ISSN: 2689-7636

Research Article
Open Access Peer-Reviewed

Department of Engineering Seismology, Institute for Earth's Physics, Magurele-Bucharest MG-6, POBox MG-35, Romania

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The long-standing interest in the vibrations of a solid sphere is related to the seismic vibrations of the Earth [1,2]. After a relatively short burst of energy in an earthquake the Earth continues to vibrate freely for a long time. Though with a liquid outer core and a viscous mantle, the Earth is still approximated by a solid sphere. Great progress in studying the vibrations of a homogeneous and isotropic elastic sphere has been made since the early days when Lamb introduced the vector spherical harmonics (Hansen vectors) [3-5]. The relevant eigenfrequencies were computed numerically as early as 1898 [6]. We discuss in this paper a natural simplification of this problem, which arises from the fact that a large radius of the sphere is a natural cutoff. Apart from giving formally the general solution of vibrations generated by the seismic tensorial force, we show that a large radius simplifies appreciably the boundary conditions, leading readily to the estimation of the eigenfrequencies (normal modes). The particular case of a fluid sphere is treated to a larger extent.

As it is well known, the vibrations of the Earth following an earthquake are of great importance in revealing the inner structure of the crust, mantle, and even the inner cores of the Earth. These vibrations imply a large number of modes, usually classified as spheroidal and toroidal, with periods in a wide range from 10^{−3} − 10^{−4}s to hours. They attenuate slowly in time, leading to the thermalization of the residual energy of the earthquake. Usually, they are studied numerically, from the recorded data. We give in this paper a thorough description of the vibrations of a sphere in the limit of a large radius, which simplifies greatly the problem. This simplification allows us to perform analytical calculations to a great extent. First, we present the eigenvibrations for a homogeneous and isotropic elastic solid sphere, with general boundary conditions. Second, we introduce the assumption of a large radius, as appropriate for Earth's vibrations. We show that the toroidal vibrations are easily amenable to analytical calculations, while useful quantitative estimations can be made for the spheroidal vibrations. As a useful example, we include the analysis of the vibrations of a fluid sphere. Further on, we investigate the effects of static and dynamic gravitation, a problem with a higher degree of difficulty. Also, the effects of the Coriolis and centrifugal forces are analyzed, with emphasis on their well-known frequency splitting. Finally, we discuss the possibility of estimating the temperature gained by the Earth, as a consequence of the thermalization energy released in an earthquake.

The elastic vibrations of a homogeneous and isotropic solid are described by the equation

$\mu curl\text{\hspace{0.05em}}curlu-\mathrm{(}\lambda +2\mu \mathrm{)}grad\text{\hspace{0.05em}}divu-\rho {\omega}^{2}u\mathrm{=}F\mathrm{(}\omega \mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(1)}$

where *u* is the local displacement, *ρ* is the density, *µ* and *λ* are the Lame elastic moduli, *ω* is the frequency, and F(*ω*) is the force [7]. The components of the seismic tensorial force are

$${F}_{i}\mathrm{(}\omega \mathrm{)=}{M}_{ij}\mathrm{(}\omega \mathrm{)}{\partial}_{j}\delta \mathrm{(}r-{r}_{0}\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(2)}$$

where M_{ij}(*ω*) is the Fourier transform of the seismic moment, r_{0} is the position of the point where the force is placed and *i, j*, … = 1,2,3 are cartesian labels [8,9]. An equivalent form of equation (1) is

${c}_{2}^{2}\Delta u+\mathrm{(}{c}_{1}^{2}-{c}_{2}^{2}\mathrm{)}grad\text{\hspace{0.05em}}divu+{\omega}^{2}u\mathrm{=}-f\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(3)}$

where
${c}_{1}\mathrm{=}\sqrt{\mathrm{(}\lambda +2\mu \mathrm{)/}\rho}$
is the velocity of the longitudinal elastic waves,
${c}_{2}\mathrm{=}\sqrt{\mu \mathrm{/}\rho}$
is the velocity of the transverse elastic waves, and f(*ω*) = *F*(*ω*) / *ρ* (also, m_{ij}(*ω*) = *M*_{ij} (*ω*) / ρ). As it is well known, equation (3) is separated into two inhomogeneous Helmholtz equations

$\begin{array}{c}{c}_{1}^{2}\Delta \Phi +{\omega}^{2}\Phi \mathrm{=}-\phi \text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{c}_{2}^{2}\Delta A+{\omega}^{2}A\mathrm{=}-h\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\end{array}\text{(4)}$

by
$u\mathrm{=}grad\Phi +curlA\text{,}divh\mathrm{=0}$
,
$f\mathrm{=}grad\phi +curlh\text{,}divh\mathrm{=0}$
, where *ϕ* and *h* are given by
$\Delta \phi \mathrm{=}divf\text{,}\Delta h\mathrm{=}-curlf$
(Helmholtz potentials). We get

$$\begin{array}{c}\phi \mathrm{=}-\frac{1}{4\pi}{m}_{ij}{\partial}_{i}{\partial}_{j}\frac{1}{|r-{r}_{0}|}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{h}_{i}\mathrm{=}\frac{1}{4\pi}{\epsilon}_{ijk}{m}_{kl}{\partial}_{j}{\partial}_{l}\frac{1}{|r-{r}_{0}|}\end{array}\text{(5)}$$

(where *εijk* is the antisymmetric tensor of rank three), such that we are led to consider the equation

${c}^{2}\Delta F+{\omega}^{2}F\mathrm{=}\frac{1}{r}\text{(6)}$

with solution

$F\mathrm{(}r\mathrm{)=}\frac{1-\mathrm{cos}kr}{{\omega}^{2}r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{k}^{2}\mathrm{=}{\omega}^{2}\mathrm{/}{c}^{2}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(7)}$

This solution results immediately from the vibration Green function $G\mathrm{=}-\frac{\mathrm{cos}kr}{4\pi {c}^{2}r}$ of the Helmholtz equation ${c}^{2}\Delta G+{\omega}^{2}G\mathrm{=}\delta \mathrm{(}r\mathrm{).}$ We get a particular solution of equation (3)

$$\begin{array}{c}{u}_{i}^{p}\mathrm{=}\frac{1}{4\pi}{m}_{ij}{\partial}_{j}\Delta {F}_{2}\mathrm{(}|r-{r}_{0}|\mathrm{)}+\\ \\ +\frac{1}{4\pi}{m}_{jk}{\partial}_{i}{\partial}_{j}{\partial}_{k}\left[{F}_{1}\mathrm{(}|r-{r}_{0}|\mathrm{)}-{F}_{2}\mathrm{(}|r-{r}_{0}|\mathrm{)}\right]\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\end{array}\text{(8)}$$

