T-II-2new

Variational methods: Sound waves in solid and fluid materials

Elasticity in solids

Under the action of external forces a solid body moves. But it can also be deformed, i.e. it can change volume and shape. The first succesful attempt to describe this phenomenon is provided by the theory of elasticity. The theory is based on two assumptions:

1) the solid body can modeled as a continuum medium

2) the energetic cost of the deformation is kept at the lowest order in perturbation.

The fundamental equations of elasticity have been established by Cauchy and Poisson in the twenties of the 19th century, well before the discovery of the atomic structure of matter. Here we consider a perfectly isotropic continuum medium and show how sounds propagate in it.

The strain tensor

Each point of the body can be identified with its postion in the space:

${\vec {r}}=(x_{1}=x,x_{2}=y,x_{3}=z)$

When the body is deformed a point located in ${\vec {r}}$ moves to a new location ${\vec {r}}_{\text{new}}$. The displacement, ${\vec {u}}={\vec {r}}_{\text{new}}-{\vec {r}}$, differs from point to point of the body and is then a function of the original position ${\vec {r}}$. The displacementvectorial field

${\vec {u}}({\vec {r}})=\left(u_{1}({\vec {r}}),u_{2}({\vec {r}}),u_{3}({\vec {r}})\right)$

encodes then all informations about the deformation of the body. However large displacements does not always correspond to large deformation as the body can move rigidly without deformation. To quantify how much a body is deformed we consider two points at an infinitesimal distance $d\ell$.

• Assuming small deformations, show that the distance $d\ell _{\text{new}}$ after the deformation can be written as:

$d\ell _{\text{new}}^{2}\approx d\ell ^{2}+2\sum _{i,k}\epsilon _{ik}dx_{i}dx_{k}$

The tensor $\epsilon _{ik}$ is called strain tensor, is symmetric and defined as

$\epsilon _{ik}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}\right)$

• Consider now a volume element $dV=dx_{1}\,dx_{2}\,dx_{3}$ and show that at the first order in perturbation one has

$dV_{\text{new}}\approx dV\left(1++\sum _{i}\epsilon _{ii}\right)$

we conclude that the sum of the diagonal elements of the strain tensor identifies with the relative change in volume $(dV_{\text{new}}-dV)/dV$ of the deformation (namley the amount of compression or rarefaction).

The Hooke law à la Landau

We want to evaluate the elastic energy associated with a small deformation.

We first note that in abscence of external force and at thermal equlibrium the body does not display any deformation. This observation as two consequences:

1) When the strain tensor is zero for all points, the energy displays a minimum.

2) The first terms of the energy expansion must be quadratic in the strain tensor. Linear terms are incompatible with the condition of a minimum

We finally note that the energy is a scalar quantity, so each term of its expansion mus be a scalar. For an isotrope medium there anly two independent scalar terms that can be associated to the strain tensor:

$E_{\text{el}}=\int d{\vec {r}}\left[{\frac {K}{2}}\sum _{i}\epsilon _{ii}^{2}+\mu \sum _{ik}\left(\epsilon _{ik}-{\frac {1}{3}}\delta _{ik}\sum _{l}\epsilon _{ll}\right)^{2}\right]$

From Eq () we observe that the first term accounts for the change in volume. The constant $K>0$ is the bulk modulus and has the dimension of a pressure. The second term accounts for the shear as it is the square os a tensor with trace equal to zero. The constant $\mu >0$ is the shear modulus and has also the dimension of a pressure. The two elastic moduli are material dependent and fully describe its elastic properties.

We write in two Step the elastic energy of a string with an elastic constant $k$. The shape of the string is defined by the function $y(x)$, where $0<x<1$ and $y(0)=y(1)=0$.

Step one: the discrete model

• Write the elastic energy of a chain of N identical springs as a function of their position $y_{1},y_{2},\ldots ,y_{N}$ (See figure).
• Determine the elastic constant of each spring in order to have a string with an elastic constant $k$

Step two: The continuum limit

• Take the limit $N\to \infty$ and express the elastic energy as a functional of $y(x)$. (Input: use $1/N\to {\text{d}}x$ and ${\frac {y_{i+1}-y_{i}}{1/N}}\to {\frac {{\text{d}}y(x)}{{\text{d}}x}}$)

Gravity

We write now the gravitational potential as a functional of $y(x)$. We will use two different methods. We suppose that the total mass of the line in $M$.

• Suppose that springs are massless and $N$ identical masses are located at positions $y_{1},y_{2},\ldots$

• Combine your finding for elasticity and gravity and show that the shape of the elastic line is parabolic

Dynamics

We want to study the motion of the elastic line in absence of gravity and in presence of small oscillations. The shape of the line is now time dependent $y(x,t)$.

