1. If a displacement field

\(\varvec{u}\) is irrotational

\([{\varvec{\nabla }}\varvec{\times u}\!=\!0]\), it can be represented as the gradient of a scalar potential

$$ \varvec{u}={\varvec{\nabla }}\phi . $$

Show that a general irrotational wave moving in the

*x*-direction can be described by a function

\(\phi (x\!-\!c_1t,y, z)\) satisfying the equation

$$ \frac{\partial ^2 \phi }{\partial {y}^2}+\frac{\partial ^2 \phi }{\partial {z}^2}=0, $$

where

\(c_1\) is given by Eq. (

19.6).

2. Table

19.1 shows representative values for the elastic constants

\(E,\nu \) and density

\(\rho \) for a selection of materials. Estimate the corresponding values of the dilatational and shear wave speeds

\(c_1, c_2\). Comment on the implications of your results for practical applications. For example, would you expect elastodynamic effects to be significant in the interaction between a car tyre and the road, or the impact between a ship and an iceberg.

Table 19.1 Physical properties of various materials

3. A rigid body with the sinusoidal profile of Fig. 6.6 is pressed against an elastic half-plane by a mean traction \(\bar{p}\) that is just sufficient to ensure that half of the interface is in contact—i.e. \(a\!=\!L/4\). The body now slides without friction over the surface at speed *V*. Find the minimum value of \(V\!<\!c_R\) for which full contact will occur.

4. Use the Smirnov–Sobolev transform of Sect. 6.6 to prove Churilov’s result (19.43) for the steady-state surface displacement due to a point force *P* moving at speed \(V\!<\!c_R\) over the surface of an elastic half-space.

5. An SH-pulse described by the displacement

$$ u_y=C\left\{ a^2-\left( x^\prime -c_2t\right) ^2\right\} \;\;\;\;\;-a<x^\prime -c_2t<a $$

impinges on a plane interface between two half-planes of identical materials, where the coordinate direction

\(x^\prime \) is defined in Fig.

19.8 and the angle of incidence is

\(\alpha \). The half-planes are pressed together by a uniform normal pressure

\(p_0\), but no external shear tractions are applied. If the coefficient of friction is

*f*, find the maximum value of

*C* [

\(=C_0\)] for which there is no slip. Then find the shift

\(h_0\) caused by the passage of the pulse, if

\(C\!=\!2C_0\).

6. Two granite rock masses can be approximated by half-spaces that are pressed together by a uniform pressure of 10 MPa. An SH-pulse impinges on the interface with an angle of incidence of 55

\(^{\circ }\). The stresses at a given point associated with the incident pulse have the step function form

$$ \sigma _{xy}^\prime (t)=45\left[ H(t)-H(t+0.5)\right] \text{ MPa } $$

*in the coordinate system* \(x^\prime y\) *aligned with the direction of propagation of the pulse* [

*see Figure* 19.8], where

*t* is time in seconds. Find the shift at the interface due to the pulse if the coefficient of friction is 0.6. The mechanical properties of granite are given in Table

19.1 above. How would your answer change if the interface also transmits a uniform tangential traction of 3 MPa.

7. Use the Green’s function (

19.61) to find the displacement derivative

\(\partial u_y/\partial x\) due to the traction distribution

moving at subseismic speed

\(c\!<\!c_2^{(2)}\) over the surface of the half-plane

\(z\!<\!0\).

Use this result and the arguments of Sect. 19.3.1 to obtain the bilateral solution for an incident wave defined by Eq. (19.52), and hence verify (i) that the reflected wave has the same amplitude as the incident wave, and (ii) that the phase lag is given by (19.81).

8. A frictionless rigid power-law punch defined by the gap function \(g_0(x)\!=\!C|x|^m\) indents an elastic half-plane at a speed *V*(*t*) which is a function of time *t*. State the kinematic boundary conditions and hence show that the problem is self-similar if and only if \(V(t)\!\sim \! t^{m-1}\).

9. Bedding and Willis (1976) report that the maximum ratio of tangential to normal tractions in the ‘no slip’ solution to the superseismic wedge indentation problem is 0.16. Suppose that the coefficient of friction \(f\!<\!0.16\), so that we must anticipate regions of slip and stick. State the boundary conditions for the problem, including inequalities, and hence show that the resulting problem is also self-similar.