For a fluid, where *c _{2}* = 0 (

$${u}^{p}\mathrm{=}-\frac{m}{4\pi {c}_{1}^{2}}grad\frac{\mathrm{cos}{k}_{1}|r-{r}_{0}|}{|r-{r}_{0}|}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(9)}$$

To apply these results to a sphere we need to use expansions in a series of (orthogonal) vector spherical harmonics, defined by[10]

$\begin{array}{c}{R}_{lm}\mathrm{=}{Y}_{lm}{e}_{r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {S}_{lm}\mathrm{=}\frac{\partial {Y}_{lm}}{\partial \theta}{e}_{\theta}+\frac{1}{\mathrm{sin}\theta}\frac{\partial {Y}_{lm}}{\partial \phi}{e}_{\phi}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {T}_{lm}\mathrm{=}\frac{1}{\mathrm{sin}\theta}\frac{\partial {Y}_{lm}}{\partial \phi}{e}_{\theta}-\frac{\partial {Y}_{lm}}{\partial \theta}{e}_{\phi}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\end{array}\text{(10)}$

*l* ≠ 0, where *Y _{lm}* are spherical harmonics and

${u}^{p}\mathrm{=}{\displaystyle \sum _{lm}}\mathrm{(}{f}_{lm}^{p}{R}_{lm}+{g}_{lm}^{p}{S}_{lm}+{h}_{lm}^{p}{T}_{lm}\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(11)}$

where
${f}_{lm}^{p}$
,
${g}_{lm}^{p}$
and
${h}_{lm}^{p}$
are functions only of the radius *r*. A similar series holds also for the free solution uf of equation (1).

The explicit form of the coefficients
${f}_{lm}^{p}$
,
${g}_{lm}^{p}$
and
${h}_{lm}^{p}$
is extremely cumbersome. We prefer to work formally with equation (1) and series expansions of the full solution u = u^{p} + u^{f} and the force F(*ω*), with coefficients *f _{lm}*,

$\begin{array}{c}{f}^{{\text{'}}^{\prime}}+\frac{2}{r}{f}^{\mathrm{\text{'}}}+\frac{\rho {\omega}^{2}}{\lambda +2\mu}f-\left[2+\frac{\mu l\mathrm{(}l+\mathrm{1)}}{\lambda +2\mu}\right]\frac{1}{{r}^{2}}f+\\ \\ +\frac{\mathrm{(}\lambda +3\mu \mathrm{)}l\mathrm{(}l+\mathrm{1)}}{\mathrm{(}\lambda +2\mu \mathrm{)}{r}^{2}}g-\frac{\mathrm{(}\lambda +\mu \mathrm{)}l\mathrm{(}l+\mathrm{1)}}{\mathrm{(}\lambda +2\mu \mathrm{)}r}{g}^{\mathrm{\text{'}}}\mathrm{=}-\frac{{F}^{r}}{\lambda +2\mu}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {g}^{{\text{'}}^{\prime}}+\frac{2}{r}{g}^{\mathrm{\text{'}}}+\frac{\rho {\omega}^{2}}{\mu}g-\frac{\mathrm{(}\lambda +2\mu \mathrm{)}l\mathrm{(}l+\mathrm{1)}}{\mu {r}^{2}}g+\\ \\ +\frac{\mathrm{2(}\lambda +2\mu \mathrm{)}}{\mu {r}^{2}}f+\frac{\lambda +\mu}{\mu r}{f}^{\mathrm{\text{'}}}\mathrm{=}-\frac{{F}^{s}}{\mu}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {h}^{{\text{'}}^{\prime}}+\frac{2}{r}{h}^{\mathrm{\text{'}}}+\frac{\rho {\omega}^{2}}{\mu}h-\frac{l\mathrm{(}l+\mathrm{1)}}{{r}^{2}}h\mathrm{=}-\frac{{F}^{t}}{\mu}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\end{array}\text{(12)}$

where, for the sake of simplicity, we dropped out the suffixes *lm*.

We turn now to the boundary conditions. The force *P* acting (inwards) on the surface *r = R* of the sphere, where *R* is the radius of the sphere, with the spherical components *P _{α}* = (α =

$\begin{array}{c}2\mu {u}_{\theta r}\mathrm{=}{P}_{\theta}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}2\mu {u}_{\phi r}\mathrm{=}{P}_{\phi}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ 2\mu {u}_{rr}+\lambda divu\mathrm{=}{P}_{r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\end{array}\text{(13)}$

where divu is written in spherical coordinates,

$divu\mathrm{=}{\displaystyle \sum _{lm}}\left[\frac{1}{{r}^{2}}\frac{d}{dr}\mathrm{(}{r}^{2}{f}_{lm}\mathrm{)}-\frac{{g}_{lm}}{r}l\mathrm{(}l+\mathrm{1)}\right]{Y}_{lm}\text{(14)}$

(by using the properties of the vector spherical harmonics equations[10]). We compute the strain tensor *u _{αβ}* in spherical coordinates[7]

$\begin{array}{c}{u}_{rr}\mathrm{=}\frac{\partial {u}_{r}}{\partial r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{u}_{\theta \theta}\mathrm{=}\frac{1}{r}\frac{\partial {u}_{\theta}}{\partial \theta}+\frac{{u}_{r}}{r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{u}_{\phi \phi}\mathrm{=}\frac{1}{r\mathrm{sin}\theta}\frac{\partial {u}_{\phi}}{\partial \phi}+\frac{{u}_{\theta}}{r}\mathrm{cot}\theta +\frac{{u}_{r}}{r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ 2{u}_{\theta \phi}\mathrm{=}\frac{1}{r}\left(\frac{\partial {u}_{\phi}}{\partial \theta}-{u}_{\phi}\mathrm{cot}\theta \right)+\frac{1}{r\mathrm{sin}\theta}\frac{\partial {u}_{\theta}}{\partial \phi}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}2{u}_{r\theta}\mathrm{=}\frac{\partial {u}_{\theta}}{\partial r}-\frac{{u}_{\theta}}{r}+\frac{1}{r}\frac{\partial {u}_{r}}{\partial \theta}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ 2{u}_{\phi r}\mathrm{=}\frac{1}{r\mathrm{sin}\theta}\frac{\partial {u}_{r}}{\partial \phi}+\frac{\partial {u}_{\phi}}{\partial r}-\frac{{u}_{\phi}}{r}\end{array}\text{(15)}$

