= Solution
Assume the coefficients are sufficiently differentiable in the region considered, with no material discontinuity or <caustic> at the front. Let $t=T(\mathbf x)$ describe arrival time, $\mathbf p=\nabla T$ the slowness vector, and $\mathbf n$ the unit <elastic wave> propagation normal. For any field, its jump is the value immediately behind the front minus that immediately ahead. If the first nonzero jump occurs in the kth time <derivative>, $k\geq2$, write
$$
[\partial_t^ku_i]=A_i(\mathbf x),\qquad u_i=A_iF_k(t-T)+B_iF_{k+1}(t-T)+\cdots,\quad F_j(s)=s_+^j/j!.
$$
A smooth background may be added without changing these leading jumps. Since $F_j'=F_{j-1}$, <differentiation> expresses spatial jumps in terms of $p_j$ and time-derivative jumps. The restriction $k\geq2$ lets <displacement field> and its first time <derivative> remain continuous and avoids an imposed impulsive source at the front.
Insert this local expansion in $\rho\partial_t^2u_i=\partial_j(c_{ijpq}\partial_pu_q)$. The coefficient of $F_{k-2}$ is
$$
\left(c_{ijpq}p_jp_p-\rho\delta_{iq}\right)A_q=0.
$$
For isotropic <elasticity> the slowness acoustic <matrix> is $\mu|\mathbf p|^2I+(\lambda+\mu)\mathbf p\mathbf p^T$. Its longitudinal <eigenvalue> is $(\lambda+2\mu)|\mathbf p|^2$. The P branch therefore gives the <eikonal equation> and polarization
$$
\boxed{|\nabla T|=\alpha^{-1},\qquad\mathbf n=\alpha\nabla T,\qquad A_i=a^{[k]}n_i.}
$$
The next coefficient, $F_{k-1}$, is
$$
\left(c_{ijpq}p_jp_p-\rho\delta_{iq}\right)B_q=c_{ijpq}p_j\partial_pA_q+\partial_j(c_{ijpq}p_pA_q).
$$
The <matrix> on the left is symmetric and annihilates $\mathbf n$. Project onto $n_i$ to obtain its solvability condition. Put $a=a^{[k]}$. The two projected terms can be evaluated without dropping <gradients> of <mass density> or moduli:
$$
n_ic_{ijpq}p_j\partial_p(an_q)=\rho\alpha\,\mathbf n\cdot\nabla a+\frac{\lambda a}{\alpha}\nabla\cdot\mathbf n,
$$
$$
n_i\partial_j(c_{ijpq}p_pan_q)=\nabla\cdot(\rho\alpha a\mathbf n)-\frac{\lambda a}{\alpha}\nabla\cdot\mathbf n.
$$
These identities use $|\mathbf n|^2=1$, hence $n_q\partial_pn_q=0$, and $(\lambda+2\mu)/\alpha=\rho\alpha$. Their curvature terms cancel, leaving
$$
2\rho\alpha\,\mathbf n\cdot\nabla a+a\nabla\cdot(\rho\alpha\mathbf n)=0.
$$
Multiplying by $a$ proves the requested <P-wavefront discontinuity transport> law,
$$
\boxed{\partial_q\left[\rho\alpha\left(a^{[k]}\right)^2n_q\right]=0.}
$$
The <wave amplitude> here is real for a real jump; a complex <wave amplitude> uses the corresponding modulus-squared law. An <elastic ray> follows $d\mathbf x/ds=\mathbf n$, with arclength $s$ and $dt/ds=1/\alpha$. If $J$ is the infinitesimal cross-sectional area of a neighboring ray tube, $d\log J/ds=\nabla\cdot\mathbf n$. Thus $\rho\alpha a^2J$ is constant along that tube, the <ray-tube conservation for a P-wave jump>. This defines the geometrical spreading and shows why the <wave amplitude> changes with material properties even before attenuation is considered. At a <caustic> the simple ray-tube <wave amplitude> is singular and a uniform local description is needed.
Back to article page
