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For the seismic moment tensor $M=e_1{e}_2^{\mathsf{T}}+e_2{e}_1^{\mathsf{T}}$, source differentiation of the outgoing elastodynamic Green tensor gives $A_P=n^{\mathsf{T}}Mn$ and ${\mathbf{A}}_S=(I-n{n}^{\mathsf{T}})Mn$. In the $x_1{x}_2$ plane the signed radial P wave and tangential S wave factors are $\sin 2\varphi$ and $\cos 2\varphi$. The full S wave magnitude squared is $n_1^2+n_2^2-4n_1^2{n}_2^2$.

Opposite forces separated along the unit direction $\mathbf{e}$, with fixed force times separation $D$, have this distributional body force limit. Its net force is zero. A symmetric seismic moment tensor is a sum of such dipoles along its principal directions.

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.

Use fixed spatial coordinates in a stationary heterogeneous Earth and define the operatorHere the elastic stiffness tensor has its minor and major symmetries, and the physical body force per unit volume is $q_i=\rho f_i$. The retarded elastodynamic Green tensor $G_i^j(\mathbf{x},\mathbf{\xi },t)$ is the displacement in direction $i$ due to a unit physical force impulse in direction $j$ at $\mathbf{\xi }$:It satisfies the actual Earth's homogeneous boundary conditions, including zero traction on the free surface and continuity of displacement and traction at bonded internal interfaces. Integration through $t=0$ gives $\rho (\mathbf{x}){\partial }_t{G}_i^j(0^{+})={\delta }_i^j\delta (\mathbf{x}-\mathbf{\xi })$, with $G(0^{+})=0$. No spherical symmetry is required.

For zero initial conditions, form the convolutionApplying $\mathcal{L}$ to this expression uses the defining Dirac delta function to give $\rho (\mathbf{x})f_i(\mathbf{x},t)$. Causality gives zero initial conditions, and each Green column supplies the required boundary conditions. The positive-energy uniqueness argument then identifies this expression with the physical response. The source factor is $\rho (\mathbf{\xi })$ because $f$ is force per unit mass, whereas the Green tensor was normalized to unit force.

A force dipole is a limiting pair of opposite point forces separated by a small vector. Put $+F_j$ at $\mathbf{\xi }+\epsilon \mathbf{d}/2$ and $-F_j$ at $\mathbf{\xi }-\epsilon \mathbf{d}/2$, and hold $D_{jk}=\epsilon F_j{d}_k$ fixed as $\epsilon \to 0$. Expansion of the two Dirac delta functions givesIts resultant force is zero, and integration by parts gives its radiated displacement as ${\int }_0^t{D}_{jk}(\tau ){\partial }_{{\xi }_k}{G}_i^j(\mathbf{x},\mathbf{\xi },t-\tau )d\tau$. Thus the source derivative of the elastodynamic Green tensor represents a dipole with force direction $j$ and separation direction $k$.

For the surface source choose a normal $\mathbf{n}$ from the minus side to the plus side and defineThe same normal is used in both tractions. If the elastic coefficients are continuous through the surface, the distributional gradient of a piecewise smooth displacement is ${\partial }_j{u}_i=({\partial }_j{u}_i{)}_{\mathrm{reg}}+b_i{n}_j{\delta }_{\Sigma }$. Its stress therefore has a singular part $c_{ijkl}{b}_k{n}_l{\delta }_{\Sigma }$. Taking the divergence includes the regular stress jump, so a field satisfying the source-free equations on both sides has equivalent sourceIn particular, the signs follow from the chosen jump convention, not from an unspecified orientation. The elastodynamic surface-jump representation is consequentlywith the Green tensor evaluated at $(\mathbf{x},\mathbf{\xi },t-\tau )$. The first term is a surface distribution of forces, and the second is a surface distribution of dipoles.

One can derive the same formula without multiplying discontinuous material coefficients by a surface delta. Apply integration by parts on the two sides of $\Sigma$ to the field $u$ and the reciprocal Green field $v$. Major symmetry cancels the volume strain terms; the spatial identity iswhen the temporal terms are handled by the retarded time convolution. More explicitly, take $v_i(\mathbf{\xi },\tau )=G_i^a(\mathbf{\xi },\mathbf{x},t-\tau )$, extend the zero-initial-data solution causally to $\tau <0$, and integrate over all source times. Integration by parts cancels the two temporal terms, with zero endpoint terms; the causal supports subsequently reduce the representation to $0\le \tau \le t$. The Green impulse contributes $-u_a(\mathbf{x},t)$ on the left. The two outward face normals are $\mathbf{n}$ and $-\mathbf{n}$, so their sum on the right is $-[u_i]t_i(v)+v_i[t_i]$. Moving it to the other side gives $[u_i]t_i(v)-v_i[t_i]$. The same identity applied to two force Green fields with interchanged source and receiver proves the reciprocal relation $G_i^j(\mathbf{x},\mathbf{\xi },t)=G_j^i(\mathbf{\xi },\mathbf{x},t)$; hence the displayed surface formula follows. On a bonded material interface the Green displacement and common Green traction are continuous: $[u_i]t_i(v)$ is then the unambiguous version of the dipole term, evaluated with the matching one-sided coefficients. External free-surface terms vanish because both tractions vanish.

