geology 5640/6640 introduction to seismology 4 mar 2015 © a.r. lowry 2015 last time: evanescent...

Geology 5640/6640Introduction to Seismology

4 Mar 2015

© A.R. Lowry 2015

Last time: Evanescent Waves; Surface Waves• A wave arriving with incident angle ≥ the critical angle (e.g., j1 ≥ 1/2) still requires continuity of stress & displacement. Transmission is imaginary, & solution is e.g.:

These exponentially decaying “trapped” waves are called evanescent waves.

• Surface waves (Love, Rayleigh) are important because: Cylindrical spreading (A 1/ ) these are the largest arrivals outside the near-field They are responsible for most of the damage in EQ’s Dispersive: Different wavelengths travel at different velocities and sample different depths!

Read for Fri 6 Mar: S&W 86-100 (§2.7–2.8)

€

uyT (z) = B'e

−kxrβ2* z

r

Your midterm take-home exam is now posted on the course website…

Due Friday, March 20 at the beginning of class

Note: Not all of the relationships you’ll need have been covered in course notes, but if you’ve been reading the text you’ll know where to look!

• Surface waves can cause damage to taller buildings at relatively large distances.

QuickTime™ and a decompressor

are needed to see this picture.

• Surface waves are dispersive: different wavelengths sample different depth ranges of Earth structure!

Types of Surface Waves:

1) Rayleigh waves2) Lamb waves3) Love waves4) Stoneley waves

Basically, a surface wave describes any wave that can only propagate along a surface (e.g., when one of the media has zero shear strength).

TAMU

Stoneley (“tube”) waves: Occur under certain conditions along elastic-fluid or elastic-elastic interfaces. These are a source of noise in borehole active source seismology (e.g. wireline logging, or vertical seismic profiling (VSP).

However, they are alsoan important source ofsignal in boreholeactive sourceseismology!

Lamb waves are waves within a plate-like object having two free-surfaces, and are analogous to Rayleigh waves in the Earth… Useful e.g. for engineering-related defect detection.

WikicommonsDavid Greve, Carnegie Mellon

Here, image of a defect behaving like a source disturbance.

Love waves are composed of SH motion only. These do not propagate in a uniform flat medium, but do occur in a curved Earth, or a layered medium.

Rayleigh waves are an interference pattern composed of P-SV motions. These require shear- and normal-stress ij = 0 at an interface (so could occur e.g. along a fracture as well as at the Earth’s surface).

Let’s consider Rayleigh waves first:

Rayleigh waves are a combination of “trapped” (i.e., evanescent) P- and SV-waves at a free-surface interface: That is, shear and normal stress at the interface must equal zero. To get a feeling for these, let’s recall the P and SV solutions to the wave equation.

P-wave:

SV-wave:

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φ=Ae−i ωt−kxx−kxrα z( )

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ψ =Be−i ωt−kxx−kxrβ z( )

Recall that r is the ratio of the vertical to horizontal wavenumbers, and cx is the apparent horizontal velocity of a wave.

We here note that , and so:

(And note that for these to propagate in combination in a surface wave, cx must be the same for both!)

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rα =kzαkx

= cxη α

= cx1

α 2− p2

=cx

2

α 2−cx

2p2

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p =kxω

=1

cx

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rα =cx

2

α 2−1 . Similarly,

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rβ =cx

2

β 2−1

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= 1

cx