Layout 6


From the Somigliana waves to the evanescent waves

Pietro Caloi

Annali di Geofisica, Vol. 25, n. 4, 1972.

ANNALS OF GEOPHYSICS, VOL. 53, N. 1, FEBRUARY 2010

ABSTRACT

The Rayleigh equation has real coefficients; therefore, also the case of
complex conjugated roots may be explained physically. The Author proves
that the Somigliana waves may be formed for Poisson ratio values until
0.30543; for gradually less rigid media, they are missing altogether and
degenerate into evanescent waves.

1. In some previous notes [Caloi 1966a, 1966b, 1967,
1969] I have dealt with the physical interpretation of the
roots of the Rayleigh equation which are above the unit, for
the values of the σ coefficient of Poisson to which
correspond three real roots for the Rayleigh equation. And I
have proved that those roots have an exact physical meaning:
they permit the theoretical interpretation of sizable groups
of seismic oscillations which I named waves of Somigliana.
I found then the limits within which the Somigliana waves
originate, within the real roots above the unit, and I
emphasized how interesting it was to include the study of
such waves into the research of stratifications building up the
Earth's crust.

However, as was already noted by Somigliana in this
third contribution to the propagation of seismic waves
[Somigliana 1918], due to the homogeneity of equations of
motion and to the fact that the Rayleigh equation has real
coefficients, also the case of complex, conjugated roots may
be explained physically with the separation of the real from
the imaginary part of roots.

This is what I am undertaking as follows.

2. First of all, let us try to find analytically the value of
σ separating the real field from the complex field for roots
above the unit.

This value has already been obtained empirically in the
previous note [Caloi 1969].

As is known, the Rayleigh equation is expressed in its
most known form with the usual meaning of symbols
[Caloi 1966a]:

from which, after having made [Caloi 1966a]

follows
(1)

Remembering that                                , we put

whence (1) becomes

(2)

Let us see how the roots of this equation vary when σ
varies between 0 and     , that is for

Now we free (2) from the second degree term in χ. To this
end we put

Equation (2) changes then into

We make

thus obtaining

If p and q are real, we know from the mathematical
analysis that the condition for the three roots of (3) to be real
is expressed in the relation

Considering the values of p and q, we have in our case

59

2 16 1 1 ,v
v

v
v

v
v

2
2

3
2 4

1
2

3
2

2
2

3
2

=- - -c c cm m m

,v v3
2

2
2= |

8 8 3 2 16 1 0 .v
v

v
v

3 2

1
2

2
2

1
2

2
2

+ =| | |- - - -c cm m

2 1
1 2

v
v

1
2

2
2

=
v

v

-
-
^ h

1
2 1

1 ;v
v

1
2

2
2

= =
v

f -
-^ h

2
1 1 .G Gf

8 8 1 2 16 0 .3 2 + + =f f| | |- -^ h

3
8 .= +d|

3
8 6 5 3

16 5 9
28 0 .3 + + =d f d f- -^ `h j

3
8 6 5 , 3

16 5 9
28p q= =f f- -^ `h j

0 .p q3 + + =d d

4 27 0 .
q p32

+ G

2
1



For the roots to be real, we must have

(3)

It is easily found that the value of ε which annulls Δ(ε) is

And since

it follows

which practically coincides with the value previously
obtained (σ = 0,26305). It is in correspondence to this value
that the two roots above the unit of the Rayleigh equation
coincide; in fact one obtains χ2,3= 3.5754 [Caloi 1969].

Therefore, in order to have a real root above the unit
and two complex conjugated roots, it must be

to which corresponds for σ the field of variability:

3. There remained now to calculate a series of complex,
conjugated root couples for σ values within the above limits.

Calculations have been made for the following σ values:
0.265; 0.27; 0.30; 0.305; 0.35; 0.40; 0.50.

