Elastic Waves in Solids: Longitudinal and Transverse Modes, the Cauchy Stress Tensor

Intuition Beginner

Strike a steel rail with a hammer. Two kinds of waves race through the metal. One is a compression wave -- regions of the steel squeeze together and spring apart, like pushing a Slinky end-to-end. The other is a shear wave -- layers of metal slide sideways past each other, like flicking a rope up and down.

These are longitudinal and transverse waves. In a longitudinal wave, the motion of the material points is along the direction the wave travels. In a transverse wave, the material moves perpendicular to the travel direction. Sound in air is purely longitudinal because air has no resistance to shearing -- it cannot hold a shape. But a solid has both kinds of stiffness: it resists being compressed and it resists being bent. So both wave types propagate.

Seismologists call these P-waves and S-waves. The P stands for "primary" (also "push-pull") because these compression waves arrive first at a seismometer. The S stands for "secondary" (also "shake") because the shear waves arrive later. The reason is simple: the longitudinal wave is faster. In a typical rock, the P-wave speed is about 6 km/s and the S-wave speed is about 3.5 km/s. After an earthquake, the P-wave hits your station first, and seconds later the more destructive S-wave arrives. Earthquake early-warning systems exploit this time gap -- the few seconds between the P-wave detection and the S-wave arrival can trigger automatic shutdowns of trains, gas lines, and nuclear reactors.

The tool that unifies the description of all these waves is the stress tensor ${\sigma }_{ij}$. This is a $3\times 3$ matrix that encodes the force per unit area acting on every possible surface through every point of the body. The diagonal entries are normal stresses (compression or tension pulling perpendicular to a surface). The off-diagonal entries are shear stresses (forces parallel to the surface). The tensor is symmetric (${\sigma }_{ij}={\sigma }_{ji}$) because otherwise the material would spontaneously spin -- conservation of angular momentum forces this symmetry.

The wave speeds depend on the material's elastic constants. For an isotropic solid (one whose properties are the same in every direction), there are only two independent constants: the Lame parameters $\lambda$ and $\mu$. The longitudinal wave speed is $c_L=\sqrt{(\lambda +2\mu )/\rho }$ and the transverse wave speed is $c_T=\sqrt{\mu /\rho }$, where $\rho$ is the mass density. Since $\lambda >0$ and $\mu >0$, the longitudinal wave is always faster -- typically about 1.7 to 1.9 times faster in common materials.

Visual Beginner

Mode	Particle motion	Wave speed	Example
Longitudinal (P-wave)	Parallel to propagation	$c_L=\sqrt{(\lambda +2\mu )/\rho }$	Sound in a solid
Transverse (S-wave)	Perpendicular to propagation	$c_T=\sqrt{\mu /\rho }$	Vibrating string

Worked example Beginner

For steel with density $\rho =7800{kg/m}^3$, Young's modulus $E=200\mathrm{GPa}$, and Poisson's ratio $\nu =0.3$, compute the longitudinal and transverse wave speeds.

The Lame parameters are $\mu =E/(2(1+\nu ))=200/(2\times 1.3)\approx 76.9\mathrm{GPa}$ and $\lambda =E\nu /((1+\nu )(1-2\nu ))=200\times 0.3/(1.3\times 0.4)\approx 115.4\mathrm{GPa}$.

Longitudinal speed: $c_L=\sqrt{(115.4+2\times 76.9)\times 10^9/7800}=\sqrt{269.2\times 10^9/7800}=\sqrt{3.45\times 10^7}\approx 5875m/s$.

Transverse speed: $c_T=\sqrt{76.9\times 10^9/7800}=\sqrt{9.86\times 10^6}\approx 3140m/s$.

The ratio $c_L/c_T\approx 1.87$. In an earthquake 100 km away, the P-wave arrives in about 17 seconds and the S-wave in about 32 seconds. That 15-second gap is what early-warning systems use.

