Geology 5640/6640 Introduction to Seismology 23 Mar 2015 A.R. Lowry 2015 Last time: Normal Modes Free oscillations are stationary waves that result from.
Why study normal modes? A lm n are the excitation amplitudes, analogous to A n in the 1D (string) example… So from measurements of u one can get information about the source (provided the eigenfrequencies lm n are known!) Conversely, given a source function A lm n and known lm n, one can predict u … The modes form the basis vectors to describe displacements if one wants to model synthetic seismograms. The frequencies lm n depend on density, shear modulus, and compressibility modulus of the Earth… so are used to get Earth structure.
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Last time: Normal Modes• Free oscillations are stationary waves that result from interference of propagating waves• For a string (length L, velocity v) fixed at the endpoints, all propagating waves have eigenfrequencies n = nv/L:
• The Amplitudes An in this equation relate to the source that excited the string:
• Propagating waves in the string can be represented by these normal modes.• In the Earth, the equation is a leetle more complicated…
Read for Wed 25 Mar: S&W 119-157 (§3.1–3.3)
€
u x,t( ) = Anun x,ωn ,t( )n=1
∞
∑ = An sin ωn xv
⎛ ⎝ ⎜
⎞ ⎠ ⎟cos ωn t( )
n=1
∞
∑
€
An = sin ωn xs
v ⎛ ⎝ ⎜
⎞ ⎠ ⎟F ωn( )
Geology 5640/6640Seismology
Last time: Normal Modes (Continued)• On a sphere, free oscillations are described in terms of spherical harmonics as:
Here n is radial order (0 for fundamental; > 0 for overtones); l (colatitude) and m (longitude) are surface orders; Alm
n
describe source displacement; lmn are eigenfrequencies;
& yln(r) (at depth) and xlm
(surface) are eigenfunctions.
• Spherical harmonics are basis functions on a sphere: orthonormal and can completely describe any function.
23 Mar 2015
€
v u (r,θ ,φ,t) =n= 0
∞
∑l= 0
∞
∑ Almn
m= 0
∞
∑ yln (r)
r x lm (θ ,φ)e iω lm
n t
Why study normal modes?
Almn are the excitation amplitudes, analogous to An in the 1D
(string) example… So from measurements of u one can get information about the source (provided the eigenfrequencies lm
n are known!)
Conversely, given a source function Almn and known lm
n, one can predict u… The modes form the basis vectors to describe displacements if one wants to model synthetic seismograms.The frequencies lm
n depend on density, shear modulus, and compressibility modulus of the Earth… so are used to get Earth structure.
€
v u (r,θ ,φ,t) =n= 0
∞
∑l= 0
∞
∑ Almn
m= 0
∞
∑ yln (r)
r x lm (θ ,φ)e iω lm
n t
Recall PREM is derived from normal modes!
Toroidal and spheroidalToroidal and spheroidal
€
uT (r,θ ,φ) =n= 0
∞
∑l= 0
∞
∑ n Alm
m=−l
l
∑ nWl (r)Tlm (θ ,φ)e i nω l
m t
Using spherical harmonics (base on a spherical surface), we can separate the displacements into Toroidal (torsional) and spheroidal modes (as done with SH and P/SV waves):
T :
€
uS (r,θ ,φ) =n= 0
∞
∑l= 0
∞
∑ n Alm
m=−l
l
∑ nU l (r) Rlm (θ ,φ)+nVl (r) Sl
m (θ ,φ)[ ]e i nω lm t
S : Radialeigenfunction
Surfaceeigenfunction
Characteristics of the modesCharacteristics of the modes
• No radial component: tangential only, normal to the radius: motion confined to the surface of n concentric spheres inside the Earth.• Changes in the shape, not of volume
Not observable using a gravimeter (but…)
• Do not exist in a fluid: so only in the mantle (and the inner core?)
• Horizontal components (tangential) et vertical (radial)• No simple relationship between n and nodal spheres
• 0S2 is the longest (“fundamental”)
• Affect the whole Earth (even into
the fluid outer core !)
Toroidal modes nTml : Spheroidal modes nSm
l :
n, l, m …n, l, m …
S : n : no direct relationship with nodes with depthl : # nodal planes in latitudem : # nodal planes in longitude
! Max nodal planes = l0S02
T : n : nodal planes with depthl : # nodal planes in latitudem : # nodal planes in longitude
! Max nodal planes = l - 10T03
0S0 : « balloon » or
« breathing » :
radial only
(20.5 minutes)
0S2 : « football » mode
(Fundamental, 53.9 minutes)
0S3 :
(25.7 minutes)
Spheroidal normal modes: examples:Spheroidal normal modes: examples:
Animation 0S2 from Hein Haakhttp://www.knmi.nl/kenniscentrum/eigentrillingen-sumatra.html
Animation 0S0/3 from Lucien Saviothttp://www.u-bourgogne.fr/REACTIVITE/manapi/saviot/deform/