Exact plane-wave solutions of the isotropic linear elastic model, with longitudinal and transverse speeds

Let ${\rho }_0>0$ and Lam'e parameters $\lambda ,\mu \in \mathbb{R}$ with $\mu >0$ and $\lambda +\frac{2\mu }{3}>0$, let $a,k\in {\mathbb{R}}^3$ and $\omega \in \mathbb{R}\setminus \{0\}$, and define the plane wave

$\xi :\mathbb{R}\times {\mathbb{R}}^3\to {\mathbb{R}}^3,\xi (t,x):=a\cos (⟨k,x{⟩}_2-\omega t),$

with the Euclidean inner product $⟨\cdot ,\cdot {⟩}_2$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). For each fixed $t$, the spatial displacement map is ${\xi }_t(x):=\xi (t,x)$. Define the strain and stress fields using the spatial derivative at fixed time:

$e(t,x):=\frac{1}{2}(D_x\xi (t,x)+(D_x\xi (t,x){)}^{\mathsf{T}}),\sigma (t,x):=\lambda (\mathrm{tr}e(t,x))I_3+2\mu e(t,x)\mathrm{.}$

These are the pointwise constitutive formulas of the isotropic linear elastic model (The small-strain isotropic linear elastic constitutive law and its strain energy); in particular $D_x\xi$ is the total derivative of $x\mapsto {\xi }_t(x)$, not the derivative of the full spacetime map.

Then the componentwise identity

${\rho }_0{\mathrm{\partial }}_t^2{\xi }_i={\sum }_{j=1}^3{\mathrm{\partial }}_j{\sigma }_{ij}(i=1,2,3)$

holds on all of $\mathbb{R}\times {\mathbb{R}}^3$ if and only if

$$\begin{array}{cccc}& {\rho }_0{\omega }^{2}a=\mu \mathrm{\mid }k{\mathrm{\mid }}_2^{2}a+(\lambda +\mu )⟨k,a{⟩}_{2}k\mathrm{.}& & \text{(D)}\end{array}$$

When (D) holds and $a,k$ are nonzero, define the phase speed by $c_{\mathrm{p}\mathrm{h}}:=\mathrm{\mid }\omega \mathrm{\mid }\mathrm{/}\mathrm{\mid }k{\mathrm{\mid }}_2$. If $a=\alpha k$ for some $\alpha \mathrm{\ne }0$ (longitudinal polarization), then ${\rho }_0{\omega }^2=(\lambda +2\mu )\mathrm{\mid }k{\mathrm{\mid }}_2^2$ and $c_{\mathrm{p}\mathrm{h}}^2=(\lambda +2\mu )\mathrm{/}{\rho }_0$. If $⟨k,a{⟩}_2=0$ (transverse polarization), then ${\rho }_0{\omega }^2=\mu \mathrm{\mid }k{\mathrm{\mid }}_2^2$ and $c_{\mathrm{p}\mathrm{h}}^2=\mu \mathrm{/}{\rho }_0$. The constitutive positivity hypotheses imply $\lambda +2\mu =(\lambda +2\mu \mathrm{/}3)+4\mu \mathrm{/}3>0$ and $\mu >0$, so both branch phase speeds are real and positive. (When $k=0$, (D) has no solution with $a\mathrm{\ne }0$ and $\omega \mathrm{\ne }0$.)

• The theorem verifies exact solutions of the adopted constant-coefficient PDE; it does not assert that every solution of the elastic model is a plane wave, that a wave packet propagates unchanged, or that any material realises the model. General initial-value problems for the elastic system are not solved here.

• The identity (D) is the dispersion relation of the plane-wave ansatz. The two eigenvalue branches computed in the last step are the longitudinal and transverse speeds; they require $\lambda +2\mu >0$ and $\mu >0$ respectively, and no claim is made about $\lambda$ alone.