The Rayleigh equations pertinent to the above σ values
are:

The corresponding roots are (Table 1):

4. I asked myself whether all σ values in the interval
0.26305 ÷ 0.5, were leading to roots to which correspond
Somigliana waves. Those roots are complex and
conjugated and some of their values have been given
under 3. They are complex, hence also the values of [Caloi
1966a, 1966b],

(4)

Let us indicate a general complex root as follows

The formulas (10) will then be written

The relation

allows to obtain

After the value of σ, included in the above interval, has
been assigned, the Rayleigh equation furnishes the
corresponding couple of complex, conjugated roots. Thus r
and c are obtained and thence tang e1, tang e2 and v3
pertaining to the chosen σ value.

After separating the real part from the imaginary one,
we can thus arrive at the real values of e1 (if existing), of e2
and of v3. Let us indicate the latter

while putting

where the negative value for C is being taken.
On the basis of the r and c values taken from the

previous Table 1 we obtain Table 2 by varying σ as follows:

FROM THE SOMIGLIANA WAVES TO THE EVANESCENT WAVES

60

4 27 3
4 45 28 12 10 .

q p 3
6

3
2 3

2

+ +f f- -^ ^h h; E

( ) 45 28 12 10 0 .2 3= + Gf f fD - -^ ^h h

0.6785=f

1 2
1 . 1=v f-

0.26308=v

4 27 0 ,
q p32

+ =2

0.2630 0.5 .5 1vG

for σ = 0.265
for σ = 0.27
for σ = 0.3
for σ = 0.305
for σ = 0.35
for σ = 0.4
for σ = 0.5

3.1277 χ³ – 25.022 χ² + 59.065 χ – 34.043 = 0
3.1739 χ³ – 25.391 χ² + 60.174 χ – 34.782 = 0
3.50     χ³ – 28.0     χ² + 68.0     χ – 40.0     = 0
3.564   χ³ – 28.512 χ² + 69.536 χ – 41.024 = 0
4.333   χ³ – 34.66   χ² + 88.0     χ – 53.33   = 0
3          χ³ – 24        χ² + 64        χ – 40        = 0

χ³ – 8          χ² + 24        χ – 16        = 0

σ χ1 χ2,3

0.265
0.27
0.3
0.305
0.35
0.4
0.5

0. 8498
0.85125
0.86009
0.86154
0.8740
0.8877
0.9128

3.5752 ± i 0.16221
3.5743 ± i 0.3120
3.5714 ± i 0.7284
3.5690 ± i 0.7896
3.5625 ± i 1.1791
3.5562 ± i 1.5406
3.5436 ± i 2.2302

, tang 1 , tang 1.v v e v
v

e3 2
2

1
1
2

2
2

2
2= = =| | |- -

r i c .2,3 = +|

tang 1 ,

tang 1

e r v
v

i c v
v

e r i c, v v r i c .

2
1

1
2

2
2

1
2

2
2

2
2 3 2

= +

= + = +

-

-

,R i r i cC+ = +

2 , 2R
r c r C r c r

2 22
1

2 2
1

2

= + + = + -` `j j

v v R3 2=l

,v v C3 2=

�σ v2
2/v1

2 tang2 e1 e1 tang
2 e2 e2 R

0.265

0.27

0.3

0.305

0.35

0.4

0.5

0.31973

0.31507

0.2857

0.2805

0.23077

0.16667

0

0.14310

0.12615

0.02035

0.0011045

– 0.1779

– 0.4073

– 1

20˚45´.25

19˚33´

8˚07´

1˚51´

–

–

–

2.5752

2.5743

2.5714

2.5690

2.5625

2.5562

2.5436

58˚04´.25

58˚04´

58˚03´

58˚02´

58˚00´

57˚58´.5

57˚55´

1.8913

1.8924

1.8995

1.9006

1.9125

1.9277

1.99605



61

It follows from the analysis of the Table 2 that the
Somigliana waves may vary for σ values until 0.31; for
gradually less rigid medium they are missing altogether and
degenerate into ordinary transversal waves. The values of
efficient angles, for longitudinal incidence, presuppose
incidences nearing rapidly the right angle. So far as
transversal incidence is concerned, the efficient angles
increase slightly as the rigidity of the medium decreases and
reach a limit angle of incidence of about 32˚ to which
corresponds the total reflection.

