Post on 26-Mar-2021
PEAT8002 - SEISMOLOGYLecture 15: Free Oscillations II
Nick Rawlinson
Research School of Earth SciencesAustralian National University
Free oscillations IINormalized surface harmonics
At the end of the last lecture, we showed that the (r , θ)components of free oscillation (the spheroidal mode) on ahomogeneous sphere were described by the Legendreequation.Here, we briefly discuss the spherical harmonicrepresentation of spheroidal modes.The normalised surface harmonic defined for l ≥ 0,−l ≤ m ≤ l is given by:
Y ml (θ, φ) = K m
l F (φ)Pml (cos θ) = K m
l Pml (cos θ) exp[imφ]
Recalling that F (φ) = A exp[imφ], and using K ml = AK m
l .The normalisation constant K m
l is chosen so that theintegral over the surface of the sphere of each harmonictimes itself is unity, i.e.∫ 2π
0dφ
∫ π
0[Y m
l (θ, φ)]2 sin θdθ = 1
Free oscillations IINormalized surface harmonics
It turns out that for this condition to be satisfied,
K ml = (−1)m
√2l + 1
4π
√(l −m)!
(1 + m)!
The spherical harmonics form a complete orthonormalbasis on the sphere:∫ 2π
0dφ
∫ π
0[Y m
l (θ, φ)][Y m′
l ′ (θ, φ)] sin θdθ = δll ′δmm′
Any function defined on the surface of the sphere may beexpressed as a sum of spherical harmonics (analogous toFourier series in 1-D, e.g. to represent a waveform):
h(θ, φ) =lmax∑l=0
l∑m=−l
Aml Y m
l (θ, φ)
Free oscillations IINormalized surface harmonics
Each of the harmonics present in a particular solutiondescribes a surface pattern with a particular pattern ofsymmetries.For normal modes, the pattern continues down into theEarth modulated by R(r).Spherical harmonics are the usual representation forglobal geophysical datasets (e.g. gravity, magnetics).The animations below show the family of oscillations withan angular order number of l = 3, experienced by anhomogeneous sphere.
Free oscillations IINormalized surface harmonics
The example below is for l = 3, m = 0.
Free oscillations IINormalized surface harmonics
The example below is for l = 3, m = 1.
Free oscillations IINormalized surface harmonics
The example below is for l = 3, m = 2.
Free oscillations INormalized surface harmonics
The example below is for l = 3, m = 3.
Free oscillations IIThe radial eigenfunctions
In the previous section, we separated the (θ, φ) part of thesolution to leave the radial equation:
ddr
(r2 dR
dr
)+
[(ω
c
)2− l(l + 1)
]R = 0
The general solution to this class of equation is jl(ωr/c),where jl(x) is the spherical Bessel function of order l .A general formula for this function is:
jl(x) = x l[−1x
ddx
]l (sin xx
)with x = ωr/c.
Free oscillations IIThe radial eigenfunctions
For l = 0,
j0(x) =sin x
x, so R(r) =
cωr
sin( c
ωr
)For l = 1
j1(x) = x(−1x
)d
dx
(sin x
x
)= −
[− sin x
x2 +cos x
x
]=
1x2 (sin x − x cos x)
Free oscillations IIThe radial eigenfunctions
Thus, we can write R(r) as
R(r) =( c
ωr
)2 [sin
(ωrc
)−
(ωrc
)cos
(ωrc
)]Boundary conditions at the surface determine permissablevalues of oscillation frequency ω.Note that the amplitude is undetermined, unless we specifythe total energy in the oscillation.
Free oscillations IIPressure modes in a homogeneous fluid sphere
For the free oscillations of a homogeneous fluid sphere,the shear potentials are zero (a fluid does not supportS-waves - µ = 0.)Thus, we are only concerned with oscillations in dilatationθ, which are proportional to oscillations in pressure p,since θ = ∇2Φ = p/κ.From before, we had that (with c = α =
√λ/ρ):(ω
α
)2∇2Φ +∇4Φ = 0
so (ω
α
)2p +∇2p = 0
so pressure satisfies Helmholtz’ equation.
