Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 2

Let $u(x,z,t)$ be displacement along the source line and let $F(t)$ be its force per unit length. The shear-horizontal wave equation isRotational symmetry about the line makes $u=u(r,t)$. Integrate over a disk of radius $r$ and use the divergence theorem:At a fixed regular time, a logarithmic singularity and its time derivatives are locally integrable; the inertial integral is $O(r^2|\log r|)$ and vanishes as $r\downarrow 0$. The leading balance is consequently $r{u}_r=-F(t)/(2\pi \mu )$, giving the SH line-source near-field singularityThe reference length $r_{\ast }$ only changes the bounded term; boundary conditions and source history determine that term. This is a local fixed-time statement, interpreted distributionally in time if the source force has jumps. It establishes logarithmic spatial dependence directly, without extrapolating far-field ray spreading into the source.

Away from the line source, insert a leading wavefront form $u\sim a(\mathbf{x})W(t-T(\mathbf{x}))$ into the shear-horizontal wave equation and match the two most singular orders. The first gives the eikonal equation $|\nabla T|=1/\beta$. The next givesEquivalently, SH ray-tube amplitude transport conserves $\rho \beta a^{2}J$, where $J$ is the transverse width of a two-dimensional ray tube. A line source has straight radial elastic rays and $J=rd\theta$ for fixed angular width, soThe same geometrical spreading applies to the first nonzero derivative discontinuity or leading wavefront singularity; the source history determines which waveform $W$ and normalization $C$ occur. In three dimensions a point source instead has $J\propto r^2$ and amplitude proportional to $r^{-1}$.

This does not contradict the SH line-source near-field singularity. Ray transport describes an arrival-front limit with $t-r/\beta$ small, or the high-frequency field outside the immediate source neighborhood. The logarithm describes $r\downarrow 0$ at fixed elapsed time and includes the tail behind the two-dimensional wavefront. For example, a causal step force $F_{0}H(t)$ has the exact radial displacementFor $r\ll \beta t$, this is $F_0\log (2\beta t/r)/(2\pi \mu )$, the near-source logarithm. For $t=r/\beta +\tau$ with $0<\tau \ll r/\beta$, it isThe coefficient of the leading arrival singularity therefore has exactly the ray-theory dependence. A force impulse gives the corresponding $r^{-1/2}H(\tau )/\sqrt{\tau }$ leading field. The limits describe different regions of one causal solution; neither is a uniform approximation to the other.

Choose interface $z=0$, source $(0,-h)$ in medium 1 and transmitted medium 2 at $z>0$. Write ${\mu }_j={\rho }_j{\beta }_j^2$, take $p$ to be signed tangential slowness, and defineThe propagating range is $|p|<1/\max ({\beta }_1,{\beta }_2)$; the absolute value is needed if both azimuthal directions are included. Snell law for elastic waves preserves $p$. The incident ray reaches the interface after time $t_1$ at horizontal position $x_I$, withIn the transmitted medium its velocity is ${\beta }_2(\sin {\theta }_2,\cos {\theta }_2)=({\beta }_2^{2}p,{\beta }_2^2{q}_2)$. Therefore the refraction of a cylindrical SH wavefront givesTime is measured from source emission. For a later emission add its time to the clock. These straight transmitted rays and their common-time endpoints determine the refracted wavefront; critical incidence and an evanescent transmitted field are excluded by the strict slowness bound.

At a welded interface, elastic-interface boundary conditions require continuous SH displacement and shear traction. Incident and reflected vertical slownesses have signs $q_1,-q_1$, givingThus the local displacement coefficient isIt multiplies the leading waveform discontinuity as well as a harmonic displacement. At the interface the incident amplitude from SH ray-tube amplitude transport is $C/\sqrt{r_1}$, where $r_1=h/({\beta }_1{q}_1)$ is the incident path length and $C$ is the source normalization from the previous part.

The transmitted geometrical spreading is not simply its total path length. Differentiate the ray position with respect to $p$ at fixed travel time and project onto the unit direction transverse to the transmitted ray, ${\mathbf{m}}_2=({\beta }_2{q}_2,-{\beta }_{2}p)$. Since $d{x}_I/dp=h/({\beta }_1^2{q}_1^3)$, the transverse width per unit $p$ isThe derivative of $t_1$ contributes only in the ray direction and drops out of this projection. At entry,Within uniform medium 2 the leading amplitude varies as $J_2^{-1/2}$, so the required refraction of a cylindrical SH wavefront amplitude isThe local transmission coefficient and ray-tube spreading together incorporate both impedance contrast and refraction. Indeed the normal energy transmission fraction is ${\mu }_2{q}_2|T_{\mathrm{SH}}{|}^2/({\mu }_1{q}_1)$, while the change of transverse width supplies the refraction factor in the conserved tube flux. If the media coincide, $q_1=q_2$ and $T_{\mathrm{SH}}=1$, reducing this expression to $C/\sqrt{{\beta }_{1}t}$ as required. The formula holds for the stated propagating slownesses, without a critical or head-wave contribution.