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SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ... 61 (fault plane solution), is an important aid in the study of source processes as well as the study of large-scale tectonics of a source region. Often the decision between the two options for the rupture surface can be made based on geological arguments. The moment function of an earthquake with a smooth rupture, is, to a good approximation, a step function with non-vanishing rise time T and final value M0 , the moment of the earthquake (see Fig.

In the following, we consider the general form ∂2u 1 ∂2u = 2 2. 2) 40 CHAPTER 3. 1)). Another form is u(x, t) = F t − x x +G t+ . 3) have to be interpreted as waves propagating in the positive and negative x -direction, respectively. 3) for x = x1 can be written as u(x1 , t) = F t − x1 c = F1 (t). For another distance x2 > x1 u(x2 , t) = F t − x2 c =F t− x2 − x1 x1 − c c = F1 t − x2 − x1 c . This means that for time t at distance x2 the same effects occur as at distance x1 at the earlier time t − (x2 − x1 )/c.

R r 58 CHAPTER 3. BODY WAVES The y- and z-displacement are treated similarly. Therefore,  u′′x  u′′y  u′′z z(x−x0 ) ′ = − 4πρα t− 3 r3 K r α   (x − x0 )/r y/r ǫ·  z/r. 20) As expected, the P -displacement of the single couple is also longitudinal. The force dipole is defined strictly by the limit ǫ → 0, combined with a simultaneous increase of K(t), so that lim K(t) ǫ = M (t) ǫ→0 remains finite (but non-zero). M (t) is called moment function of the dipole with the dimensions of a rotational moment.

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Analytical investigation of internally resonant second harmonic lamb waves in nonlinear elastic isotropic plates by Martin Fritz Mueller

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