Plane elastic waves in the adopted model: the dispersion condition and the longitudinal and transverse speeds

Adopt the small-displacement isotropic linear elastic branch of Adopted isotropic linear elastic constitutive model with its small-strain regime and strain energy with reference density ${\rho }_0>0$ of dimension $\mu {\mathrm{\ell }}^{-3}$ and Lamé parameters $\lambda ,{\mu }_s$ of dimension $\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$, satisfying ${\mu }_s>0$ and $\lambda +\frac{2{\mu }_s}{3}>0$ (Continuum body data, the fixed inertial frame and the dimensions of the fields). Let (E) be the displacement equation of The displacement equation of the linear elastic branch: rho_0 partial_t^2 xi = rho_0 b + mu_s Laplacian xi + (lambda + mu_s) grad div xi with body force $b\equiv 0$. Let $x\in {\mathbb{R}}^3$ denote position of dimension $\mathrm{\ell }$ and $t\in \mathbb{R}$ time of dimension $\tau$. Let $k\in {\mathbb{R}}^3\setminus \{0\}$ be a wave vector of dimension ${\mathrm{\ell }}^{-1}$, $a\in {\mathbb{R}}^3$ a polarization amplitude of dimension $\mathrm{\ell }$, and $\omega \in \mathbb{R}\setminus \{0\}$ an angular frequency of dimension ${\tau }^{-1}$. The phase is dimensionless. Define the plane displacement field

$\xi (t,x):=a\cos (⟨k,x{⟩}_2-\omega t),$

with the Euclidean inner product $⟨\cdot ,\cdot {⟩}_2$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). Then $\xi$ satisfies (E) on all of $\mathbb{R}\times {\mathbb{R}}^3$ if and only if

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

Moreover, if this plane wave satisfies (E) (equivalently, if (D) holds), then for $a\mathrm{\ne }0$: if $a$ is parallel to $k$ (longitudinal), then ${\rho }_0{\omega }^2=(\lambda +2{\mu }_s)\mathrm{\mid }k{\mathrm{\mid }}_2^2$ and its phase speed is $\mathrm{\mid }\omega \mathrm{\mid }\mathrm{/}\mathrm{\mid }k{\mathrm{\mid }}_2=c_L$, where $c_L:=\sqrt{(\lambda +2{\mu }_s)\mathrm{/}{\rho }_0}>0$; if $a$ is orthogonal to $k$ (transverse), then ${\rho }_0{\omega }^2={\mu }_s\mathrm{\mid }k{\mathrm{\mid }}_2^2$ and its phase speed is $\mathrm{\mid }\omega \mathrm{\mid }\mathrm{/}\mathrm{\mid }k{\mathrm{\mid }}_2=c_T$, where $c_T:=\sqrt{{\mu }_s\mathrm{/}{\rho }_0}>0$. Positivity follows from the branch hypotheses, and $[c_L]=[c_T]=\mathrm{\ell }{\tau }^{-1}$ (Continuum body data, the fixed inertial frame and the dimensions of the fields, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). No assertion is made that every solution of (E) is a superposition of plane waves, and no measured modulus or speed is claimed.

• Exact solutions of the model's linear equation. The waves displayed are exact solutions of (E) with $b\equiv 0$, not approximations, but they are special solutions: a general initial-value problem must be solved by other means, and the smooth fixed-end and chain items below supply existence results for particular geometries. Existence for general data in ${\mathbb{R}}^3$ is not asserted on this page.

• Reading of the speeds. For a material with measured moduli the speeds $c_L=\sqrt{(\lambda +2{\mu }_s)\mathrm{/}{\rho }_0}$ and $c_T=\sqrt{{\mu }_s\mathrm{/}{\rho }_0}$ are model predictions; comparing them with measured sound speeds tests the model and its idealizations, and any such comparison uses the moduli as qualified empirical premises. No measured value is asserted here.