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Unformatted text preview: CHAPTER 12 RAYLEIGH SURFACE WAVES 12.1 LﬂTRODUCTION It has been shown that in unbounded media only two types of waves
(dilatational and shear) can exist. When there is a bounding surface,
however, Aunﬁace Waveé may also occur (recall discussion in previous chapter).
These surface waves, which are similar to gravitational surface waves in
liquids, were first investigated by Lord Rayleigh, and hence are sometimes called Rayﬂeégh wave/s . 12.2 EBOPAGATION OF A WAVE ALONG A FREE SURFACE Consider a plane wave propagating along the x—direction in the semi— infinite body z_3 O. ~ﬂ~ free surface WK Z The motion is to be two—dimensional so that all quantities will be independent of y. 12—2 Recalling the displacement potential representation (10.53) we write 1 \ u=ﬂ+ﬂ i 3x 32 l l __392_§2 w — az 3X (12.1) i V = O a i: (Os—1p,0) : where ¢ = ¢(x,z) and w = ¢(X,z). E The dilatation and rotation are then given by _m m=2
— 3X + az V ¢ :
. _ l.§E._ 2E =_l 2 §
%‘2%z a) 2V1 (ma) ;
(0 =11) =0
x z Using (12.1) in the equations of motion (10.33), gives  3 32¢ 3 32¢
p{8x 2 + 3z 2}
8t at I! a 2 a 2
(K+2U)§;V¢+U§;VW 2 2 (12.3)
8 8 ¢ 8 3 w 8 2 3 2
p{ ——— — ———~—& = (A + 2n) ——v ¢ — u~—— v w
Bz atz 8 3t2 82 3X
If we take
2
3 A + 2 2 2 2
*%=—EAV¢=%V¢
at 02.4)
2
3 w u 2 2 2
~—=—vw=cvw,
at2 p 2
equations (12.3) will be satisfied identically.
Now if we consider harmonic wave motion, we take
¢ = F(z)ei(wt  kX)
(12.5)
¢ = G(Z)ei(wt ~ kx) ’ where k = w/cs, with cS = surface wave speed. Substituting (12.5) into (12.4) gives 2
F"'(z) — (k2 — w—Z) F(z) = 0
C1 (12.6)
2 (n2
G"(z) — (k — —~§) G(z) = 0
C
_2 The solutions of (12.6) are F(z) = Ae qz + BeqZ 12.7
G(Z) = A,e—sz + B,esz , ( )
where
q2 = k2 — 9—— > 0
C1 (12.8)
2
Sz=k2@—2‘>0 ,
C2 amiA,AH BamiB'amaummmnm.
Now on physiéal grounds we must take B and B' to be zero, since for say
energy conservation we cannot have the wave amplitudes increasing without bounds. Hence we have ¢ = Ae—qz + i(wt — kx) (12.9) ¢ = A,e—sz + i(wt — kx) Now the following boundary conditions on the free surface must be satisfied
OZ = TZX = sz = 0 @ Z = 0 (12.10)
Using (10.58) with (12.1) yields
2 2 2
= Eli .§_Q _ 3 w
oz (K + 2n) 2 + A 2 2p 8X82
Sz 8x
T = (2 §39 gig +'§Ey) (12 11)
zx u 8X82 2 2 '
8x 32
T = 0 Zy 123 12—4 Using relations (12.9) and (12.11) in conditions (12.10) give Am +1211)q2 — 11.2] — 2A'uisk = 0
2 2 (12.12)
2qkiA + (s + k )A' = o . Eliminating the ratio A/A' between (12.12) or setting the coefficient determinant to zero, gives
4qusk2 = [(1 + Zu)q2  Ak2](s2 + k2) Squaring both sides of the above equation and simplifying produces the result KS6 — 8KS4 + (24 — l6K2)KSZ + (16162 — 16) = 0 , (12.13)
where 2 s C22k2 C22
2 (12.14) K—z _ C2 = 1~2v
‘ 2 2—2v ‘
C1 Relation (12.13), usually known as the Rayﬂeigh cquafion, represents
a cubic in K82 to determine the surface or Rayleigh wave speed cs. Note
that this wave velocity will be independent 06 the ﬁhequency and is a constant
for a given material. Also, for the plane wave case considered, there will
be no attenuation of the wave along the free surface (x~direction). The amplitude will decay, however, with depth into the solid (z—direction). The attenuation factors q and s are now given by 2
g~—=1I<_2K2.
k2 s
2
E5 = 1 — K 2
k S Note also that this attenuation is frequency dependent, i.e., high frequency waves will attenuate more rapidly with depth than will low frequency waves. 12—5 The displacements u and w follow from (12.9), (12.12) and (12.1) to be u = A(_ike—qz + 29k81 esz)el(wt ~ kx) s2 + k2
2 .