Check your understanding Beginner

Formal definition Intermediate+

The displacement field and the strain tensor

Let $\mathbf{u}(\mathbf{x},t)$ be the displacement of a material point from its reference position $\mathbf{x}$. The infinitesimal strain tensor is

$${\epsilon }_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right).$$

This symmetric second-order tensor measures the local deformation (stretching and shearing) of the material. The diagonal components ${\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{33}$ are normal strains (fractional length changes along each axis). The off-diagonal components ${\epsilon }_{12},{\epsilon }_{13},{\epsilon }_{23}$ are half the engineering shear strains. The trace ${\epsilon }_{kk}=\nabla \cdot \mathbf{u}$ is the volumetric strain (fractional volume change).

The antisymmetric part of the displacement gradient, ${\omega }_{ij}=\frac{1}{2}(\partial u_i/\partial x_j-\partial u_j/\partial x_i)$, represents an infinitesimal rigid rotation -- it produces no deformation and stores no elastic energy. Only the symmetric part ${\epsilon }_{ij}$ enters the constitutive law.

The Cauchy stress tensor

The Cauchy stress principle states that the internal force per unit area (the traction vector) on any surface with unit normal $\hat{n}$ at point $\mathbf{x}$ is

$$\mathbf{t}(\hat{n},\mathbf{x})=\sigma (\mathbf{x})\hat{n},$$

where $\sigma$ is the Cauchy stress tensor. Cauchy's theorem proves that the existence of $\sigma$ follows from balance of linear momentum and the tetrahedron argument (see Master section). Symmetry ${\sigma }_{ij}={\sigma }_{ji}$ follows from balance of angular momentum (absence of body torques).

The physical meaning of the components: ${\sigma }_{ij}$ is the $j$-component of the traction on the face with outward normal ${\hat{e}}_i$. The diagonal entries are normal stresses (positive = tension, negative = compression). The off-diagonal entries are shear stresses. For a fluid at rest, ${\sigma }_{ij}=-p{\delta }_{ij}$, reducing to the scalar pressure $p$.

The Cauchy momentum equation

Newton's second law for an infinitesimal volume element gives

$$\boxed{\rho {\ddot{u}}_i=\frac{\partial {\sigma }_{ij}}{\partial x_j}+f_i,}$$

where $f_i$ is the body force per unit volume (e.g., gravity). This is the Cauchy momentum equation -- the fundamental balance law of continuum mechanics. It states that the mass density times acceleration equals the divergence of the stress plus the body force. For a fluid, this reduces to the Euler equation (inviscid) or Navier-Stokes equation (viscous).

Hooke's law and the elastic stiffness tensor

For small strains, stress is a linear function of strain:

$${\sigma }_{ij}=C_{ijkl}{\epsilon }_{kl},$$

where $C_{ijkl}$ is the fourth-order elastic stiffness tensor. Summation over repeated indices $k,l$ is implied. The stiffness tensor has the minor symmetries $C_{ijkl}=C_{jikl}=C_{ijlk}$ (from symmetry of $\sigma$ and $\epsilon$) and the major symmetry $C_{ijkl}=C_{klij}$ (from the existence of a strain energy density). These reduce the 81 components of a general fourth-order tensor in three dimensions to 21 independent constants -- the same number as for a fully anisotropic crystal.

Isotropic linear elasticity

For an isotropic material (properties independent of direction), the stiffness tensor reduces to just two independent constants -- the Lame parameters $\lambda$ and $\mu$:

$$C_{ijkl}=\lambda {\delta }_{ij}{\delta }_{kl}+\mu ({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}).$$

Hooke's law becomes

$${\sigma }_{ij}=\lambda {\epsilon }_{kk}{\delta }_{ij}+2\mu {\epsilon }_{ij}.$$

The parameter $\mu$ is the shear modulus (resistance to shearing deformation). The parameter $\lambda$ is related to resistance to volume change. Inverting gives the strain in terms of stress:

$${\epsilon }_{ij}=\frac{1+\nu }{E}{\sigma }_{ij}-\frac{\nu }{E}{\sigma }_{kk}{\delta }_{ij},$$

where Young's modulus $E=\mu (3\lambda +2\mu )/(\lambda +\mu )$ and Poisson's ratio $\nu =\lambda /(2(\lambda +\mu ))$.