Free oscillations IIPressure modes in a homogeneous fluid sphere
The external boundary condition (at r = a, the surface ofthe sphere) is zero traction σrr = λθ = 0. Since θ = P/κ,this equates to p = 0 at r = a.When we solved Helmholtz’ equation, we assumed that thedisplacement function was separable i.e.f (r , θφ) = R(r)Q(θ)F (φ). In this case, when we are dealingwith oscillations in pressure p, thenp(r , θ, φ) = R(r)Q(θ)F (φ).The boundary condition is therefore satisfied if R(a) = 0If l = 0, then
R(r = a) =cωa
sin(ωa
c
)= 0
noting that ωa/c 6= 0.
Free oscillations IIPressure modes in a homogeneous fluid sphere
From the above expression, it is clear that the boundaryconditions can only be satisfied if:
ωac
= (n + 1)π, n = 0, 1, 2, ...
which defines a discrete spectrum of oscillations.The permitted frequencies are therefore given by:
ωn =c(n + 1)π
awhere c = α.The fundamental mode occurs for n = 0.
Fundamentalmode
n=0
Free oscillations IIPressure modes in a homogeneous fluid sphere
n ≥ 1 gives overtones or higher modes.Nodal surfaces occur at the boundary, and we have ninternal nodal surfaces where the sign of the displacementreverses.The above equation shows that the frequency of theoscillation increases with n.The eigenfunctions associated with solutions of theHelmholtz equation are labelled with the angular and radialorder number l and n, as nSl (spheroidal modes), or nTl(torsional modes), and the frequencies associated withthose eigenfunctions are labelled as nωl .Note that, although the frequency increases withincreasing l , the azimuthal order number m does notappear in the radial equation, and therefore cannot affectthe frequency of oscillation.
Free oscillations IIPressure modes in a homogeneous fluid sphere
Solutions are said to be degenerate with respect toazimuthal order number (all vibrate with the samefrequency), and the degeneracy is said to be (2l + 1)-foldsince −l ≤ m ≤ l .However, very small frequency separation is observed formodes differing in m due to planetary rotation anddepartures from spherical symmetry.The complete solution for the oscillations in pressure of thefluid sphere may consist of a sum of modes:
p(r , θ, φ) =∑
n
∑l
∑m
[nAm
l jl(
nωl rα
)Y m
l (θ, φ)]
where the amplitude coefficients nAml of the individual
modes are determined by the excitation function whichcauses the oscillations (i.e. the magnitude, location andtype of earthquake).
Free oscillations IIPressure modes in a homogeneous fluid sphere
If the Earth was composed of a fluid with wavespeed α = 5km/s, and a = 6, 371 km, then the angular frequency of thefundamental mode (n = 0, l = 0 ⇒ m = 0) is :
0ω0 =απ
a
and the corresponding period is
T =2π
0ω0=
2πaαπ
=2aα
Substitution of the above values give T = 2548 s ≈ 43mins.The period of the overtones will be 43/(n + 1) minutes forn ≥ 1.
Free oscillations IIPressure modes in a homogeneous fluid sphere
The l = 0 mode is sometimes referred to as the “breathing"mode since angular dependence is zero. This correspondsto the planet everywhere moving in and out at the sametime:
Y 00 (θ, φ) = K 0
0 P00(cos θ) exp[i(m = 0)φ] = K 0
0
since P00(cos θ) = 1. Thus, the normalised surface
harmonic is constant.In the Earth, the fundamental spheroidal mode 0S0 has aperiod of 20 minutes (different because we neglectedshear rigidity and variation of elastic parameters withdepth).The 0S1 mode is not physically realizable because it wouldcorrespond to the entire planet moving in one direction atfrequency ω. The 0S2 mode (football mode) is observedand has the longest period of any fundamental modeat 54 minutes.
