Radiation Patterns in a Homogeneous Whole Space

Radiation Patterns in a Homogeneous Whole Space#

Far-Field Body Waves and Focal Geometry#

Learning Objectives#

By the end of this lecture and lab, students should be able to:

  • Derive and interpret the far-field P and S displacement expressions for a point moment tensor.

  • Explain how the radiation pattern emerges from tensor–geometry interactions.

  • Predict nodal planes and polarity quadrants for a double-couple source.

  • Distinguish P and S radiation symmetries.

  • Connect radiation patterns to focal mechanisms (“beach balls”).

1. From Moment Tensor to Far-Field Waves#

For a homogeneous whole space, the far-field P-wave displacement from a moment tensor source is:

\[ u_i^P(\mathbf{x}, t) = \frac{1}{4\pi \rho \alpha^3} \frac{x_i x_j x_k}{r^3} \frac{1}{r} \dot{M}_{jk}(t - r/\alpha) \]

(ITS Eq. 9.21)

Assumptions#

  • Homogeneous, isotropic medium

  • Point source

  • Far-field (1/r term retained, 1/r² neglected)

  • Body waves only

What breaks this?#

  • Near-field observations

  • Strong heterogeneity

  • Extended finite fault sources

  • Surface wave dominance

2. Geometry Controls Amplitude#

For a double-couple in the (x₁,x₂) plane with slip in x₁:

\[ u_P \propto \sin 2\theta \cos \phi \]

(ITS Eq. 9.24)

Immediate Observations#

  • Nodal planes occur where amplitude = 0

  • Four quadrants (compressional/dilatational)

  • Purely geometric dependence

  • Independent of distance (after 1/r scaling)

3. S-Wave Radiation#

Far-field S displacement:

\[ u_i^S = \frac{1}{4\pi \rho \beta^3} (\delta_{ij} - \gamma_i \gamma_j) \gamma_k \frac{1}{r} \dot{M}_{jk}(t - r/\beta) \]

(ITS Eq. 9.25)

Key difference:

  • No nodal planes

  • Six nodal points

  • Vector polarization matters

4. Physical Interpretation#

Radiation pattern is:

A projection of the moment tensor onto the takeoff direction.

It reflects:

  • Fault normal

  • Slip direction

  • Symmetry of M

  • Not Earth structure (in this ideal case)

5. Beach Balls as Projections#

Beach balls are:

  • Lower hemisphere projections

  • Shaded = compressional quadrant

  • Defined by nodal planes

Important:

Radiation pattern → polarity map → focal mechanism

Check Your Understanding#

  1. Why does far-field displacement scale as 1/r?

  2. Why do double-couples produce four quadrants?

  3. Why does isotropic source have no nodal planes?

  4. Why are S-wave nodal structures fundamentally different?

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