The Navier equation

Substituting Hooke's law into the Cauchy momentum equation and expressing strain in terms of the displacement field gives the Navier equation (also called the Lame-Navier equation):

$$\boxed{\rho \ddot{\mathbf{u}}=(\lambda +\mu )\nabla (\nabla \cdot \mathbf{u})+\mu {\nabla }^2\mathbf{u}+\mathbf{f}.}$$

This is a vector wave equation -- three coupled PDEs for the three components of $\mathbf{u}$. It is the central equation of linear elastodynamics. Setting $\ddot{\mathbf{u}}=0$ gives the equilibrium (elastostatic) equation $0=(\lambda +\mu )\nabla (\nabla \cdot \mathbf{u})+\mu {\nabla }^2\mathbf{u}+\mathbf{f}$, which governs static deformation problems (beam bending, structural analysis).

Key derivation Intermediate+

Separation into longitudinal and transverse waves

The Navier equation for the displacement field $\mathbf{u}$ (with $\mathbf{f}=0$) is:

$$\rho \ddot{\mathbf{u}}=(\lambda +\mu )\nabla (\nabla \cdot \mathbf{u})+\mu {\nabla }^2\mathbf{u}.$$

Decompose $\mathbf{u}$ into its irrotational (longitudinal) and solenoidal (transverse) parts via the Helmholtz decomposition: $\mathbf{u}=\nabla \varphi +\nabla \times \mathbf{A}$ where $\nabla \cdot \mathbf{A}=0$.

Scalar potential (P-wave). The irrotational part $\nabla \varphi$ has $\nabla \times (\nabla \varphi )=0$ (the curl of a gradient vanishes) and $\nabla \cdot (\nabla \varphi )={\nabla }^2\varphi$. Substituting into the Navier equation:

$$\rho \nabla \ddot{\varphi }=(\lambda +\mu )\nabla ({\nabla }^2\varphi )+\mu {\nabla }^2(\nabla \varphi )=(\lambda +2\mu )\nabla ({\nabla }^2\varphi ).$$

Taking the gradient outside: $\nabla [\rho \ddot{\varphi }-(\lambda +2\mu ){\nabla }^2\varphi ]=0$. Choosing the integration constant to be zero gives

$$\rho \ddot{\varphi }=(\lambda +2\mu ){\nabla }^2\varphi ,$$

a wave equation with speed $\boxed{c_L=\sqrt{(\lambda +2\mu )/\rho }}$.

Vector potential (S-wave). The solenoidal part $\nabla \times \mathbf{A}$ has $\nabla \cdot (\nabla \times \mathbf{A})=0$ (the divergence of a curl vanishes). Substituting:

$$\rho \nabla \times \ddot{\mathbf{A}}=\mu {\nabla }^2(\nabla \times \mathbf{A}).$$

This gives

$$\rho \ddot{\mathbf{A}}=\mu {\nabla }^2\mathbf{A},$$

a wave equation with speed $\boxed{c_T=\sqrt{\mu /\rho }}$.

Plane-wave solutions

Look for solutions of the form $\mathbf{u}(\mathbf{x},t)=\mathbf{p}f(\mathbf{n}\cdot \mathbf{x}-ct)$ where $\mathbf{n}$ is the propagation direction, $\mathbf{p}$ is the polarisation direction, and $f$ is an arbitrary waveform.

Longitudinal wave: $\mathbf{p}\parallel \mathbf{n}$, speed $c_L$. The displacement is along the propagation direction. The particle motion is compression-rarefaction. The wave is irrotational ($\nabla \times \mathbf{u}=0$) and involves volume change ($\nabla \cdot \mathbf{u}\ne 0$).

Transverse wave: $\mathbf{p}\perp \mathbf{n}$, speed $c_T$. The displacement is perpendicular to the propagation direction. The particle motion is shearing. The wave is solenoidal ($\nabla \cdot \mathbf{u}=0$) and involves no volume change. There are two independent transverse polarisations (in the plane perpendicular to $\mathbf{n}$).

Since $\lambda >0$ and $\mu >0$, we have $c_L>c_T$ always. The ratio $c_L/c_T=\sqrt{(\lambda +2\mu )/\mu }=\sqrt{2(1-\nu )/(1-2\nu )}$ depends only on Poisson's ratio. For $\nu =0.25$ (a common value for rocks), $c_L/c_T=\sqrt{3}\approx 1.73$.