by using the spherical components

$\begin{array}{c}{u}_{r}\mathrm{=}{\displaystyle \sum _{lm}}{f}_{lm}{Y}_{lm}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {u}_{\theta}\mathrm{=}{\displaystyle \sum _{lm}}{g}_{lm}\frac{\partial {Y}_{lm}}{\partial \theta}+{\displaystyle \sum _{lm}}\frac{{h}_{lm}}{\mathrm{sin}\theta}\frac{\partial {Y}_{lm}}{\partial \phi}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {u}_{\phi}\mathrm{=}{\displaystyle \sum _{lm}}\frac{{g}_{lm}}{\mathrm{sin}\theta}\frac{\partial {Y}_{lm}}{\partial \phi}-{\displaystyle \sum _{lm}}{h}_{lm}\frac{\partial {Y}_{lm}}{\partial \theta}\end{array}\text{(16)}$

of the expansion of the displacement vector and the definition of the vector spherical functions (equations (10)). Similarly, we decompose the force *P* in vector spherical harmonics (with coefficients *P ^{r,s,t}*) and identify its spherical components. The boundary conditions given by equations (13) lead to

$\begin{array}{c}2\mu {f}^{\mathrm{\text{'}}}+\lambda \left[\frac{2}{r}f+{f}^{\mathrm{\text{'}}}-\frac{g}{r}l\mathrm{(}l+\mathrm{1)}\right]{|}_{r\mathrm{=}R}\mathrm{=}{P}^{r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ \mu \left(\frac{g}{r}-{g}^{\mathrm{\text{'}}}-\frac{f}{r}\right){|}_{r\mathrm{=}R}\mathrm{=}-{P}^{s}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ \mu \left(\frac{h}{r}-{h}^{\mathrm{\text{'}}}\right){|}_{r\mathrm{=}R}\mathrm{=}-{P}^{t}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\end{array}\text{(17)}$

where we dropped the subscripts *lm*.

The solutions *f*, *g* and *h* of equations (12) consist of free solutions (solutions of the homogeneous equations (12)) plus particular solutions. The homogeneous third equation (12), which describes toroidal vibrations, is the equation of the spherical Bessel functions
${j}_{l}\mathrm{(}kr\mathrm{)},k\mathrm{=}\sqrt{\rho {\omega}^{2}\mathrm{/}\mu}\mathrm{=}\omega \mathrm{/}{c}_{2}$
. For *F ^{t}* = 0 and

${j}_{l}\mathrm{(}kR\mathrm{)=}kR{j}_{l}^{\mathrm{\text{'}}}\mathrm{(}kR\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(18)}$

this equation has an infinity of solutions *β _{ln}*, labeled by an integer

${\omega}_{ln}\mathrm{=}\frac{{c}_{2}}{R}{\beta}_{ln}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(19)}$

We can get an approximate estimate of the numbers *β _{ln}* by using the asymptotic expression of the spherical Bessel functions[11]

${j}_{l}\mathrm{(}kr\mathrm{)}\simeq \frac{1}{kr}\mathrm{cos}\left[kr-\mathrm{(}l+\mathrm{1)}\frac{\pi}{2}\right]\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}kr\gg 1\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(20)}$

For *kR* ≫ 1 equation (18) becomes

$\mathrm{tan}\left[kR-\mathrm{(}l+\mathrm{1)}\frac{\pi}{2}\right]\mathrm{=}-\frac{2}{kR}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(21)}$

which have the approximate zeroes

${\beta}_{ln}\simeq n\pi +\mathrm{(}l+\mathrm{1)}\frac{\pi}{2}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(22)}$

where *n* is any (large) integer. We can see that the frequencies are dense for large
$R\left(\Delta {\omega}_{ln}\mathrm{=}\pi {c}_{2}\mathrm{/}R\right)$
. The free toroidal solution is a superposition of
${j}_{l}\mathrm{(}{k}_{ln}r\mathrm{)}$
, where
${k}_{ln}\mathrm{=}{\omega}_{ln}\mathrm{/}{c}_{2}\mathrm{=}{\beta}_{ln}\mathrm{/}R$
, with undetermined coefficients.

In general, for *F ^{t} ≠ 0* and

We pass now to the spheroidal components which involve the functions *f and g* in equations (12) and (17). We note that the two coupled equations (12) of for the functions *f* and *g* include Bessel operators for spherical Bessel functions. We can get a simplified picture of these equations for large values of *r*. Indeed, it is easy to see that in the limit *wr* / *c _{1,2}* ≫

$$\begin{array}{c}f\simeq \frac{A}{r}\mathrm{cos}\mathrm{(}\omega r\mathrm{/}{c}_{1}+{\phi}_{1l}\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}g\simeq \frac{B}{r}\mathrm{cos}\mathrm{(}\omega r\mathrm{/}{c}_{2}+{\phi}_{2l}\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\end{array}\text{(23)}$$

where the coefficients *A* and *B* and the phases ϕ_{1l,2l} remain undetermined. In this limit the boundary conditions are
${f}^{\prime}{|}_{r\mathrm{=}R}\mathrm{=}{g}^{\prime}{|}_{r\mathrm{=}R}\mathrm{=0}$
and the eigenfrequencies are given by
$${\omega}_{nl}R\mathrm{/}{c}_{\mathrm{1,2}}+{\phi}_{1l\mathrm{,2}l}\mathrm{=}n\pi $$
, where n is any integer (the roots of the equation
$$\mathrm{sin}\mathrm{(}\omega R\mathrm{/}{c}_{\mathrm{1,2}}+{\phi}_{1l\mathrm{,2}l}\mathrm{)=0}$$
).

