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Linear models of dissipation whose
is almost frequency independent *)
M . C A PU TO ( * * )
R ic ev ut o il 31 M agg io 1966
SUMMARY . La bo ra to r y exper im en ts and fie ld observa t i ons ind i ca t e
that t l i e
Q
of ma ny non f e r ro ma gne t i c ino rgan i c so li ds i s a lmos t f r equ enc y
ind ep en de nt in the range 10 ' to 10~
2
cps; a l tho ugh no s ing le subs tance has
been inve s t i g a t ed o ve r the en ti r e f r eq uen cy spec t rum. One o f the purposes
o f th is inv es t iga t ion is to f ind the an a ly t ic exp ress ion o f a l inear d iss ipat iv e
m e c h a n i s m w h o s e
Q
i s a lmos t f r equ enc y indepen den t o v e r la rg e f r eq uen cy
ranges . Th i s w i l l be ob ta ined by in t roduc in g f rac t i ona l de r i va t i ve s in the
st ress s t ra in re la t io n .
S ince the a im o f th is research is to a lso con tr ib ute to e luc id at ing the
diss ipat in g me chan ism in the earth f re e modes, w e sha ll t reat the cases o f
d iss ipa t ion in the f ree pur e ly to rs iona l m odes o f a she ll and the pu re ly
radia l v ib ra t io n o f a so l id sphere .
Th e theo ry i s check ed w i th the new va lues de t e rm ined f o r the
Q
of
the sphero ida l f ree mod es o f the earth in the range be twe en 10 and 5 minutes
inte gra ted w i th the Q o f the R a i le gh wa ve s in the range betw een 5 and 0.6
minutes .
An o th er check o f the theo r y is mad e w i th the exp er im en ta l va lues
of the
Q
o f the long i tu din a l w ave s in an a l lum in im i rod, in the range bet -
ween 10~
5
and 10~
3
seconds.
In bo th c l i cks the theo ry represen t s the observed phen omen a ve r y
s a t i s f a c t o r y .
RIASSUNTO. 1 risult at i delle, r icerc he i l i lab ora tor io e del le oss erv a-
z ion i in fen om en i natura l i indica no che il Q d i par ecch i so l id i in orga n ic i
non f e r ro ma gne t i c i ind ip end en te da l le f r equ enz e ne l l ' i n t e rva l l o IO
- 2
, IO
7
c ic l i al seco ndo ; per qu anto nessuna sostanza s ia s ta ta s tudiata in tut to
( * ) Th is pa per was pre sen ted at the 1966 annual me et in g o f A G U in
W a s h i n g t o n D C .
( * * ) De pa r tm en t o f Geophys i cs , Un i ve rs i t y o f B r i t i sh Co lumb ia , Canada .
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384
M . C A P U T O
questo int erv allo di freq ue nze . Tino deg li seopi della presente r icerca
quello di trovare l 'espressione analit ica di un modello di dissipazione l i-
neare in cui Q sia indipendente dalla frequenza in un vasto intervallo di
f requenze . Questo sar ot tenuto introducendo der ivate di ordine f raz ionar io
ne l le re laz ioni f ra s forzo e de formazione.
Poich uno degli scopi di questa r icerca anche di contr ibuire ad una
m iglior com prens ione dei meccan ismi di dissipazione d ell 'energ ia nelle oscil-
lazion i l ibere delia T err a, in questa nota si appliche r la legg e di dissipazione
citata al caso delle oscil lazioni torsionali l ibere di uno strato sfer ico.
La teoria esposta viene poi applicata allo studio dei valor i di Q osservat i
nelle onde di Rayleigh e nelle oscil lazioni sferoidali della Terra.
Un 'altr a app licazion e della teoria fat ta allo studio dei valo r i di Q
osservati nelle onde longitudinali di una sbarra di alluminio.
In entrambe le appl icaz ioni la teor ia rappresenta in maniera soddi-
s facente i f enomeni osservat i .
INTRODUCTION
In a hom ogen eou s iso t r op ic e las t ic fi eld , the e las t ic pro per t ies o f
the subs tance a re spec i f i ed b y a desc r ip t io n o f the s t ra in an d s t r e
sses iu a l i m ite d p or t io n of the f i e ld s ince the s tra ins and stresses ar e
l in ea r l y r e l a t ed by t w o pa r a me t e r s w h i ch de s c r ibe t h e e l a st i c p r ope r t i e s
o f the f i e ld . I f the e las t ic he ld i s not hom ogen eous nor i so t ro p ic the
pro per t ies o f th e f i e ld a re spec i f i ed in a s imi la r m ann er by a la rg e r
num be r o f pa r a me t e r s w h i ch a l s o depend on the pos i t i on .