W = A( Sgk 2 e—sz _ qe—qz)e1(wt kx) ,
s + k or taking the real part of the above u = Ak(e_qZ — ~§2ﬂ§—§e—Sz)sin(wt — kx)
S + 1‘ (12.15) —qz Zkz —sz
W = Aq(e — ———— e )cos(wt — kz) . E
2 2 z
s + k g This type of motion is shown schematically in the following figure. —————>—
Instantaneous wave surface Direction of wave propagation Instantaneous particle velocity V Particle path ‘ . 5. Schematic drawing for Rayleigh surface waves. 1
123 MERICAL EXAMPLE FOR v = 2 CASE Consider now the special case of v = %~which implies that A = u. Let us investigate the surface wave behavior in detail. For the case V = %',
K—2 = %~, and the roots of (12.13) are
l
KSZ=4, 2+—Z—~ , 2—3— . (12.16) ‘
/§ ‘5 The first two values of K82 lead to imaginary values for q/k and s/k, and hence must be rejected. The third value gives 12—6 S .9194
(12.17) .9194 c2 7i
II H
(D
O H The displacements follow from (12.15) to be Ak(eO.8475kz _ .5773e0.3933kz) E?
H (12.18)
0.8475kz O.3933kz Aq(e — 1.732e ) . f E
II Graphically, these displacements vary with 2 as shown in the following figure. Finally we also present two generalized figures for cs/cl and cS/c2 as a function of arbitrary Poisson's ratio. 7 0 0'2 0'4 0 0.2 0.4
v v The ratios of cs/cl, 02/0,? CS/Cg for various values of Poisson’s ratio. 12.4 QVE WAVES IN A LAYERED HALFSPACE For the Rayleigh waves examined in the previous section the material motion was only in the x—z plane. It can be shown that surface wave motion
in the y—direction (SH—waves) is impossible in a homogeneous half—space.
However SH surface waves may exist in a layered medium. Consider the layered semi—infinite body 12~7 12—8 with SH—shear waves propagating, i.e. u = w = O
(12.19)
v = v(x,z,t) ,
and with the following boundary conditions:
i
TZy(X,~h9t) = O
v(x,0,t) = V'(X,O,t) (12.20) sz(x,0,t) = r;y(x,0,t) Take the incident, reflected and refracted wave motion to be ik2(xcose + zsine — czt) i.e., the SH—shear wave speed must be higher in the bottom media. Hence for this case w will be imaginary; therefore let w = im, so (12.22) reads
CI cosh m ='— cose ,
c (
l
V = Ale é
. . i
r 1k2(xcos6 — 251n6 — czt) (12.21)
V = A e
2 a
ik'(xcos¢)+zsinw — c't)
, 2 2
v = Be
where k2 = %— , kg = gT. Note primed quantities refer to the semi—infinite
2 2 g
body while unprimed quantities correspond to the layer media. g
Now equations (12.20)2 3 imply that
9
C V
kéicosw = k2 c089 =$> cosw = 2cose . (12.22)
' 2 l
cé E
Consider the case of where E—~cose > 1, which can only be true if cé > c2, E
2 ;
1
i
i
1
i 2 and (12.21)3 then reads —kéz sinhm + ﬂkzx c036 — wt)
V' = Be 12—9 Consequently the displacements in the two media become i(kx + 82 — wt) i(kx — Bz — wt)
e v = A1 + A2e 5
' (12.23) i
— ' k — wt
Vy = Be 8 2 e1( x ) ’
i
where
k = k2 cose , k ='%— 1
CL2 1/2 ‘
B = k2 Sine = k(——§ — l) > O
2 C2
CL 1/2 '
B'=k(l——§) >0 , 5
6'2 Note that we are assuming that C; > CL > oz. The motion in the layer given E
E
by (12.23% is usually referred to as a Love wave.
Next using form (12.23) into the boundary conditions (12.20) yields 1 i e_lBh — A e18h = 0 A1 2 Al + A2 — B = O (12.24)  _ v v =
luB(Al A2) + B H B 0 For a non—trivial solution of (12.24), the determinant of the coefficients must vanish, which gives u'B' + uBtath = O or .
c 2 c 2 C 2 ’
uv 1—1*§+ u if]. tan[kh —Ii§—1]=o (12.25) C2 C2 Hence the solution of (12.25) would give CL = cL(k), which is the bpeed 06 paopagation 06 Love wave4 in the layered media. We then see that Love waves § are diapenéivc, i.e., frequency dependent wave speed. The behavior of cL(k) is shown in the figure below. 12—10 ...
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This note was uploaded on 10/03/2011 for the course MCE 565 taught by Professor Staff during the Spring '11 term at Rhode Island.
 Spring '11
 Staff

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