Worked example: granite

For granite with $\rho =2700{kg/m}^3$, $E=50\mathrm{GPa}$, $\nu =0.25$:

$\mu =E/(2(1+\nu ))=50/(2\times 1.25)=20\mathrm{GPa}$, $\lambda =E\nu /((1+\nu )(1-2\nu ))=50\times 0.25/(1.25\times 0.5)=20\mathrm{GPa}$.

$c_L=\sqrt{(20+40)\times 10^9/2700}=\sqrt{2.22\times 10^7}\approx 4714m/s$.

$c_T=\sqrt{20\times 10^9/2700}=\sqrt{7.41\times 10^6}\approx 2722m/s$.

The ratio $c_L/c_T=\sqrt{3}\approx 1.73$ as expected for $\nu =0.25$.

Reflection at a boundary

A P-wave incident on a planar interface between two elastic media produces both a reflected P-wave and a reflected S-wave (mode conversion), plus transmitted P and S waves. The reflection and transmission coefficients are determined by four boundary conditions: continuity of the two displacement components and continuity of the two traction components at the interface. This four-fold coupling between P and S modes at boundaries is what makes seismic wave modelling rich and challenging.

For the special case of normal incidence, mode conversion vanishes and the problem reduces to the acoustic case. With impedances $Z_1={\rho }_1{c}_{L1}$ and $Z_2={\rho }_2{c}_{L2}$:

$$R=\frac{Z_2-Z_1}{Z_1+Z_2},T=\frac{2Z_2}{Z_1+Z_2}.$$

When $Z_2\gg Z_1$ (wave hits a much stiffer medium), $R\approx 1$ -- most energy reflects. When $Z_1\approx Z_2$ (well-matched media), $R\approx 0$ -- most energy transmits.

Bridge. The Helmholtz decomposition and the separation into P-waves and S-waves builds toward the analysis of seismic wave propagation in geophysics, where the speed contrast between P and S waves is the primary diagnostic for Earth's internal structure. The foundational reason the separation works is that the isotropic elastic stiffness tensor has exactly two independent parameters, which generate two and only two wave speeds. This generalises to anisotropic media where the Christoffel equation produces three wave speeds (one quasi-longitudinal, two quasi-transverse). The central insight is that the number of independent wave modes equals the number of degrees of freedom in the stiffness tensor, and the Navier equation is the load-bearing structure that connects the constitutive law (Hooke's law) to the wave speeds. This is dual to the analysis of electromagnetic waves in anisotropic media, where the dielectric tensor plays the role of the stiffness tensor.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not contain the Cauchy stress tensor, the strain tensor, the Navier equation, or the theory of elastic waves. The closest infrastructure is the matrix algebra and the analysis of PDEs. Formalising continuum mechanics would require: the definition of stress as a linear map from normals to tractions; Cauchy's theorem (existence of the stress tensor from momentum balance); the strain-displacement relation; Hooke's law as a fourth-order constitutive tensor acting on the second-order strain tensor; and the Navier equation as a second-order PDE system for the vector displacement field. The plane-wave analysis requires spectral theory for the Christoffel matrix. The Rayleigh-wave dispersion relation requires solving a characteristic equation with transcendental coefficients. None of this infrastructure exists in Mathlib. The strain energy density $W=\frac{1}{2}{C}_{ijkl}{\epsilon }_{ij}{\epsilon }_{kl}$ would require tensors over EuclideanSpace (Fin 3) Real with the appropriate symmetries formalised as type-class instances. This unit ships without a Lean module.

Advanced results Master

Rayleigh surface waves

At a free surface of an elastic half-space ($z\ge 0$, with $z=0$ the surface), the Navier equation admits a solution that decays exponentially with depth and propagates along the surface: the Rayleigh wave. Seek a solution of the form $\mathbf{u}\propto e^{i(kx-\omega t)}$ with amplitude decaying as $e^{-\alpha z}$ for the longitudinal part and $e^{-\beta z}$ for the transverse part. The stress-free boundary conditions ${\sigma }_{zz}={\sigma }_{xz}=0$ at $z=0$ produce a $2\times 2$ system whose vanishing determinant gives the Rayleigh equation:

$${\left(2-\frac{c_R^2}{c_T^2}\right)}^2=4\sqrt{1-\frac{c_R^2}{c_L^2}}\sqrt{1-\frac{c_R^2}{c_T^2}}.$$

Squaring and rearranging gives a cubic in $(c_R/c_T{)}^2$. For $\nu =0.25$, $c_R\approx 0.9194c_T\approx 0.53c_L$. Rayleigh waves are the most destructive in earthquakes because their energy is confined to a depth of about one wavelength beneath the surface, concentrating the shaking. They are also dispersive when the half-space is layered, which is the basis for using Rayleigh-wave dispersion to infer subsurface velocity structure (a technique called MASW -- multichannel analysis of surface waves).