• Write the kinetic energy associated to the $N$ masses located at $y_{1}(t),y_{2}(t),\ldots ,y_{N}(t)$. Take the continuum limit.

The action associated to the elastic chain is

$\int _{t_{1}}^{t_{2}}dt\int _{0}^{1}dx\,{\mathcal {L}}(x,t)$

where ${\mathcal {L}}(x,t)$ is the Lagrangian denisty

• Write explicitly ${\mathcal {L}}(x,t)$ .
• Write the equation of motion of the chain using the minimum action principle. This equation is the D'Almbert equation.

Sound waves in ideal gas

Consider a uniform ideal gas of equilibrium mass density $\rho _{0}$ and equilibrium pressure $P_{0}$. Let us investigate the longitudinal oscillations of such a gas. Of course, these oscillations are usually referred to as sound waves.

Kinetic Energy

In a first time we can start from the discrete description of a system of N atoms at positions ${\vec {u}}_{1}(t),{\vec {u}}_{2}(t),\ldots ,{\vec {u}}_{N}(t)$.

• Write the kinetic energy, $T$, for this system as a function of mass of the atom $m$.
• Consider the limit of small oscillations. Take the continuum limit: ${\vec {u}}_{i}(t)\to {\vec {u}}({\vec {r}},t)$ and show that the kinetic energy writes

$T={\frac {\rho _{0}}{2}}\int \,d{\vec {r}}\,{\dot {\vec {u}}}^{2}({\vec {r}},t)$

Potential Energy

In order to write the full Lagrangian we need to find the potential energy associated to the oscillations. During the oscillations the mass density can have small fluctuations

$\rho ({\vec {r}},t)=\rho _{0}+\delta \rho ({\vec {r}},t)=\rho _{0}\left(1+\sigma ({\vec {r}},t)\right)$

The scalar field $\sigma ({\vec {r}},t)$ represents the relative fluctuation of the density.

As a consequence of these fluctuations, the volume which contains a given mass $M$ expands from the initial value $V_{0}$ to a new value $V_{0}+\delta V$. The pressure inside the domain also fluctuates $P=P_{0}+\ldots$. The variation of potential energy is given by the work associated to the volume expansion:

$V_{0}\,{{\mathcal {V}}({\vec {r}},t)}-\int _{V_{0}}^{V_{0}+\delta V}\,P\,d{\vec {r}}=-P_{0}\delta V-{\frac {1}{2}}\left({\frac {\partial P}{\partial V}}\right)_{0}\delta V^{2}+\ldots$

where ${\mathcal {V}}({\vec {r}},t)$is the density of potential energy.

Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,

$PV^{\gamma }=$ constant

where $\gamma$ the ratio of specific heats (i.e., the ratio of the gas's specific heat at constant pressure to its specific heat at constant volume). This ratio is approximately $1.4$ for ordinary air.

• Show that the total potential energy, $W=\int \,d{\vec {r}}\,{\mathcal {V}}({\vec {r}},t)$ writes

$W=P_{0}\int \,d{\vec {r}}\,\left(\sigma +{\frac {\gamma }{2}}\sigma ^{2}\right)$

• Using the continuity equation of the gas, $\partial _{t}\rho =-\rho _{0}{\vec {\nabla }}\cdot \partial _{t}{\vec {u}}({\vec {r}},t)$, show that $\sigma ({\vec {r}},t)=-{\vec {\nabla }}\cdot {\vec {u}}({\vec {r}},t)$ and write the potential energy as a function of ${\vec {u}}({\vec {r}},t)$

Lagrangian description

• Show that the Lagrangian density writes

${\mathcal {L}}({\vec {r}},t)={\frac {\rho _{0}}{2}}{\dot {\vec {u}}}^{2}+P_{0}{\vec {\nabla }}\cdot {\vec {u}}-{\frac {\gamma }{2}}P_{0}({\vec {\nabla }}\cdot {\vec {u}})^{2}$

• Show that the associated equation of motion

$\rho _{0}{\frac {\partial ^{2}{\vec {u}}}{\partial t^{2}}}-\gamma P_{0}{\vec {\nabla }}{\vec {\nabla }}\cdot {\vec {u}}=0$

• Using the continuity relation $\sigma =-{\vec {\nabla }}\cdot {\vec {u}}$ show that the latter equation corresponds to the D'Alembert Equation for the scalar field $\sigma$

This scalar field behaves exaclty like an elastic scalar field.

• Determine the sound velocity.