that the l’j correspond 1 in (12) pertains to quantities
to transmitted
waves of both types and are not
associated with E waves, the subscript
2 to those
On
the taammieeion of thermoelaeticpI8ne wavea aerosf~a plane boundary.
39
where (14)
The expressions in (13) and (14) yield some interesting information. For example, we note that, in the first of the expressions for R, (the reflected E wave) and in the expression for !!ll (the transmitted E wave), the factors preceding the bracket represent, in each case, the amplitude ratio appropriate to the classicai elastic solution ; the second portion of the bracketed expressions represents the correct,ion terms, these being of 0 (5), or, by virtue of (P), of 0 (Q*). In the classical solution, when the two media in contact have matched impedances (i.e. p,J1)c(l) = pt2)cS(~)), no reflected wave is generated, all of the energy being transmitted into the second medium. Such is not the case here, for, when the impedances are matched (or very nearly matched), the amplitude ratio of the refected wave is non-zero; it is exhibited as the second of the expressions for R,. 1n addition, of course, energy is reflected back in the form of a T wave. From a practical point of view, these quantities will assume importance only at high frequencies, being, again, of O(e). Under certain circumstances, however, the reflected E wave may be totally extinguished. One such case is given in the third of the expressions for RI. It requires, in effect, that the quantities* pcs and /? (1 $- E)/Ebe the same for both media. A second such case can be extracted from the second of the expressions for R,. Again we require that the mechanical impedances match, i.e. that pcs be the same for the two solids ; in addition, F, given by the second of (lb), must vanish, which occurs when EW//~T,, (1 + 6) is the same for both solids. However, since w is the same by virtue of the boundary conditions and T,,, the equilibrium temperature, has been assumed (in the boundary conditions) the same, this condition is seen to be identical with the one discussed above. Several other special cases are of interest. Thus, if the coefficient of thermal expansion, G(,is zero for the first medium (i.e. the one in which the incident E wave originates), a T wave is not reflected, while if u = 0 for the second medium, a T wave is not transmitted. However, in either of these cases, the corrections on both the reflected and the transmitted E waves vanish (to the present order of approximation). Finally, it can be shown that the expressions in (13) reduce to the classical results when either solid is thermally nonconducting. More precisely, if one of the media is nonconducting, the amplitude of the corresponding T wave vanishes identically, but the amplitude of the T wave in the contiguous medium can be shcwn to be of 0 ([?), or 0 (Q), this being also the order of magnitude of the correction on the amplitudes of the two E waves. *These donot,here,representratios ; the subscriptshave merely beenomittedfor convenience. REE.EREN CES hADWICE,P.
1960
DERESIEWICZ, H.
1960 1961 1962 1957 1958
LESSEN, M.
LOCKEXT,F. J.
in Solid Mechnniq Vol. I, pp. 307-312 (North-Holland Publishing Co., Amsterdam). J. Mech. Phys. Solids 8, 164. J. Mech. Phys. Solids 9, 191. J. Mech. Phys. Solids 10, 179. Proc. 9th kt. Congr. Appl. Mech. (Brussels), 7, 154. J. Mech. Phys. Solids 7, 71. Progress