Love waves

In a layered medium (a soft layer of thickness $h$ over a stiffer half-space), horizontally polarised shear waves (SH-polarised) can be trapped in the layer by total internal reflection and propagate as guided modes -- Love waves. The displacement is horizontal and transverse to the propagation direction. The dispersion relation for a single layer over a half-space is

$$\tan \left(qh\right)=\frac{{\mu }_{2}s}{{\mu }_{1}q},$$

where $q^2={\omega }^2/c_{T1}^2-k^2$ and $s^2=k^2-{\omega }^2/c_{T2}^2$ with $c_{T1}<c_{T2}$ (soft layer over stiff half-space). Love waves are dispersive (unlike body waves in a homogeneous medium), with phase speed depending on frequency. At long periods the phase speed approaches $c_{T2}$ (the deeper material controls the wave); at short periods it approaches $c_{T1}$ (the wave is trapped in the surface layer). Love's 1911 analysis of these waves was one of the earliest successes of theoretical seismology.

Stoneley interface waves

At the interface between two solids, a wave analogous to the Rayleigh wave propagates along the interface: the Stoneley wave. It exists only when the elastic contrast between the two media is small enough (the ratio of shear moduli and densities must fall within a certain range). The dispersion relation is more complex than the Rayleigh equation because both media contribute exponentially decaying fields. Stoneley waves are important in borehole logging and in the analysis of wave propagation across geological interfaces.

Anisotropic elasticity and the Christoffel equation

In an anisotropic medium, the elastic stiffness tensor $C_{ijkl}$ has up to 21 independent constants (for the most general triclinic crystal). The plane-wave ansatz $\mathbf{u}=\mathbf{p}f(\mathbf{n}\cdot \mathbf{x}-ct)$ leads to the Christoffel equation

$${\Gamma }_{ik}{p}_k=\rho c^2{p}_i,{\Gamma }_{ik}=C_{ijkl}{n}_j{n}_l.$$

The Christoffel matrix $\Gamma$ is symmetric and positive definite (from the positive-definiteness of the strain energy), so it has three real positive eigenvalues $\rho c_1^2\le \rho c_2^2\le \rho c_3^2$ with three mutually orthogonal eigenvectors. For a given propagation direction $\mathbf{n}$, there are three wave modes:

1. The fastest is quasi-longitudinal (polarisation roughly along $\mathbf{n}$).
2. The two slower modes are quasi-transverse (polarisations roughly perpendicular to $\mathbf{n}$).

In general, the polarisation directions are neither exactly parallel nor perpendicular to $\mathbf{n}$ -- they are the eigenvectors of $\Gamma$, not the geometric axes. This "quasi" character is a hallmark of anisotropy. Only for isotropic media (and for certain high-symmetry directions in anisotropic crystals) do the modes become purely longitudinal and purely transverse.

The symmetry class of the crystal determines the number of independent elastic constants: cubic (3), hexagonal/transversely isotropic (5), tetragonal (6 or 7), orthotropic (9), monoclinic (13), triclinic (21). The Earth's mantle is approximately transversely isotropic due to the alignment of olivine crystals, and seismic anisotropy is a major tool for mapping mantle flow patterns.

Elastodynamics and the PDE connection

The Navier equation $\rho \ddot{\mathbf{u}}=(\lambda +\mu )\nabla (\nabla \cdot \mathbf{u})+\mu {\nabla }^2\mathbf{u}+\mathbf{f}$ is a system of three coupled second-order linear PDEs with constant coefficients (for a homogeneous medium). From the perspective of the PDE methods chapter 09.02.17 pending, it is:

• Hyperbolic: its characteristic surfaces are the wavefronts of the P-wave and S-wave, and signals propagate at finite speed.
• Linear: the superposition principle applies; general solutions are built from plane-wave and Green's-function solutions.
• Constant-coefficient: Fourier analysis (plane-wave decomposition) fully solves the Cauchy problem in an infinite domain.

