Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2015 / Paper 1 / 1 / b

Assume a stable, lossless elastic body with the usual symmetry of the elasticity tensor and boundary conditions that eliminate the surface terms in integration by parts, such as a traction-free outer surface. Its elastic stiffness operator is then self-adjoint. When density varies spatially, the precise eigenproblem is generalized:Here $\alpha$ abbreviates the pair of mode labels. Equivalently, ${\rho }^{-1}\hat{H}$ is a self-adjoint operator in the mass inner product. Stability makes ${\omega }_{\alpha }^2$ real and nonnegative. Modes at different eigenvalues are orthogonal, and an orthonormal basis can be selected within each degenerate eigenspace. Completeness permits expansion of a suitable displacement field in these mass-weighted elastic normal modes.

Insert the expansion and project onto ${\mathbf{u}}_{\alpha }^{\ast }$. Mass normalization diagonalizes both the acceleration term and the restoring term. For a symmetric stress glut tensor supported away from the boundary,The symmetry of the stress glut tensor removes the antisymmetric part of the displacement gradient, leaving the strain. The result is one independent forced oscillator per mode, with the stated excitation integral. If the modes instead have mass norm $N_{\alpha }$, that integral is divided by $N_{\alpha }$. A source touching the boundary requires the corresponding surface forcing to be retained rather than silently dropping it.

Rigid translations and rotations can be zero-frequency modes of a free body. Their strain vanishes, so an internal symmetric seismic moment tensor does not excite them through this coupling. Attenuation, radiative boundaries or other non-self-adjoint effects require modifications of the simple conservative expansion.

For a mode of positive frequency, differentiation of the causal convolution givesThe first boundary term vanishes because $\sin 0=0$; the second supplies $f(t)$ because $\cos 0=1$. Both initial values vanish. This verifies the forced-oscillator equation and its initial conditions, including a step source in the piecewise or distributional sense.

The spatial Dirac delta function reduces the excitation toFor an event at $t_s\ge 0$, put $\tau =t-t_s$. Integrating the oscillator kernel from $t_s$ to $t$ gives the normal-mode response to a step seismic moment:Non-unit mass normalization introduces the same factor $1/N_{\alpha }$ as above. Complex modes must be combined with their conjugate partners to produce a real displacement. The formal $\omega \to 0$ limit is $a=C{\tau }^{2}H(\tau )/2$; for rigid-body modes here, $C=0$ because their strain is zero.

The step produces a static component plus continuing oscillations in this lossless model. It does not predict physical damping. A point source is appropriate when rupture dimensions are small compared with the wavelengths being analysed and rupture duration is short compared with their periods. Long-period observations of a relatively small earthquake can meet these conditions; high-frequency finite-fault directivity generally cannot. A finite source-time function replaces the ideal instantaneous step when its duration matters.