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Write $n=(x-\xi )/R$ for the receiver direction. With the printed time convention $e^{i\omega t}$, an outgoing spherical wave has phase $e^{-i\omega R/v}$. The leading terms of the elastodynamic Green tensor are of order $1/({\alpha }^{2}R)$ and $1/({\beta }^{2}R)$, while the remaining terms have orders $1/(\omega v{R}^2)$ and $1/({\omega }^2{R}^3)$. Thus the far-field approximation requiresThis is an absolute asymptotic comparison of the tensor terms. At an angular node of a leading radiation component, the smaller near field can still dominate that component. The PDF prints an opposite phase sign in the near-field shear exponential; that would be incoming with this time convention. The outgoing leading terms, which determine all the requested far-field patterns, are unambiguous.

For a unit time-harmonic point force in direction $d$, the far-field P and S radiation from a point force isThe normalization here follows the given elastodynamic Green tensor; a separate force-density convention would include its corresponding density factor. If $\theta$ is the angle between $n$ and $d$, the signed radial P wave amplitude is $\cos \theta$, and the magnitude of the S wave amplitude is $\sin \theta$. The P wave has two oppositely signed lobes and a nodal plane perpendicular to $d$. The S wave is polarized in the projection of $d$ onto the receiver's tangent plane, has nodes along $\pm d$, and has its maximum on the perpendicular great circle. Rotating its planar section about $d$ gives the full toroidal S wave magnitude pattern.

For a point source with seismic moment tensor $M$, convolution with the derivatives of the Dirac delta function gives $u_i=M_{jk}{\partial }_{{\xi }_k}{G}_i^j$. In the far field, differentiating the outgoing phase contributes $i\omega n_k/v$; differentiating $R^{-1}$ or $n$ is a lower-order effect. Thus the point-moment elastodynamic displacement is

The double couple has $M=e_1{e}_2^{\mathsf{T}}+e_2{e}_1^{\mathsf{T}}$. Its double-couple radiation pattern therefore has signed radial P wave factor and tangential S wave vectorIn the $x_1{x}_2$ plane, $n=(\cos \varphi ,\sin \varphi ,0)$, giving $A_P=\sin 2\varphi$ and $A_S=\cos 2\varphi$, where the signed S wave component is along $(-\sin \varphi ,\cos \varphi ,0)$. The P wave nodal planes are $n_1=0$ and $n_2=0$, and the alternating lobes peak on their bisectors. The planar S wave lobes peak along the coordinate axes. For the complete three-dimensional pattern, write $n=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$:The full S wave has additional zeros at $\pm e_3$ as well as at the planar bisectors.

For the explosion source, $M=I$. Then $n^{\mathsf{T}}Mn=1$ and $(I-n{n}^{\mathsf{T}})Mn=0$. The explosion radiation in an isotropic elastic solid is therefore an isotropic radial P wave, with no S wave radiation. A dilatational point source is a gradient source; its far-field force is parallel to the propagation direction, so the transverse projection vanishes.

Each radius in the sketch is the magnitude of the angular factor, separately normalized for each mode. Blue and orange distinguish signs of the indicated radial or tangential component. The point-force sections contain its axis; the double-couple sections are in the $x_1{x}_2$ plane. The formulas above specify the full three-dimensional patterns.

Treat a small patch as the point tensile-crack seismic source at ${\mathbf{y}}_0$, with $M(t)=KV(t)$ and $K=\lambda I+2\mu \mathbf{n}{\mathbf{n}}^{\mathsf{T}}$. Let $\mathbf{e}=(\mathbf{x}-{\mathbf{y}}_0)/R$, $\gamma =\mathbf{e}\cdot \mathbf{n}$, $\mathbf{a}=K\mathbf{e}$, $m=\mathbf{e}\cdot K\mathbf{e}$ and $s=\mathrm{tr}K$. Source convolution and an integration by parts give the point-moment elastodynamic displacementThe derivatives act on angular factors and on retarded times as well as on powers of $R$; keeping only the explicit $1/R$ Green-function terms would miss the static displacement.

Set $V_P=V(t-R/\alpha )$, $V_S=V(t-R/\beta )$ andConvolving the elastodynamic Green tensor with the opening history givesUse ${\partial }_{j}R=e_j$, ${\partial }_j{e}_i=({\delta }_{ij}-e_i{e}_j)/R$, ${\partial }_j{V}_P=-e_j{\dot{V}}_P/\alpha$, andFor example, $K_{kj}{\partial }_j(e_i{e}_k/R)=[a_i+e_i(s-3m)]/R^2$, while differentiating the near-field angular tensor gives $[6a_i+3e_{i}s-15e_{i}m]/R^4$. Combining these with the endpoint terms in ${\partial }_{R}I$ yieldsFor the normal crack, $\mathbf{a}=\lambda \mathbf{e}+2\mu \gamma \mathbf{n}$, $m=\lambda +2\mu {\gamma }^2$, and $s=3\lambda +2\mu$. Explicitly, the three nonradiative vector coefficients in the last two lines areThus those lines are ${\mathbf{C}}_P{V}_P/({\alpha }^2{R}^2)+{\mathbf{C}}_S{V}_S/({\beta }^2{R}^2)+{\mathbf{C}}_{I}I/R^4$. The small-source approximation assumes the patch size is small relative to both $R$ and the wavelengths of interest; a finite crack is obtained by integrating the same point-patch expression over its surface with its local opening and normal.