The Green's function (the response to a point impulse $\mathbf{f}={\mathbf{F}}_0\delta (\mathbf{x})\delta (t)$) is the Stokes solution for elastodynamics, derived by George Gabriel Stokes in 1849. It consists of a near-field term (decaying as $1/r^2$, with both longitudinal and transverse character) and two far-field terms (decaying as $1/r$, one radiating at speed $c_L$ and one at speed $c_T$). The Stokes solution is the foundation of seismic source theory: an earthquake is modelled as a point force (or a moment tensor) acting in the Earth, and the resulting displacement field is the convolution of the source time function with the elastodynamic Green's function.

Energy estimates for the Navier equation follow from the identity derived in Exercise 4. The total elastic energy $E=\int (T+W)dV$ satisfies

$$\frac{dE}{dt}=-{\oint }_{\partial V}\mathbf{S}\cdot \hat{n}dA+\int \mathbf{f}\cdot \dot{\mathbf{u}}dV,$$

where $\mathbf{S}=-\sigma \cdot \dot{\mathbf{u}}$ is the Umov-Poynting vector. Without body forces and with appropriate boundary conditions (fixed or free), $E$ is conserved. This energy identity is the starting point for the well-posedness theory of the Navier equation: existence and uniqueness of solutions, continuous dependence on initial data, and finite-speed propagation. These results parallel the theory for the scalar wave equation studied in 09.02.06 but require more technical machinery because the Navier equation is a system, not a scalar equation.

Seismic wave theory

Seismology applies elastic wave theory to the Earth. The Earth is approximately spherically symmetric (to first order), and the P-wave and S-wave speeds increase with depth through the crust, upper mantle, transition zone, and lower mantle. The preliminary reference Earth model (PREM) gives $c_L(z)$ and $c_T(z)$ as functions of depth.

Key seismological observations:

• Shadow zone: P-waves from a shallow earthquake are absent in the angular range $103^{\circ }$--$142^{\circ }$ from the epicentre. This shadow zone is caused by refraction at the core-mantle boundary (CMB), where the P-wave speed drops from $\sim 13.7$ km/s in the lower mantle to $\sim 8.0$ km/s in the liquid outer core. The existence of the shadow zone was the evidence used by Oldham (1906) and Gutenberg (1914) to infer the existence of the liquid core.

• S-wave cutoff: S-waves do not propagate through the liquid outer core ($\mu =0$). S-waves from earthquakes are observed only at distances $<103^{\circ }$, providing direct evidence that the outer core is liquid. The solid inner core was inferred from PKIKP phases by Lehmann (1936).

• Surface waves: Rayleigh and Love waves dominate the seismogram at periods of 10--300 seconds and are the primary tool for studying crust and upper-mantle structure. Their dispersion (frequency-dependent speed) is inverted to obtain velocity-depth profiles.

• Normal modes: The Earth rings like a bell after a large earthquake, with discrete normal-mode frequencies determined by the radial velocity structure. The longest-period mode (${}_0{S}_0$, the "football mode") has a period of about 20.5 minutes. Normal-mode seismology provides the most tightly constrained models of Earth's deep interior.

Proof: Cauchy's stress theorem

Proposition (Cauchy's stress theorem). Let $\mathbf{t}(\hat{n},\mathbf{x})$ be the traction vector on a surface with outward normal $\hat{n}$ at point $\mathbf{x}$ in a continuum body. If balance of linear momentum holds and $\mathbf{t}$ depends continuously on $\hat{n}$, then there exists a unique second-order tensor field $\sigma (\mathbf{x})$ such that $\mathbf{t}=\sigma \hat{n}$.

