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All wavevectors lie in the $xz$ plane and have the same tangential component. The incident P-wave travels upward toward $z=0$. In the lower solid there are a downward reflected P-wave, polarized parallel to its wavevector, and a downward reflected SV-wave, polarized in the $xz$ plane perpendicular to its wavevector. In the upper solid there are corresponding upward transmitted P- and SV-waves. The SH polarization decouples from this in-plane incident field. Their angles obey the Snell law for elastic and acoustic waves.

For a welded interface, the two in-plane displacement components are continuous:The two in-plane traction components are also continuous:These four independent scalar conditions determine the four reflected and transmitted amplitudes. This is mode conversion at a planar elastic interface.

Taking the divergence of the Cauchy momentum equation and commuting constant-coefficient derivatives givesThus the dilatation is a wave of speedTaking the curl, and using $\nabla \times \nabla f=0$, givesso the rotation propagates atThese are respectively the speeds of P-waves and S-waves.

The Reynolds number isWhen it is large, viscosity is negligible in most of the flow, but the inviscid outer solution cannot generally satisfy the no-slip boundary condition at a rigid wall. A thin boundary layer resolves this mismatch: tangential gradients remain of outer scale $\mathcal{L}$, normal gradients have the much shorter scale $\delta$, and the small viscous coefficient is offset by $u_{yy}=O(\mathcal{U}/{\delta }^2)$. Matching a viscous inner solution to an inviscid outer solution retains the leading viscous effects without solving the full equations uniformly everywhere.

At any point choose orthonormal coordinates. The antisymmetries and pair symmetry of the Riemann curvature tensor leave only $R_{1212}$ independent. In these coordinates,while $R=2R_{1212}$. Since this is a tensor identity, it follows in every coordinate system that the Ricci tensor in two dimensions satisfiesThe Einstein tensor consequently vanishes identically. Hence the two-dimensional vacuum field equation contains no local gravitational dynamics, while coupling the unmodified equation to matter would require $T_{\alpha \beta }=0$. This is the basic degeneracy of two-dimensional general relativity.

Contracting the stated Einstein field equations with $g^{\alpha \beta }$ in $n$ dimensions givesThus $R=-2T/(n-2)$, and substitution back yields the Trace-reversed Einstein field equations in D dimensions

Because the one-particle spectrum is bounded above, a negative temperature is possible. A population-inverted configuration with more particles in the $+ϵ$ level than in the $-ϵ$ level haswhich requires $\beta <0$. Equivalently, above the maximum-entropy energy $E=0$, adding energy reduces the number of compatible microstates, so $\partial S/\partial E=1/T<0$. The limiting configuration with every particle at $+ϵ$ has $E=Nϵ$, zero entropy, and $T\to 0^{-}$.