The condition *w**r* / *c _{1,2}* ≫

We can see that there are two branches of spheroidal eigenfrequencies (corresponding to the velocities *c _{1,2}*), which are dense (continuous) for large

The numerical solution of equations (12) indicates that the lowest mode (the fundamental mode) is *S _{lm}* with

From studies of propagation of the seismic waves, it was inferred the Earth's solid inner core [23,24] of ≃1000*km* and the outer liquid core of radius radius ≃2000*km*. The inner-outer core discontinuity is called the Bullen, or Lehmann, discontinuity. The temperature of the inner core is radius ≃6000*km* (iron and nickel) and the pressure is ≃1012 *dyn /cm2*. The buoyancy at this boundary could be the source of convection currents that generate the Earth's magnetic field (geodynamo effect). The next layers are a viscous mantle of thickness ≃3000*km* and the solid crust of thickness≃70*km*. The boundary between mantle and crust is known as the Mohorovic discontinuity.

For a fluid sphere, the shear modulus *µ* is zero (*µ*= 0); equations (12) become

$\begin{array}{c}{f}^{{\text{'}}^{\prime}}+\frac{2}{r}{f}^{\mathrm{\text{'}}}+{k}^{2}f-\frac{2}{{r}^{2}}f-\frac{d}{dr}\left[\frac{l\mathrm{(}l+\mathrm{1)}g}{r}\right]\mathrm{=}-\frac{{F}^{r}}{\lambda}\\ \\ \frac{1}{r}{f}^{\mathrm{\text{'}}}+\frac{2}{{r}^{2}}f-\frac{l\mathrm{(}l+\mathrm{1)}}{{r}^{2}}g+{k}^{2}g\mathrm{=}-\frac{{F}^{s}}{\lambda}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\end{array}\text{(24)}$

where *k ^{2} = ρω^{2} / λ = ω^{2} / c^{2}*; the boundary condition reads

$\left[\frac{2}{r}f+{f}^{\mathrm{\text{'}}}-\frac{g}{r}l\mathrm{(}l+\mathrm{1)}\right]{|}_{r\mathrm{=}R}\mathrm{=}\frac{{P}^{r}}{\lambda}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(25)}$

Let us introduce *div u*, given by equation (14), which includes

$d\mathrm{=}{f}^{\mathrm{\text{'}}}+\frac{2}{r}f-\frac{g}{r}l\mathrm{(}l+\mathrm{1)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(26)}$

Then the boundary condition becomes

$d{|}_{r\mathrm{=}R}\mathrm{=}\frac{{P}^{r}}{\lambda}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(27)}$

the second equation (24) reads

$\frac{d}{r}+{k}^{2}g\mathrm{=}-\frac{{F}^{s}}{\lambda}\text{(28)}$

and the first equation (24) is

${d}^{\mathrm{\text{'}}}+{k}^{2}f\mathrm{=}-\frac{{F}^{r}}{\lambda}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(29)}$

Hence, we have

$g\mathrm{=}-\frac{d}{{k}^{2}r}-\frac{{F}^{s}}{\lambda {k}^{2}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}f\mathrm{=}-\frac{{d}^{\mathrm{\text{'}}}}{{k}^{2}}-\frac{{F}^{r}}{\lambda {k}^{2}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(30)}$

Now we introduce these functions in equation (26) and get

${d}^{{\text{'}}^{\prime}}+\frac{2{d}^{\mathrm{\text{'}}}}{r}+{k}^{2}d-\frac{l\mathrm{(}l+\mathrm{1)}}{{r}^{2}}d\mathrm{=}-\frac{{\mathrm{(}{F}^{r}\mathrm{)}}^{\mathrm{\text{'}}}}{\lambda}-\frac{2{F}^{r}}{\lambda r}+\frac{{F}^{s}}{\lambda r}l\mathrm{(}l+\mathrm{1)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(31)}$

For free vibrations this is the Bessel equation for spherical Bessel functions
$d\mathrm{=}{j}_{l}\mathrm{(}kr\mathrm{)}$
; the boundary condition (27) leads to the eigenfrequencies
${\omega}_{ln}\mathrm{=(}c\mathrm{/}R\mathrm{)}{\beta}_{ln}$
,
${j}_{l}\mathrm{(}{\beta}_{ln}\mathrm{)=0}$
. In a fluid, we have only pressure *p*, and the stress tensor is
${\sigma}_{ij}\mathrm{=}-p{\delta}_{ij}$
(
${\sigma}_{ij}\mathrm{=2}\mu {u}_{ij}+\lambda {u}_{kk}{\delta}_{ij}$
with
$\mu \mathrm{=0}$
); therefore, for a fluid
$p\mathrm{=}-\lambda {u}_{ii}\mathrm{=}-\lambda divu$
; the equations written above for *d* are in fact equations for the pressure *p*. It is convenient to introduce the decomposition in Helmholtz potentials
$u\mathrm{=}grad\Phi +curlA\mathrm{,}$
$divA\mathrm{=0}$
and
$F\mathrm{=}grad\phi +curlh$
$divh\mathrm{=0}$
, ; then,
$p\mathrm{=}-\lambda \Delta \Phi $
and the equation of motion
$\rho \ddot{u}\mathrm{=}\lambda grad\cdot divu+F\mathrm{=}-gradp+F$
becomes
$\rho \ddot{\Phi}\mathrm{=}\lambda \Delta \Phi +\phi $
, where the potential
$\phi $
is given by
$\Delta \phi \mathrm{=}divF$
and
$h\mathrm{=0}$
,
$A\mathrm{=0}$
. For vibrations this equation reads
${c}^{2}\Delta \Phi +{\omega}^{2}\Phi \mathrm{=}-\frac{1}{\rho}\phi $
and for
$F\mathrm{=}-Mgrad\delta \mathrm{(}r-{r}_{0}\mathrm{)}$
we get
$\phi \mathrm{=}-M\delta \mathrm{(}r-{r}_{0}\mathrm{)}$
,
$\Phi \mathrm{=}-m\frac{\mathrm{cos}k|r-{r}_{0}|}{4\pi {c}^{2}|r-{r}_{0}|}$
(
$m\mathrm{=}M\mathrm{/}\rho $
) and the solution
$${u}^{p}$$
given by equation (9).