Proof (Cauchy's tetrahedron argument). Consider a tetrahedron with three faces having outward normals $-{\hat{e}}_1,-{\hat{e}}_2,-{\hat{e}}_3$ and a fourth face with normal $\hat{n}$. Let the areas of the coordinate faces be $\Delta S_i=|n_i|\Delta S$ where $\Delta S$ is the area of the slant face. Balance of linear momentum on the tetrahedron states:

$$\mathbf{t}(\hat{n})\Delta S+\sum _{i=1}^3\mathbf{t}(-{\hat{e}}_i)\Delta S_i+\mathbf{f}\Delta V=\rho \mathbf{a}\Delta V,$$

where $\Delta V$ is the volume, $\mathbf{f}$ is the body force per unit volume, and $\mathbf{a}$ is the acceleration. Define $\mathbf{t}(-{\hat{e}}_i)=-{\sigma }^T{\hat{e}}_i$ where ${\sigma }_{ij}$ is the $j$-component of the traction on the face with normal ${\hat{e}}_i$. As the tetrahedron shrinks to a point ($\Delta V\to 0$), the volume terms ($\mathbf{f}\Delta V$ and $\rho \mathbf{a}\Delta V$) vanish faster than the surface terms ($\Delta V\sim h^3$ vs. $\Delta S\sim h^2$). Taking the limit:

$$\mathbf{t}(\hat{n})=\sum _{i=1}^3{n}_i({\sigma }^T{\hat{e}}_i)={\sigma }^T\hat{n}.$$

Relabelling ${\sigma }^T\to \sigma$ (or equivalently defining ${\sigma }_{ij}$ as the traction component on the $i$-face in the $j$-direction) gives $\mathbf{t}=\sigma \hat{n}$. Uniqueness follows from the linearity in $\hat{n}$: if ${\sigma }_1\hat{n}={\sigma }_2\hat{n}$ for all $\hat{n}$, then $({\sigma }_1-{\sigma }_2)\hat{n}=0$ for all $\hat{n}$, which forces ${\sigma }_1={\sigma }_2$. $\square$

Synthesis

The Cauchy stress tensor and the Navier equation constitute the load-bearing structure of linear elastodynamics: the stress tensor encodes the internal forces, Hooke's law provides the constitutive closure between stress and strain, and the Navier equation governs the resulting wave motion. The central insight is that the symmetry of the stress tensor (angular momentum balance) and the linearity of Hooke's law (small-strain assumption) together produce exactly two wave modes with well-defined speeds in isotropic media, and the Helmholtz decomposition provides the tool that separates them. This generalises from isotropic to anisotropic media (the Christoffel equation produces three modes), and the foundational reason the decomposition works is that the elastic stiffness tensor has the same symmetries as the strain tensor -- a consequence of energy conservation. The Rayleigh, Love, and Stoneley waves are boundary-guided modes that emerge from the same Navier equation with boundary conditions. Putting these together, seismology reads the Earth's interior structure from the arrival times, polarisations, and amplitudes of these waves, making elastic wave theory the backbone of geophysical imaging. The PDE structure of the Navier equation (hyperbolic, linear, constant-coefficient) connects directly to the methods of the PDE chapter 09.02.17 pending, and the energy identity for elastic waves is the concrete realisation of the abstract energy estimates that underpin well-posedness theory for hyperbolic systems.

Connections Master

• Continuum mechanics field theory 09.07.01 provides the Lagrangian framework from which the stress tensor and strain tensor are derived as the continuum limit of discrete lattice forces. The Cauchy momentum equation is the Euler-Lagrange equation for the displacement field with the elastic Lagrangian density.

• Field Lagrangians 09.02.06 give the Lagrangian density for the elastic medium from which the Navier equation is derived by the Euler-Lagrange equations. The three-dimensional elastic Lagrangian density $\mathcal{L}=\frac{1}{2}\rho |\dot{\mathbf{u}}{|}^2-\mu {\epsilon }_{ij}{\epsilon }_{ij}-\frac{1}{2}\lambda ({\epsilon }_{kk}{)}^2$ is the direct descendant of the one-dimensional string Lagrangian.

• PDE methods 09.02.17 pending provide the general theory of hyperbolic PDEs, energy estimates, and Green's functions of which the Navier equation is a specific instance. The Stokes elastodynamic Green's function and the finite-speed propagation property are special cases of the general theory.

• Ideal fluid mechanics 09.07.03 is the special case $\mu =0$ of the Navier equation; the stress tensor reduces to ${\sigma }_{ij}=-p{\delta }_{ij}$ (isotropic pressure only, no shear), and only longitudinal (acoustic) waves propagate.