For a fault in isotropic material, the Lamé parameters giveTangential slip obeys $\mathbf{b}\cdot \mathbf{n}=0$, and ordinary fault traction is continuous across the two faces. Therefore the source is a double-couple fault source withFor a uniform patch of area $A$ with $\mathbf{b}=b{\mathbf{e}}_1$ and $\mathbf{n}={\mathbf{e}}_3$, its integrated seismic moment tensor isThe $13$ dipole is a force in the slip direction separated in the normal direction; the $31$ dipole is a normal force separated in the slip direction. Their equal and opposite torques cancel. The symmetric trace-free seismic moment tensor has eigenvalues $M_0,-M_0,0$, with zero resultant force and torque. A temporal stress drop on a fault is not a jump of traction between its two simultaneously matching faces; it does not add a single-force layer to this tangential-slip representation.

Complete the crack normal to an orthonormal basis $({\mathbf{e}}_1,{\mathbf{e}}_2,\mathbf{n})$ with the first two vectors in the crack plane. The seismic moment tensor is diagonal in this basis:Each term gives an elastic force dipole, $f_i=-D{e}_i{e}_j{\partial }_j\delta$: two equal and opposite forces in direction $\mathbf{e}$, separated along $\mathbf{e}$, with moment $D$. Hence there are two identical in-plane dipoles of strength $\lambda V$ and one normal dipole of strength $(\lambda +2\mu )V$. For positive opening and ordinary positive Lamé parameters, all are extensional and the normal one is larger by $2\mu V$.

More generally, stable isotropic linear elasticity requires $\mu >0$ and $3\lambda +2\mu >0$. These imply $\lambda +2\mu >|\lambda |$, so the normal dipole has greater magnitude even if an auxetic material has negative $\lambda$ and compressive in-plane dipoles. The point source has zero net force, because it differentiates a delta function, and zero net torque, because its moment tensor is symmetric. Its normal moment distinguishes opening from a pure shear dislocation.

Let $d(\mathbf{y},t)=[u^{[n]}]$ denote the signed opening, positive when the faces separate, and let $S$ be the crack surface. Its displacement jump is $\mathbf{b}=d\mathbf{n}$. The singular strain associated with this jump has moment density $c_{ijkl}{b}_k{n}_l$. For the isotropic elastic stiffness tensor, contraction givesThe equivalent distributional body force is the negative divergence of this prescribed moment density:The resulting forced elastic wave in an isotropic solid equation is $\rho u_{i,tt}={\partial }_j{\sigma }_{ij}+f_i$. This is a tensile-crack seismic source; it need not be represented as an ordinary smooth volume force. For a small nearly planar patch of area $S_0$, uniform normal $\mathbf{n}$ and opening $d(t)$, define its volume change $V(t)=S_{0}d(t)$. Its seismic moment tensor and point-source force areThe sign convention makes a positive opening a positive extensional moment. Reversing the convention for the displacement jump changes both signs together; the force is independent of reversing the orientation of the crack normal.

Integrating the equivalent dipole body force against the elastodynamic Green tensor by parts gives the displayed time convolution. For a fixed seismic moment tensor shape $M_{kj}=K_{kj}V(t)$, its $R^{-1}$ terms are $[\mathbf{e}(\mathbf{e}\cdot K\mathbf{e})\dot{V}(t-R/\alpha )/{\alpha }^3+(K\mathbf{e}-\mathbf{e}(\mathbf{e}\cdot K\mathbf{e}))\dot{V}(t-R/\beta )/{\beta }^3]/(4\pi \rho R)$. Derivatives of the angular projectors and the retarded near-field integral additionally produce $R^{-2}$ and $R^{-4}$ coefficients, yielding the final static field after the source stops.

A localized zero-resultant seismic moment tensor represents the leading force-dipole part of an elastic wave source through the displayed equivalent body force. For a prescribed displacement jump $b_i$ across a surface with normal $n_j$, its moment density is $c_{ijkl}{b}_k{n}_l$, where $c$ is the elastic stiffness tensor. Spatial integration gives the point-source moment when the source is small compared with the wavelengths.

A small planar crack of area $S$ opening normally by $d(t)$ has volume change $V=Sd$ and the displayed seismic moment tensor. Its principal dipoles are $\lambda V,\lambda V,(\lambda +2\mu )V$: two act in the crack plane and the largest acts normally for positive opening and ordinary positive Lamé parameters. The definition follows by contracting the isotropic elastic stiffness tensor with two copies of the crack normal.