A gravitational force

$FdV\mathrm{=}G\rho dV\frac{m}{{r}^{2}}\mathrm{=}\frac{4\pi}{3}G{\rho}^{2}rdV\text{(32)}$

acts upon a volume element *dV* placed at a distance *r* from the center of a sphere, where *G* = 6.67 ´ 10-8 *cm*3/ g × s2 is the universal constant of gravitation, *r* is the density of the sphere (assumed incompressible), and *m* = (4p / 3)*r**r*3 is the mass of the sphere with radius *r*. If the sphere is compressible, the gravitational potential *j* is given by the Poisson equation Dj = 4*p**G**r* and the gravitational force per unit mass is F = -*grad**j*; the condition of (hydrostatic) equilibrium (for a non-rotating sphere) reads *grad**r* = *r*F = -*r**grad**j*, such that *div*[(*grad**r*)/*r*] = -4*p**G**r*; the dependence of the pressure on the density is given by the equation of state; for a constant density the pressure for a self-gravitating sphere of radius *R* at rest with free surface is *p* = (2*p**/3*)* G**r**2*(*R*2-*r*2) (it seems that the pressure in the inner Earth's (solid) core is ≃300*GPa* = 3 ´ 1012*dyn/cm2*). Making use of equation (32), the equation of the elastic motion reads

$\rho \ddot{u}-\mu \Delta u-\mathrm{(}\lambda +\mu \mathrm{)}grad\text{\hspace{0.05em}}divu\mathrm{=}F\mathrm{=}-\gamma r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(33)}$

where $\gamma \mathrm{=(4}\pi \mathrm{/3)}G{\rho}^{2}$ . Since ${Y}_{00}\mathrm{=1/}\sqrt{4\pi}$ we may write

$F\mathrm{=}-\gamma r\mathrm{=}-\sqrt{4\pi}\gamma r{Y}_{00}{e}_{r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(34)}$

whence we can see that **F** has an expansion is series of spheroidal and toroidal functions with all the coefficients zero, except the coefficient
${F}_{00}^{r}\mathrm{=}-\sqrt{4\pi}\gamma r$
of the function ; it follows that the motion may include all the eigenmodes

$\begin{array}{c}{f}^{{\text{'}}^{\prime}}+\frac{2}{r}{f}^{\mathrm{\text{'}}}-\frac{2}{{r}^{2}}f\mathrm{=}\frac{\sqrt{4\pi}\gamma}{\lambda +2\mu}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\end{array}\text{(35)}$

It is easy to see that a particular solution of this equation is
$\mathrm{[}\sqrt{4\pi}\gamma \mathrm{/10(2}\mu +\lambda \mathrm{)]}{r}^{3}$
, while the homogeneous part of this equation has the solution *C _{1}r +C_{2}/ r^{2}*, where

${u}_{r}\mathrm{=}A{r}^{3}+{C}_{1}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}A\mathrm{=}\frac{\gamma}{\mathrm{10(2}\mu +\lambda \mathrm{)}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(36)}$

This solution must satisfy the boundary conditions at the surface of the sphere; making use of equations (16), we have the strain tensor ${u}_{rr}\mathrm{=}{u}_{r}^{\mathrm{\text{'}}}$ and ${u}_{\theta \theta}\mathrm{=}{u}_{\phi \phi}\mathrm{=}{u}_{r}\mathrm{/}r$ ; the force on the surface is $-{\sigma}_{\alpha r}{|}_{R}$ , where the stress tensor is given by ${\sigma}_{\alpha \beta}\mathrm{=2}\mu {u}_{\alpha \beta}+\lambda {u}_{\gamma \gamma}{\delta}_{\alpha \beta}$ ; for a free surface we get the boundary condition

$\mathrm{(2}\mu +\lambda \mathrm{)}{u}_{r}^{\mathrm{\text{'}}}+2\lambda \frac{{u}_{r}}{r}{|}_{r\mathrm{=}R}\mathrm{=0}\text{(37)}$

( ${\sigma}_{\alpha r}{|}_{R}\mathrm{=0}$ ), whence we determine the constant ${C}_{1}\mathrm{=}-\mathrm{[(6}\mu +5\lambda \mathrm{)/(2}\mu +3\lambda \mathrm{)]}A{R}^{2}$ and, finally, the radial displacement

${u}_{r}\mathrm{=}Ar\left({r}^{2}-\frac{6\mu +5\lambda}{2\mu +3\lambda}{R}^{2}\right)\mathrm{=}\frac{\gamma}{\mathrm{10(2}\mu +\lambda \mathrm{)}}r\left({r}^{2}-\frac{6\mu +5\lambda}{2\mu +3\lambda}{R}^{2}\right)\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(38)}$

we note that the radial displacement *u*, is negative, as expected. It is worth estimating the radial displacement at the surface due to gravitation

${u}_{r}{|}_{r\mathrm{=}R}\mathrm{=}-\frac{\gamma}{\mathrm{5(2}\mu +3\lambda \mathrm{)}}{R}^{3}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(39)}$

making use of *ρ* = 5*g* /*cm ^{3}*, λ,µ ≃ 10

$$\frac{\delta \rho}{{\rho}_{0}}\mathrm{=}A\mathrm{(3}\alpha {R}^{2}-5{r}^{2}\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\alpha \mathrm{=}\frac{6\mu +5\lambda}{2\mu +3\lambda}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(40)}$$

which is of the order unity. The proper estimation of the static effect of the self-gravitational field on the elastic sphere is to solve simultaneously the equation of elastic equilibrium (33) with F = -*r**grad**j* and the Poisson equation for the gravitational field *j*, D*j* =4*p**G**r*. With spherical symmetry we have

$F\mathrm{=}-\frac{4\pi}{3{r}^{2}}G\rho {\displaystyle {\int}_{\mathrm{0<}{r}^{\mathrm{\text{'}}}\mathrm{<}r}}d{r}^{\mathrm{\text{'}}}\rho \frac{r}{r}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(41)}$

the Poisson equation for the gravitational potential may be written as D(F/*r*) = -4*p**Ggrad**r*, such that the problem involves two equations and unknowns, *u* and *r*. Since this is a more difficult problem it is preferable to consider the density *r* as an empirically known function of *r* (a parametrization in powers of *r* can be used for *r* and a variational approach can be applied to the problem). Even so, the equations governing the influence of the gravitational field upon the elasticity of a self-gravitating sphere are difficult.