• Viscous flow 09.07.04 adds dissipative terms to the stress tensor, replacing $\mu$ (the elastic shear modulus) with $\eta$ (the dynamic viscosity), leading from the Navier equation to the Navier-Stokes equations.

• EM waves in matter have an analogous structure: the dielectric tensor plays the role of the stiffness tensor, and the wave equation for $\mathbf{E}$ has the same separation into ordinary and extraordinary modes in anisotropic crystals. The Christoffel equation for elastic waves is the direct analogue of Fresnel's equation for light in crystals.

Historical & philosophical context Master

Augustin-Louis Cauchy introduced the stress tensor in 1822 in his memoir Recherches sur l'equilibre et le mouvement interieur des corps solides ou fluides, establishing continuum mechanics on a rigorous mathematical foundation. Cauchy's key insight was that the internal forces in a continuous body could not be described by a single scalar (pressure) but required a tensor -- a linear map from surface normals to traction vectors. The tetrahedron argument that proves the existence of the stress tensor is one of the most elegant arguments in all of mechanics.

The strain tensor was formalised by George Green in 1839, and the connection between stress and strain via a linear constitutive law (the generalised Hooke's law) was developed by George Gabriel Stokes in a series of papers in the 1840s. Stokes also derived the elastodynamic Green's function (the response of an infinite elastic medium to a point impulse) in 1849, which remains the foundation of seismic source theory.

The Navier equation was first derived by Claude-Louis Navier in 1821 for one-dimensional elastic rods and generalised to three dimensions by Cauchy in 1828. Navier's original derivation used a molecular model (particles interacting via central forces) rather than the continuum stress-tensor approach; Cauchy's reformulation in terms of stress acting on surfaces was the conceptual breakthrough that generalised to fluids and to nonlinear materials.

The Lame parameters were introduced by Gabriel Lame in his Lecons sur la theorie mathematique de l'elasticite des corps solides (1852). Lame also gave the first systematic treatment of the wave speeds $c_L$ and $c_T$ and showed that the Navier equation separates into two scalar wave equations via the Helmholtz decomposition.

The distinction between longitudinal and transverse waves was known experimentally from seismology. R. D. Oldham identified P-waves and S-waves in seismograms in 1900 and noted that the P-wave always arrives first, consistent with $c_L>c_T$. Inge Lehmann discovered the solid inner core in 1936 by identifying seismic phases (PKIKP) that could only be explained by a solid region within the liquid outer core -- a triumph of elastic wave theory applied to planetary structure.

Lord Rayleigh (John William Strutt) derived the surface wave that bears his name in 1885, showing that a free surface supports a wave that decays exponentially with depth. A. E. H. Love developed the theory of the horizontally polarised surface wave in Some Problems of Geodynamics (1911), explaining observations of transverse surface waves that could not be accounted for by Rayleigh's theory. Robert Stoneley generalised the analysis to interface waves between two solids in 1924.

A philosophical point: the Cauchy stress tensor is a macroscopic construct. At the atomic scale, forces are transmitted through interatomic bonds -- discrete, directional, and quantum-mechanical. The stress tensor is the continuum limit of these microscopic forces, and it exists as a well-defined quantity only when the stress is averaged over a volume element containing many atoms. The Cauchy tetrahedron argument implicitly assumes this averaging, and the limit $\Delta V\to 0$ in the proof is understood as approaching a scale small compared to the wavelength but large compared to the atomic spacing. This dual limit is the same one that underlies the continuum hypothesis in fluid mechanics 09.07.01 and in the field-theoretic description of elastic media 09.02.06. The variational principle does not care whether the field is fundamental or emergent -- another instance of the deep insight that the Euler-Lagrange framework is agnostic about microscopic structure.

Bibliography Master

• Cauchy, A.-L., "Recherches sur l'equilibre et le mouvement interieur des corps solides ou fluides," Mem. Acad. Sci. Paris (1822).

• Navier, C.-L., "Memoire sur les lois du mouvement des fluides," Mem. Acad. Sci. Inst. France 6 (1823), 389-440.

• Stokes, G. G., "On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids," Trans. Cambridge Phil. Soc. 8 (1845), 287-319.

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