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The leading source displacement field isThe second term is one derivative smoother at the leading arrival. At each surface point, use its tangent plane, the angles from (i)–(ii), and the local solid-fluid P-wave transmission coefficient. The incident potential coefficient is $1/r(R)$, so the transmitted leading potential coefficient isTransport this coefficient along the vertical fluid rays. Their transverse area does not change with height, so ray-tube conservation for a P-wave jump gives no further geometrical spreading. The leading displacement field coefficient is consequentlyThis is the transmitted jump at a collimating solid-fluid interface. Radial derivatives of $A_{\varphi }$ multiply $f$, and therefore do not contribute to the first derivative jump carried by $f^{\prime }$. A varying transverse wave amplitude does generate smoother later terms, so this construction is a leading-front result, not a complete global plane-wave field.

Since the source is zero before $t=0$ and is $C^2$ there, $f(0)=f^{\prime }(0)=f^{\prime \prime }(0)=0$. Thus displacement field itself does not jump. If $J=[f^{(3)}(0)]$ exists and is the first nonzero jump, the leading discontinuity is the acceleration jump $[{\partial }_t^2\mathbf{u}]=-A_u(R)J{\mathbf{e}}_z$. More generally, a first jump in $f^{(m)}$ gives $[{\partial }_t^{m-1}\mathbf{u}]=-A_u(R)[f^{(m)}]{\mathbf{e}}_z$. The supplied regularity alone does not guarantee a nonzero discontinuity: a source smooth to all orders may have none. The geometrical transfer factor $A_u$ is nevertheless determined.

At $R=0$, $r=z_0$ and all three angles vanish. HenceFor large $R$, put $C_{\infty }=1-2{\beta }^2/{\alpha }^2$ and retain $s=z_0(\alpha -{\alpha }_f)$, $d=\sqrt{{\alpha }^2-{\alpha }_f^2}$. The relevant limits areWhen $C_{\infty }\ne 0$, the square-bracket factor in the denominator of $T_P$ is $C_{\infty }^2+O(R^{-1})$, soThese are signed waveform coefficients; their absolute values give wave amplitude magnitudes. The leading displacement field decays as $R^{-2}$ rather than $R^{-1}$ because grazing incidence adds a small transmission factor to spherical spreading. The leading coefficient is independent of fluid and solid mass densities, although the exact wave amplitude and its higher-order terms depend on them.

The assumptions do not exclude ${\beta }^2={\alpha }^2/2$, where $C_{\infty }=0$ and the preceding division is invalid. In this special case $\cos 2{\theta }_S={\cos }^2{\theta }_P$ exactly, and the square-bracket factor is $\cos {\theta }_P+O({\cos }^3{\theta }_P)$. Keeping both denominator terms givesIt has no $R^{-2}$ term. This exceptional case matters when requesting an asymptotic formula without an additional restriction on the elastic moduli.

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, $k\ge 2$, writeA smooth background may be added without changing these leading jumps. Since $F_j^{\prime }=F_{j-1}$, differentiation expresses spatial jumps in terms of $p_j$ and time-derivative jumps. The restriction $k\ge 2$ lets displacement field and its first time derivative remain continuous and avoids an imposed impulsive source at the front.

Insert this local expansion in $\rho {\partial }_t^2{u}_i={\partial }_j(c_{ijpq}{\partial }_p{u}_q)$. The coefficient of $F_{k-2}$ isFor isotropic elasticity the slowness acoustic matrix is $\mu |\mathbf{p}{|}^{2}I+(\lambda +\mu )\mathbf{p}{\mathbf{p}}^T$. Its longitudinal eigenvalue is $(\lambda +2\mu )|\mathbf{p}{|}^2$. The P branch therefore gives the eikonal equation and polarizationThe next coefficient, $F_{k-1}$, isThe 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:These identities use $|\mathbf{n}{|}^2=1$, hence $n_q{\partial }_p{n}_q=0$, and $(\lambda +2\mu )/\alpha =\rho \alpha$. Their curvature terms cancel, leavingMultiplying by $a$ proves the requested P-wavefront discontinuity transport law,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 $\rho \alpha a^{2}J$ 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.

The spherical source potential $f(t-r/\alpha )/r$ gives leading longitudinal displacement field $-f^{\prime }(t-r/\alpha )/(\alpha r)$. Apply the local solid-fluid P-wave transmission coefficient to its potential at a smooth interface. In hyperbolic interface collimation of P-waves, transmitted rays are parallel, so ray-tube conservation for a P-wave jump makes the leading coefficient constant along each vertical ray. This yields the displayed jump for the first nonzero derivative discontinuity of the waveform. Smooth transverse derivatives multiply $f$ rather than $f^{\prime }$ and are one order more regular. The construction presupposes the indicated one-sided derivative jump exists; a source smooth to every order may have no jump.