Let us assume a spheric, non-rotating, homogeneous, elastic Earth at equilibrium under the action of its gravitational field; we consider small elastic deformations of this equilibrium state; in the first approximation, we have a small change denoted by *K* in the gravitational potential as a consequence of the small changes in density *div(ρu)*, i.e., we have

$\Delta K\mathrm{=}-4\pi Gdiv\mathrm{(}\rho u\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(42)}$

where ρ is a known function of r. The equation of elastic motion reads

$\rho \ddot{u}-\mu \Delta u-\mathrm{(}\lambda +\mu \mathrm{)}grad\text{\hspace{0.05em}}divu\mathrm{=}-\rho gradK\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(43)}$

These two coupled (vectorial) equations are difficult to be treated by an analytical method, due to the non-uniformity of the density. For a uniform density, taking the *div* in equation (43) and using equation (42) we get for D = *div u*

$\rho \ddot{D}-\mathrm{(}\lambda +2\mu \mathrm{)}\Delta D\mathrm{=4}\pi G{\rho}^{2}D\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(44)}$

an equation which indicates that the frequency ω changes by

$\Delta \mathrm{(}{\omega}^{2}\mathrm{)=}-4\pi G\rho \text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(45)}$

for frequencies as low as ω = 10^{−4} S^{−1} the variation given by equation (45) is large. Let us use the Helmholtz decomposition
$u\mathrm{=}grad\Phi +curlA$
div**A** = 0; then, from equation (42) we have K = −4πGρΦ and from equation (43) we get
$\Delta \Phi +{k}_{1}^{2}\Phi \mathrm{=0},\Delta A+{k}_{2}^{2}A\mathrm{=0}$
. These are the same equations as those which hold in the absence of the gravitational field, except that
${k}_{1}^{2}$
is changed into
${k}_{1}^{2}\to {k}_{1}^{2}+4\pi \rho G\mathrm{/}{c}_{1}^{2}$
. Moreover, we can see that only the spheroidal modes are affected by gravitation (since *divT _{lm}* = 0). It follows that the spheroidal frequencies (i.e., the branches ω

Rotation effect

If a vector * a* rotates, its change is
$\delta a\text{}+\alpha \times a$
, where
$\delta \alpha $
is the infinitesimal rotation angle; therefore, its velocity is
$$\dot{a}+\Omega \times a$$
, where

$\rho \ddot{u}+2\rho \Omega \times \dot{u}\mathrm{=}F\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(46)}$

where * F* includes the elastic force (

In the absence of the Coriolis force in equation (46) we decompose the force * F* and the displacement

$\int}dr\cdot {r}^{2}{h}_{l}^{\mathrm{(}n\mathrm{)}}\mathrm{(}r\mathrm{)}{h}_{l}^{\mathrm{(}{n}^{\mathrm{\text{'}}}\mathrm{)}}\mathrm{(}r\mathrm{)=}{\delta}_{n{n}^{\mathrm{\text{'}}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(47)$

we recall that the toroidal functions are orthogonal, *i.e*.

$\int}do{T}_{lm}{T}_{{l}^{\mathrm{\text{'}}}{m}^{\mathrm{\text{'}}}}^{\mathrm{*}}\mathrm{=}{\delta}_{l{l}^{\mathrm{\text{'}}}}{\delta}_{m{m}^{\mathrm{\text{'}}}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(48)$

Since Ω/*ω _{ln << 1}* we solve equation (46) by a perturbation-theory method. First, we drop the labels l,m and n and use the notations
${u}_{lm}^{\mathrm{(}n\mathrm{)}}\mathrm{=}{u}_{0}$
,

$u\mathrm{=}{u}_{0}+\frac{\Omega}{{\omega}_{0}}{u}_{1}+\mathrm{...}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(49)}$

where *u _{1}* to be determined, is assumed orthogonal on

$\int}dr{u}_{1}{u}_{0}\mathrm{=0}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(50)$

A similar series is valid for the frequency

$\omega \mathrm{=}{\omega}_{0}+\frac{\Omega}{{\omega}_{0}}{\omega}_{1}+\mathrm{...}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(51)}$

Introducing these series in equation (46), with time Fourier transforms, we get

$\begin{array}{c}-\rho {\omega}_{0}^{2}{u}_{0}\mathrm{=}F\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ -\rho {\omega}_{0}\Omega {u}_{1}-2\rho \Omega {\omega}_{1}{u}_{0}-2i\rho {\omega}_{0}\Omega \times {u}_{0}\mathrm{=0}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\end{array}\text{(52)}$

the first equation (52) defines the function *u _{0}*; in the second equation (52) we take the scalar product with

${\omega}_{1}\mathrm{=}-\frac{i{\omega}_{0}}{l\mathrm{(}l+\mathrm{1)}}{\displaystyle \int}dr{e}_{z}\mathrm{(}{u}_{0}\times {u}_{0}^{\mathrm{*}}\mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(53)}$

where we put **Ω** = Ωe_{z}, e_{z} being the unit vector along the *z* -axis. Here we use e_{z} = cosθe_{r} − sinθe_{θ},
${u}_{0}\mathrm{=}{h}_{l}^{\mathrm{(}n\mathrm{)}}{T}_{lm}$
and *T _{lm}* from equations (10); we get immediately

${\omega}_{1}\mathrm{=}{\omega}_{0}\frac{m}{l\mathrm{(}l+\mathrm{1)}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(54)}$

where *m* denotes all integers from -*l* to* l*. It follows that the frequencies *w**ln*, which are degenerate with respect to *m*, are split into 2* l* + *l* branches

${\omega}_{ln}\to {\omega}_{ln}+\Omega \frac{m}{l\mathrm{(}l+\mathrm{1)}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(55)}$

using *ω _{1}* thus determined, we can get

The equation of the elastic motion for a body in rotation with a (constant) angular velocity Ω reads

$$\rho \ddot{u}+2\rho \Omega \times \dot{u}+\rho \Omega \times \mathrm{[}\Omega \times (r+u\mathrm{)]=}\mu \Delta u+\mathrm{(}\lambda +\mu \mathrm{)}grad\text{\hspace{0.05em}}divu+F\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(56)}$$

where F is an external force. We note that the centrifugal term ρ**Ω**×(**Ω**×r) is static, so we can write it as

${F}_{c}\mathrm{=}\rho \Omega \mathrm{(}\Omega \text{r}\mathrm{)}-\rho {\Omega}^{2}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(57)}$

where we denoted by * F_{c}* the centrifugal force and removed any other external force (F=0); we may neglect

