1. Geology 5640/6640 Introduction to Seismology 20 Mar 2015 © A.R. Lowry 2015 Last time: Love Waves; Group & Phase Velocity At shorter periods, Love waves can have a fundamental plus higher modes ; longer periods have fewer modes The wave velocity c x increases with period and with mode number Amplitudes u y (z) are ~sinusoidal above the turning depth; decay exponentially below (where the wave is evanescent) Dispersive waves have both group velocity U (velocity of the envelope or “beat”) and phase velocity c (velocity of individual peaks) which relate as: (hence, c > U ) Read for Fri 20 Mar: S&W 119-157 (§3.1–3.3)

2. To measure Group Velocity : Measure period as the time between successive peaks or troughs Travel-time is the time at the time of arrival of the wave group minus the origin time Divide the source-receiver distance by travel-time to get the group velocity

3. Or can get a bit more sophisticated by filtering the waveform (multiplying by a “windowing function” in the frequency domain) to isolate elements of the waveform that have a particular period, using a Fourier transform. To get phase velocity, can transform to phase  (  ) and e.g. solving for c(  ) from the difference in phase of the arrivals at two sites.

4. Gravimeters for seismological broadband monitoring: Earth’s free oscillations Michel Van Camp Royal Observatory of Belgium Note : These slides borrow heavily from a presentation by Michel van Camp, ROB…

5. Fundamental ( n = 1 ) etc. L u Free Oscillations: are stationary waves consisting of interference of propagating waves. 1 st harmonic ( n = 2 ) 2 nd harmonic ( n = 3 ) 3 rd harmonic ( n = 4 ) Consider a vibrating string attached at both ends: It can only vibrate at the eigenfrequencies for which displacement u = sin(  x/v) cos(  t) is always zero at the endpoints: I.e., only for  n = n  v/L. Thus we can write And the total displacement is given by:

6. Here A n is a weight that depends on the source displacement as: where F(  n ) describes the shape of the source and x s is the location. This example from the text is for x s = 8 and with  = 0.2.

7. Seismic normal modes  Periods < 54 min, amplitudes < 1 mm  Observable months after great earthquakes (e.g. Sumatra, Dec 2004 took about 5 months to decay) Few minutes after the earthquake Constructive interferences  free oscillations (or stationary waves) Few hours after the earthquake ( 0 S 20 ) (Duck from Théocrite, © J.-L. & P. Coudray)

8. Travelling surface waves Richard Aster, New Mexico Institute of Mining and Technology http://www.iris.iris.edu/sumatra/

9. Historic  First theories:  First mathematical formulations for a steel sphere: Lamb, 1882: 78 min  Love, 1911 : Earth  steel sphere + gravitation: eigen period = 60 minutes First Observations:  Potsdam, 1889: first teleseism (Japan): waves can travel the whole Earth.  Isabella (California) 1952 : Kamchatka earthquake (Mw=9.0). Attempt to identify a « mode » of 57 minutes. Wrong but reawakened interest.  Isabella (California) 22 may 1960: Chile earthquake (Mw = 9.5): numerous modes are identified  Alaska 1964 earthquake (Mw = 9.2)  Columbia 1970: deep earthquake (650 km): overtones  IDA Network

10. On the sphere… For a vibrating string: On the sphere: Here, n is the radial order ( n = 0 for the fundamental; n > 0 for overtones) l and m are surface orders l is the angular order; –l < m < l is the azimuthal order Radial eigenfunction Surface eigenfunction

11. A Quick Digression on Basis Functions Consider an arbitrary function f(t). It can be represented as a sum of sine and cosine waves with various frequencies via: This is the Fourier transform that we keep talking about… basically it “translates” the temporal (or spatial) description of a function into the “language” of frequency and phase. Given enough frequencies, the Fourier transform can exactly construct any arbitrary function from a sum of sines and cosines. We call e –i  t a basis function.

12. On a sphere, the analogous description of sines and cosines is called spherical harmonics. Spherical harmonics are described by Legendre polynomials and Legendre functions. Legendre polynomials are: where l denotes angular order. For a sphere, x = cos  so these describe variations with colatitude (e.g. from the source in this diagram). Legendre functions are defined by:

13. The Legendre polynomials and Legendre functions can be combined to create a set of basis functions on a sphere: Note that l describes harmonics that depend on the colatitude  and m describes harmonics that have a longitudinal (  ) dependence. As is true of all basis functions, spherical harmonics are orthonormal :