Considering as well the complex conjugated roots of the
Rayleigh equation, the efficient angles bringing about
Somigliana waves are the ones corresponding to the roots
for the following field of variability of σ:

although, practically [Caloi 1969], this is reducing to

Anyhow, the use of complex conjugated roots indicates
an enlargement and the limit of the field of variability of the
Poisson coefficient.

Concerning the propagation velocity of the Somigliana
waves (if existing), it is noted that it increases as a increases
(that is as rigidity decreases) and reaches the maximum value

for the limit value of σ = 0,305.
If in the expressions of u1, u2, w1, w2 [Caloi 1969] we put

Φ (x – v3t) under power form

and observing that                          , we will have

where p indicates the pulsation of the oscillation and v3 is a
negative constant which may be considered as the extinction
coefficient of the oscillation in time.

For σ = 0.265, for instance, v3 = – 0.043012·v2, and for σ
= 0.27, v3= – 0.08277·v2; whereas for σ = 0.305, we have v3=
– 0.2076·v2. Therefore, at equal frequencies the propagation
of the Somigliana wave is extinguished more quickly as
rigidity decreases.

5. Let us have a closer look how the Somigliana waves
change when their propagation is in elastic media whose σ
coefficient shows a trend toward the value of 0.30543.

Let us consider the case of transversal incidence, where
we have

If Φ (x – v3t) is not periodical, here applies the relation
[Caloi 1969][Caloi 1969]

χ always differs from 2.

FROM THE SOMIGLIANA WAVES TO THE EVANESCENT WAVES

Figure 1. Examples of C0,1 (T = 688 ab.), C1,2 (T = 34S ab.) and C2,3 (T = 23S ab.) waves - by the author generically indicate as Somigliana waves - record at
Somplago (on the Cavazzo Lake) by a seismograph, with free period of about 120S and optical magnification, in occasion of Alaska earthquake of July
30, 1972 (57˚,ON - 135˚.9W: H = 21.45.11,1 GMT: h = 10 km: M = 7.8) at an epicentral distance of about 8350 kms. For large earthquake (as this Alaska
earthquake) C0,1 waves can affect the outer layer of mantle, from the top of the astenosphere (low-velocity channel) to the Earth's surface (thickness of
about 70 kms).

0 0.31 ,1 1v

0. 0.31 .25 1 1v

1.9006v v3 2= $l

,e v t e3
ip x v t3=U - -^ ^h h

v v i v3 3 3= +l

,x v i v t e e3 3
ip x v t pv t3 3+ = $U - -l^ ^h h6 @

tang 1 , tang 1 .e v
v

e1
1
2

2
2

2

1
2 1 2

= =| |- - -c cm m

2
1

1

1 1
2 .

w
u

w
u

v
v

1 2

0

0

0

0

1
2

2
2

1
2

1
2

=
+

+
a a

|

|

|

-

-

-

c
c

^
m
m

h



FROM THE SOMIGLIANA WAVES TO THE EVANESCENT WAVES

62

The following Table 3 is valid:

As σ tends toward the value 0.30543, α1 is thus tending
toward the infinite. Now, in the expressions of u0, w0 - see (1)
of Caloi 1969 -, α1, the quantity characterizing the
longitudinal component, acts as a denominator in the
relative terms.

Therefore, these annull each other, namely the
contribution of the longitudinal wave in the formation of
the Somigliana wave falls away, and this degenerates into a
trasversal wave.

