Linear Model Dissipation Q Almost Frequency Independent

7/23/2019 Linear Model Dissipation Q Almost Frequency Independent

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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].


5/11

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 ,=

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Linear Model Dissipation Q Almost Frequency Independent