Th ese per f ec t ly e las t ic f i e lds a re insuf f i c ien t mo de ls f or the d escr ip t -
i on of m an y phys i c a l pheno men a be cause t he y do no t a l l ow t o e xp la in
the d iss ipa t ion o f ene rgy . A mo re com ple t e descr ip t ion o f the ac tua l
e las t ic f i e lds i s ob ta in ed b y in t rod uc ing s t ress -s tra in r e la t ion s wh ich
inc lud e a lso l inear com bin a t io ns o f t im e de r iv a t iv es o f the s t ra in
and th e s tr ess. Th e nu m er ica l coe f f i c i en ts app ear in g in the gene ra l
l in ea r c om b ina t i ons o f h i ghe r o r de r de r i v a t i v e s a r e c a l l e d v is c o -
e las t ic cons tants o f h igh er orde r .
E la s t ic fi e lds desc r ibed b y e las tic cons tants of h igher order ha ve
been d iscussed b y m an y autho rs , [ e . g . see K n o po f f , 1954; Capnto , 1966] .
Knopof f s tud ied the case in wh ich the s t r ess s t ra in r e la t ions a re o f
t he t ype
(l"
L
Tr = I g
hi
grs ei+ 'x e
rs
+ [),
m
g
hi
g e
hi
+
2/u
m
e
rs
) [1]
w he re Xm (Hid. flm are c on stan t , he ob ta in ed a con dit io n fo r these v isco -
e las t ic cons tants o f h igh er ord er an a logous to those ex is t in g f or the
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L I N E A R M O D E L S O F D I S S I P A T I O N ' W II.L SL < 3 I S A L M O S T , E T C .
3 8 5
perfect ly e last ic f ie lds and also proved that in order to have a d iss i -
pa t iv e e last ic f ie ld the stress-strain re lat ions should contain t im e der iv-
at ives o f odd order .
A g en era l izatio n of the relation [1] is
P dm
Trs
=
jm -J
0 '
g' " Ors Chi
+ 2
flm C
rS
]
[2]
W e can general i ze [2 ] to the case wh en the opera t ion - j - ^ is pe r for-
where one can also consider /, and x
m
fun ct ion s o f pos i t ion.
d
m
It
rncd w i th TO as a real n um ber 2 (see ap pe nd ix ) and also fur the r b y
subst i tu t ing the summat ion wi th an integ ra l as fo l l ows
bi b
2
f d
z
f d
z
Tr. = I /i (r, z) --- gt
(jrB
e
hi
dz + 2 / /
2
(r, z) e dz. [3]
a
2
j, i s the rad ial coo rdin ate in a spher ical co ordin ate sys tem .
Re lat io ns [1 ] and [2 ] are a special case o f [3 ] th ey are obt ain ed
by se t t ing
p
f
1
(', 0 =
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386
M . C A P U T O
and the nature of the mot ion depends on the roots o f the fo l lowing
equat ion
rj a
2
p'o q p* -f fj,
a
2
= 0 . [6]
The approximate solut ion of [6 ] , neg lect ing the term in rj, which
w e assu me t o be sm all w it h re spe ct t o /u, is
p
2
=
_
e
the solut ion which takes into account the dissipat ion is
\P\
Zo
a
2
/
7t . . n \)
p =
X |
p. I 1 + j - ^ l ^ - ( co s - s + Sin - * . ) j [ 7]
and the specif ic dissipation is
s i n ^ ,
0
. [8]
Solution of the equations of motion in spherical coordinates
W e shal l fo l lo w the me tho d descr ibed in Caputo [1966]; the opera tor
Oi introd uce d in th at pa per is
ii
Oi = V i l r ) 9]
here, according to the def init ion [3] of the stress-strain relation, these
operators wi l l be
h il
0 1
= L
F L { R
'
Z )
Y P
D Z + 2
h
r
0
2
= f
t
(r,g) dz. [ 1 0 ]
One can see that the m eth od of so lv ing the equat ions of equi l ibr ium
result ing f rom the def in it ion [1 ] o f the stress-strain re lat ion (see Caputo
1966) can be a pplie d also to th e case when th e estress-strain rela-
tion is [3].
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L I N E A R M O D E L S O F D I S S I P A T I O N ' W I I. LS L < 3 I S A L M O S T , E T C .
387
Th e La p la ce t ran s fo r m 8 (a re spher i ca l coord ina tes , d c o l a t i tud e ,=