However for σ = 0.30543, the efficient angle of
incidence of the transversal waves, is 31˚57´.4, which
coincides with the angle of total reflection of the incident
transversal wave. In fact is:

so that

which gives for i2 the above value.
Therefore, the efficient angle for transversal waves

incident at the intersurface with the outer stratification,
approximates in less rigid media the angle of incidence to
which corresponds the total reflection, which is reached, as
could be seen, for σ = 0.30543. Hence, in the field where σ
varies from 0.30543 to 0.5 there are no Somigliana waves,
since for them the total reflection of the incident transversal
waves takes place.

The formation of Somigliana waves requires a
physically finite medium beyond the surface which is hit by
the wave coming from an indefinite medium [Caloi 1967].
When the longitudinal reflected wave vanishes as progressive
ordinary wave, for satisfying the conditions at the
intersurface it is necessary to introduce an evanescent wave.
If we indicate the transversal reflected wave (oscillating, of
course, in the principal plane) by

we will have

where u and w are the horizontal and vertical motion
components.

The resultant of these movements, however, is the
socalled evanescent wave which - as the Rayleigh waves - forces
the reached particle to describe an elliptical motion [Caloi
1955, pages 300-304]. In the variability field

which means that in always more incompressible media the
Somigliana waves (possible only by transversal incidence)
degenerate into evanescent waves.

In conformity to what happens in the propagation of
the light, the velocity of evanescent waves in the second
medium is v2/sini2, where i2 is the angle of incidence.
Practically it coincides, therefore, with Rv2 where R is to be
taken from the Table 2. In case of the limit angle of incidence
(i2 = 31˚58´), is in fact v2/sini2 = 1.8894 v2, equal - at lower
than 1/1000 - to the value of     .

References
Caloi, P. (1955). Ci,j, Annali di Geofisica, VIII, 3, 293-313.
Caloi, P. (1966a). L'equazione di Rayleigh e le onde di
Somigliana. I: La teoria delle onde di Rayleigh, Atti Acc.
Naz. dei Lincei, Classe Scienze fis. mat. e nat., XLI, ser.
VIII, 1-2 Ferie.

Caloi, P. (1966b). L'equazione di Rayleigh e le onde di
Somigliana. II: La teoria di Somigliana: rettifiche,
conseguenze, Atti Acc. Naz. dei Lincei, Classe Scienze fis.
mat. e nat., XLI, ser. VIII, 5.

Caloi, P. (1967). L'equazione di Rayleigh e 1e onde di
Somigliana. III: Le Ci,j sono onde di Somigliana. Loro
importanza per lo studio della crosta terrestre, Atti Acc.
Naz. dei Lincei, Classe Scienze fis. mat. e nat., XLIII, ser.
VIII, 6.

Caloi, P. (1969). L'equazione di Rayleigh e le onde di
Somigliana. IV: Limiti d'insorgenza delle onde di
Somigliana: loro esclusiva formazione nel piano
principale, Atti Acc. Naz. dei Lincei, Classe Scienze fis.
mat. e nat., XLVI, ser. VIII, 1.

Somigliana, C. (1918). Sulla propagazione delle onde
sismiche. Nota III, Atti Acc. Naz. dei Lincei, Classe
Scienze fis. mat. e nat., XXVII, ser. V, 1.

© 2010 by the Istituto Nazionale di Geofisica e Vulcanologia. All rights
reserved.

σ

0.26305
0.265
0.27
0.3
0.305
0.30543
0.4

0.3868
0.3783
0.3552
0.14265
0.033235
0.0
imaginary

1.6048
1.60475
1.60445
1.60355
1.6028
1.6028
1.5988

1v
v

1
2

2
2 1 2

| -c m 1
1

2
|-c m

2 1
1 2 0.28013 ;v

v

1
2

2
2

= =
v

v

-
-
^ h

sin 0.52927 ,i v
v

2
1
2

2
2

= =

sine t xkz=} ~ a-^ h

sin ;

sin ,

u x e t x

w x k e t x

kz

kz

= =

= =

a ~ a

~ a

}

}

d
d

d
d

- -

- -

^
^

h
h

0.30543 0.5 ,G Gv

v3l