Free oscillations IINodal surfaces
Nodal surfaces are surfaces of zero displacement. Theyoccur for constant radius (r), co-latitude (θ) and longitude(φ).Recall that for spheroidal or torsional modes, each 1-D partof the eigenfunction has nodes where the net displacementis zero; in 3-D, these points becomes surfaces.Nodes in the radial function are actually concentricspherical surfaces where displacement changes sign whenthe spherical Bessel function is zero. The number ofspherical nodal surfaces is given by the radial ordernumber n.Similarly, the nodes in the Legendre polynomial becomenodal surfaces along conical sheets from r = 0 to r = a,along lines of constant co-latitude. The number of nodalsurfaces in this case is given by l − |m|.
Free oscillations IINodal surfaces
R(r)
r=0
r=a
l−|m|=2 m=4
θ
n=2
view from "top"
Finally, there are |m|azimuthal nodalplanes containingthe origin and pole.These surfaces ofconstant φ areequally spaced.
Free oscillations IINodal surfaces
The schematic examples below illustrate the displacementpatterns for several spheroidal modes
0S1 m=1 0S2 m=0
0S2 m=2View from top
Pole
(not realizable)
Vertical displacement
m=10S2
Free oscillations IITorsional modes in a homogeneous elastic sphere
From before, we had that:
ρω2
r∇h
2(
Tr
)+
µ
r∇2∇h
2(
Tr
)= 0
which is the equation for torsional potential in anhomogeneous elastic sphere.Multiplying both sides by r/µ and replacing β =
√µ/ρ
gives: (ω
β
)2
∇h2(
Tr
)+∇2∇h
2(
Tr
)= 0
This can be written:
∇h2
[(ω
β
)2 Tr
+∇2(
Tr
)]= 0
Free oscillations IITorsional modes in a homogeneous elastic sphere
Thus, (ω
β
)2 Tr
+∇2(
Tr
)= f (θ, φ)
and the equation ∇h2f = 0 reduces to the Legendre
equation with K = l = 0.The solution with l = m = 0 is f (θ, φ) = C, where C is aconstant. This constant does not effect u, so we choosef = 0.Therefore, (
ω
β
)2 Tr
+∇2(
Tr
)= 0
which is Helmholtz’ equation for T/r .
Free oscillations IITorsional modes in a homogeneous elastic sphere
The solutions for T/r are therefore exactly analogous tothe solutions for the pressure oscillations of the fluidsphere, except that the phase velocity is β instead of α; anadditional factor of r appears; and the boundary conditionat r = a is different.As before, the solution is separated into parts with r , θ andφ dependence, so:
T (r , θ, φ) = rjl
(nωl rβ
)Y m
l (θ, φ)
where jl is the spherical Bessel function and Y ml is the
normalised spherical harmonic.
Free oscillations IITorsional modes in a homogeneous elastic sphere
For torsional mode oscillations, the relationship betweendisplacement and potential is:
u = ∇× (T r)
=1
r2 sin θ
∣∣∣∣∣∣∣∣r r θ r sin θφ∂
∂r∂
∂θ
∂
∂φT 0 0
∣∣∣∣∣∣∣∣=
1r2 sin θ
[(0)r + r
(∂T∂θ
)θ − r sin θ
(∂T∂θ
)φ
]=
[0,
1r sin θ
∂T∂φ
,−1r
∂T∂θ
]Thus, the torsional displacement has no radial component.
Free oscillations IITorsional modes in a homogeneous elastic sphere
At the free surface (r = a), we can impose the zero tractionboundary condition. This requires that σrφ = 0.Using our stress strain relationship σij = λθδij + 2µeij , thenσrφ = 2µeij = 0 at r = a.The components of the symmetric strain tensor can bewritten:
eij =12
(∂ui
∂xj+
∂uj
∂xi
)However, the above Cartesian expression for strain mustbe expressed in terms of the spherical components ofdisplacement:
erθ =12
[∂
∂z(u · y) +
∂
∂y(u · z)
]= 0
Free oscillations IITorsional modes in a homogeneous elastic sphere
In the above expression, y is a unit vector in the φdirection, and z is a unit vector in the r direction.
dφ
u r
uφ
u r
uφ
rr
dy
y=0
y
z
At y = 0, u · z = ur , sothere is no φ component ofdisplacement.At y = dy , we mustconsider both ur and uφ,since they both havez-components ofdisplacement.