${Y}_{00}\mathrm{=}\frac{1}{\sqrt{4\pi}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{Y}_{20}\mathrm{=}\sqrt{\frac{5}{16\pi}}\mathrm{(1}-3{{\displaystyle \mathrm{cos}}}^{2}\theta \mathrm{)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(58)}$

it is easy to see that we can write *F _{c}* as a series expansion

${F}_{c}\mathrm{=}-\rho {\Omega}^{2}r\left(\alpha {R}_{00}+2\beta {R}_{20}-\beta {S}_{20}\right)\text{(59)}$

in spheroidal functions, where
$\alpha \mathrm{=2}\sqrt{4\pi}\mathrm{/3}$
and
$\beta \mathrm{=}\sqrt{16\pi \mathrm{/5}}$
. We seek a similar expansion for the displacement *u*,

$u\mathrm{=}{f}_{1}{R}_{00}+{f}_{2}{R}_{20}+g{S}_{20}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(60)}$

equations (12) lead to

$\begin{array}{c}{f}_{1}^{{\text{'}}^{\prime}}+\frac{2}{r}{f}^{\mathrm{\text{'}}}-\frac{2}{{r}^{2}}{f}_{1}\mathrm{=}-\frac{\rho {\Omega}^{2}\alpha}{\lambda +2\mu}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {f}_{2}^{{\text{'}}^{\prime}}+\frac{2}{r}{f}_{2}^{\mathrm{\text{'}}}-\frac{\mathrm{2(}\lambda +5\mu \mathrm{)}}{\lambda +2\mu}\frac{1}{{r}^{2}}{f}_{2}+\frac{\mathrm{6(}\lambda +3\mu \mathrm{)}}{\lambda +2\mu}\frac{1}{{r}^{2}}g-\frac{\mathrm{6(}\lambda +\mu \mathrm{)}}{\lambda +2\mu}\frac{1}{r}{g}^{\mathrm{\text{'}}}\mathrm{=}-\frac{2\rho {\Omega}^{2}\beta}{\lambda +2\mu}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ {g}^{{\text{'}}^{\prime}}+\frac{2}{r}{g}^{\mathrm{\text{'}}}-\frac{\mathrm{6(}\lambda +2\mu \mathrm{)}}{\mu}\frac{1}{{r}^{2}}g+\frac{\mathrm{2(}\lambda +2\mu \mathrm{)}}{\mu}\frac{1}{{r}^{2}}{f}_{2}+\frac{\lambda +\mu}{\mu}\frac{1}{r}{f}_{2}^{\mathrm{\text{'}}}\mathrm{=}-\frac{\rho {\Omega}^{2}\beta}{\mu}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\end{array}\text{(61)}$

We seek solutions of these equations of the form *f _{1,2,}g* =

${f}_{1}\mathrm{=}-\frac{\rho {\Omega}^{2}\alpha}{\mathrm{10(}\lambda +2\mu \mathrm{)}}{r}^{3}+{C}_{1}r\text{(62)}$

and

${f}_{2}\mathrm{=}{C}_{2}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{\hspace{0.05em}}\text{\hspace{0.05em}}g\mathrm{=}\frac{\rho {\Omega}^{2}\beta}{6\lambda}{r}^{3}+{C}_{3}r\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(63)}$

where *C _{1,2,3}* are constants of integration. These constants are determined by the boundary conditions given by equations (17) for a free surface. Finally, we get the displacement

$\begin{array}{c}u\mathrm{=}-\frac{\rho {\Omega}^{2}}{3\lambda}\left[\frac{\lambda}{\mathrm{5(}\lambda +2\mu \mathrm{)}}r\left({r}^{2}-\frac{5\lambda +2\mu}{3\lambda +2\mu}{R}^{2}\right)-{R}^{2}r\mathrm{(1}-3{{\displaystyle \mathrm{cos}}}^{2}\theta \mathrm{)}\right]{e}_{r}+\\ \\ +\frac{\rho {\Omega}^{2}}{3\lambda}r\left[{r}^{2}-\frac{\mathrm{2(3}\lambda +\mu \mathrm{)}}{3\lambda}{R}^{2}\right]\mathrm{sin}\theta \mathrm{cos}\theta {e}_{\theta}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\end{array}\text{(64)}$

It is worth estimating the equatorial displacement (*q* = *p*/2) for the Earth radius *R* = 6370*km*; with *r*= 5g/cm^{3} and l,m = 10^{11}*dyn*/*cm*^{2} we get *u* = *u _{r,} *≃ 10

Let us multiply by *u* the equation of the elastic motion,

$\rho \ddot{u}+\mu curl\text{\hspace{0.05em}}curlu-\mathrm{(}\lambda +2\mu \mathrm{)}grad\text{\hspace{0.05em}}divu\mathrm{=}F\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{;}\text{(65)}$

integrating by parts, we get the law of energy conservation

$\frac{\partial \mathcal{E}}{\partial t}\mathrm{=}-divS+w\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(60)}$

where

$\mathcal{E}\mathrm{=}\frac{1}{2}\rho {\dot{u}}^{2}+\frac{1}{2}\mu {\mathrm{(}curlu\mathrm{)}}^{2}+\frac{1}{2}\mathrm{(}\lambda +2\mu \mathrm{)(}divu{\mathrm{)}}^{2}\text{(67)}$

is the energy density,

${S}_{i}\mathrm{=}\mu \mathrm{(}{\dot{u}}_{j}{\partial}_{j}{u}_{i}-{\dot{u}}_{j}{\partial}_{i}{u}_{j}\mathrm{)}-\mathrm{(}\lambda +2\mu \mathrm{)}{\dot{u}}_{i}{\partial}_{j}{u}_{j}\text{(68)}$

are the components of the energy flux density and
$w\mathrm{=}\dot{u}F$
is the density of mechanical work done by the external force per unit time. It is worth noting that the energy density given by equation (67) differs from the energy density derived from the other form of the equation of motion, *e.g.*,

$\rho \ddot{u}-\mu \Delta u-\mathrm{(}\lambda +\mu \mathrm{)}grad\text{\hspace{0.05em}}divu\mathrm{=}F\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(69)}$

by the divergence of a vector; it follows that the energy density and the energy flux density are not unique (well defined).

Making use of equations (6), (7) and (10) we can write symbolically

$\begin{array}{c}curlu\mathrm{=}\frac{h}{r}l\mathrm{(}l+\mathrm{1)}R+\frac{1}{r}\frac{d}{dr}\mathrm{(}rh\mathrm{)}S+\left[\frac{f}{r}-\frac{1}{r}\frac{d}{dr}\mathrm{(}rg\mathrm{)}\right]T\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\\ \\ divu\mathrm{=}\frac{1}{{r}^{2}}\frac{d}{dr}\mathrm{(}{r}^{2}f\mathrm{)}-\frac{g}{r}l\mathrm{(}l+\mathrm{1)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\end{array}\text{(70)}$

We compute the total energy *E* by introducing these expressions for
$curlu$
and *div u* in equation (67), integrating over the solid angle and integrating by parts over the radius

$E\simeq \frac{2l+1}{8\pi}{\displaystyle \int}dr\rho {\omega}^{2}\left[{f}^{2}+l\mathrm{(}l+\mathrm{1)}{g}^{2}+l\mathrm{(}l+\mathrm{1)}{h}^{2}\right]\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(71)}$

where the summation over l is omitted (the factor 2*l* + *1* arises from the summation over *m*). The functions *ωf, ωg* and *ωh* in equation (71) are superpositions of their own normal modes (labeled by n); for large values of *R* all these eigenmodes may be taken as the spherical Bessel functions, and the eigenfrequencies are given by the zeros of the derivatives of the spherical Bessel functions; we note that these eigenmodes are orthogonal concerning the *r* -integration; the *f* -part in equation (71) is related to the velocity *c _{1}* (the combination of
$\lambda +2\mu $
of the elastic moduli), while the

Let us write the energy given by equation (71) for the normal modes as

$E\simeq \frac{1}{8\pi}{\displaystyle \sum _{lmn}}{\displaystyle \int}dr\left[\rho {\omega}_{ln}^{\mathrm{(}r\mathrm{)2}}{f}_{ln}^{2}+\rho {\omega}_{ln}^{\mathrm{(}s\mathrm{)2}}{g}_{ln}^{2}+\rho {\omega}_{ln}^{\mathrm{(}t\mathrm{)2}}{h}_{ln}^{2}\right]\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(72)}$

where the summation over *m* is restored and the coefficients *l*(*l*+1) are included in *g _{ln}* and

$E\simeq \frac{1}{4}R{\displaystyle \sum _{lmn}}\left[\rho {c}_{1}^{2}{a}_{ln}^{2}+\rho {c}_{2}^{2}{b}_{ln}^{2}+\rho {c}_{2}^{2}{c}_{ln}^{2}\right]\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(73)}$

where *R* is the radius of the sphere and *c _{1,2}* are the wave velocities. This is a simple expression, of the form

$E\mathrm{=}{\displaystyle \sum _{s}}\rho R{c}^{2}{a}_{s}^{2}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(74)}$

where is a generic notation for the normal modes.

Let us assume that energy *E* is given to the vibrating sphere; we ask how it is distributed among the normal modes. It is reasonable to assume that, after many reflections from the surface, the distribution of energy reaches an equilibrium state, in the sense that it does not depend anymore on time. This state is characterized by a probability density *w*, which is multiplicative for different spheres; *In w* is additive, and the function

$S\mathrm{=}-w\mathrm{ln}w\text{(75)}$

should have a maximum value in the equilibrium state, corresponding to a maximal " disorder"; this represents our idea of equilibrium. Obviously, the function *S* given by equation (75) is the entropy. Its maximum value for constant energy is reached for the extremum of the function *S −βwE*, where *β* is a Lagrange multiplier; we get the Boltzmann (canonical) distribution

$w\mathrm{=}const\cdot {e}^{-\beta E}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(76)}$

or, for one mode,

$w\mathrm{=}\sqrt{\beta \rho R{c}^{2}\mathrm{/}\pi}{e}^{-\beta \rho R{c}^{2}{a}^{2}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{.}\text{(77)}$

The mean energy per mode is

$\overline{e}\mathrm{=}\frac{1}{2}T\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(78)}$

as expected, and the mean value of the square amplitude is

$\overline{{a}^{2}}\mathrm{=}\frac{T}{2\rho R{c}^{2}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\mathrm{,}\text{(79)}$

where we introduced the temperature *T* = 1/*β*. The total mean energy is
$\overline{E}\mathrm{=}N\overline{e}\mathrm{=}NT\mathrm{/2}$
, where N is the total number of modes; this equality gives the temperature parameter.

Making use of the asymptotic expressions of the spherical Bessel functions (for the radial functions) we get the normal modes given by
${k}_{ln}R\mathrm{=(2}n+l+\mathrm{1)}\pi \mathrm{/2}$
; hence, we see that the normal modes are equidistant; the corresponding wavelengths are
${\lambda}_{ln}\mathrm{=4}R\mathrm{/(2}n+l+\mathrm{1)}$
. We may take, tentatively, a cutoff of short wavelengths of the order 10^{−4} cm (1*µm*, corresponding to a frequency ≃5GHz, for velocity 5*km/s*); it is reasonable to admit that below this distance the homogeneous elastic qualities of the Earth do not hold anymore. For this cutoff, we get a maximum number 2*n+l+1* of the order *N _{c}* = 10

A systematic analysis of vibrations of an elastic sphere is presented in this paper, to get analytical results. Such results are useful, on one hand, in analyzing the recorded data, and, on the other, in comparing them with the current numerical investigations. It is shown that the hypothesis of a large radius, appropriate for Earth's vibrations, simplifies the analysis to a great extent. We discuss the general formulation of the problem, the use of spherical harmonics, the approximation of a large radius, and the example of a fluid sphere. Specific results are given for toroidal and spheroidal vibrations. Also, the self-gravitation and rotation effects are analyzed in detail.

Apart from self-gravitation and rotation, the inhomogeneities may have an important effect on the vibrations of the solid sphere. For instance, from equation (1), a (uniform) change *δρ* in density causes a change *δω /ω = −δρ/2ρ* in frequency. The effect of similar changes in the elastic moduli *λ* and *µ* can be estimated by using the changes in the wave velocities *c* in the relation ω_{ln} ≃ *(c/R)β _{ln}*.

An approximate procedure is given in this paper for estimating the spectrum of eigenfrequencies (and eigenfunctions) of the vibrations of a solid sphere, with application to Earth's vibrations, as those produced by an earthquake. The procedure is sufficiently convenient to apply to other, more complex situations involving the vibrations of a solid sphere, as, for instance, the corrections brought about by self-gravitation, rotation, and inhomogeneities. The distribution of the energy among the vibrations eigenmodes is also estimated here and the concept of the earthquake " temperature" is tentatively introduced, as another means of characterizing earthquakes and estimating the earthquake's effects.

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