Free oscillations IITorsional modes in a homogeneous elastic sphere
If uzr represents the z component of ur , thencos dφ = uzr/ur .Similarly, if uzφ represents the z component of uφ, thensin dφ = uzφ/uφ
Thus,
u · z = uzr + uzφ = ur cos dφ− uφ sin dφ
= ur − uφdφ
since in general, cos(x → 0) = 1 and sin(x → 0) = x .Therefore,du · z = du · z(y = dy)− du · z(y = 0) = ur − uφdφ− ur .
Free oscillations IITorsional modes in a homogeneous elastic sphere
Since dy = r sin dφ = rdφ,
∂u · z∂y
= −uφdφ
rdφ=−uφ
r
In the radial direction, ∂uy/∂z = ∂uφ/∂r , so
∂u · z∂y
=−uφ
rand
∂u · y∂z
=∂uφ
∂r
Therefore, the boundary condition is:
σrφ = µ
[∂uφ
∂r−
uφ
r
]= 0
Free oscillations IITorsional modes in a homogeneous elastic sphere
Using our expression for the φ component of displacementderived earlier,
∂
∂r
(−1r
∂T∂θ
)− 1
r
(−1r
)∂T∂θ
= 0
∂
∂θ
[∂
∂r
(−Tr
)+
1r2 T
]= 0
∂
∂θ
[1r
∂T∂r
+2Tr2
]= 0
The term in brackets must also be equal to zero because itis invariant to changes in θ:
1r
∂T∂r
+2Tr2 = 0
Free oscillations IITorsional modes in a homogeneous elastic sphere
We will now apply this boundary condition to the radial partof the eigenfunction with l = 1 (i.e. the spherical Besselfunction j1(x)).From before, this eigenfunction has the form:
T (r) = r(
β
rω
)2 [sin
(rωβ
)−
(rωβ
)cos
(rωβ
)]The boundary condition can be written:
1r
∂T∂r
+2Tr2 =
[ddr− 1
r
]Tr
= 0
Free oscillations IITorsional modes in a homogeneous elastic sphere
Substitution of T into the above equation gives thefollowing equation at r = a:[
ddr− 1
r
]{(β
rω
)2 [sin
(rωβ
)−
(rωβ
)cos
(rωβ
)]}= 0
Simplifying this expression and substituting r = a gives:[1− 3
(β
aω
)2]
sin(
aω
β
)+ 3
(β
aω
)cos
(aω
β
)= 0
This can be written:[1− 3
x2
]sin x +
3x
cos x = 0
where x = aω/β.
Free oscillations IITorsional modes in a homogeneous elastic sphere
Dividing through by cos x yields:
tan x =3x
3− x2
This equation can be numerically solved for x . It turns outto have an infinite number of solutions, which can be seenby plotting the left hand side against the right hand side.The fundamental mode 0T0 does not occur as it wouldcorrespond to oscillatory changes in the total angularmomentum of the Earth.
0T1 has a period of approximately 25 minutes.
Free oscillations IIEigenfunctions for a spherically symmetric Earth
In the above treatment, the Earth was considered to be anhomogeneous medium, which is not a very goodapproximation.However, to a good approximation, the Earth is sphericallysymmetric - λ(r), µ(r) and ρ(r).In this case, we get a modified form of the Helmholtzequations governing torsional and spheroidal modes.Like the homogeneous case, the equations are separableinto (r , θ, φ) dependence of solution. However, the radialequation has numerical solutions only.The frequency of oscillation is still determined by the radialeigenfunctions and appropriate boundary conditions.Degeneracy with respect to the m-values still occurs.
Free oscillations IIObservations
Normal modes can be observed as very low frequencyspectral peaks in the frequency domain of the recordedwaveform following large earthquakes.The frequency of many specific modes have beenaccurately determined. They can be used to constraindepth variation of density and elastic parameters bycomparison with theoretical predictions.
Amplitude spectrumof the radialcomponent of a35-hour seismogramrecorded from theJune 9, 1994 Boliviaearthquake.