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

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

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

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

This means that the trace of the strain tensor identifies with the relative change in volume $(dV_{\text{new}}-dV)/dV$ of the body. In general all deformation can be represented as the sum of a uniform compression (change in volume without change of shape) and a shear (change of shape without change of volume):

$\epsilon _{ik}=\left(\epsilon _{ik}-{\frac {1}{3}}\delta _{ik}{\text{Tr}}(\epsilon )\right)+{\frac {1}{3}}\delta _{ik}{\text{Tr}}(\epsilon )\quad {\text{with}}\quad \delta _{ik}={\begin{cases}0\;{\text{if}}\;i\neq k\\1\;{\text{if}}\;i=k\end{cases}}\quad (2)$

The first term is a shear as its trace is zero. The second term is a compression.

Elastic waves

To describe the oscillations induced by sudden deformations one as to consider time dependent displacement vectors:

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

If one neglects the gravity the total energy of the body is given by the elastic potential energy and the kinetic term that we evaluate in the following.

Elastic energy of a deformation: the Hooke law à la Landau

To evaluate the elastic energy associated with a small deformation we note that in abscence of external force and at thermal equilibrium 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

The more general quadratic form can be written as $\int d{\vec {r}}\sum _{ijkl}\epsilon _{ij}\,C_{ijkl}\,\epsilon _{kl}$, where the tensor $C_{ijkl}$ contains all the elastic moduli. For an isotropic material, consideration of the principle of material frame indifference leads to the conclusion that the strain energy density function can be written in a simpler form (we will discuss it in class):

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

The first term accounts for the cost of the change in volume, the second for the change of shape. The positive constants $K,\mu$ are respectively the bulk and the shear modulus and have dimensions of a pressure. These two constants are material dependent and fully describe its elastic properties.

Kinetic Energy

The mass contained in an infinitesimal volume around the point located in ${\vec {r}}$ is $\rho \,dV=\rho \,d{\vec {r}}$, with $\rho$ the mass density of the body. The total kinetic energy can then be written as

$T=\rho \int d{\vec {r}}\left({\dot {u}}_{1}^{2}({\vec {r}},t)+{\dot {u}}_{2}^{2}({\vec {r}},t)+{\dot {u}}_{3}^{2}({\vec {r}},t)\right)$

The action for a simplified model

To simplify the calculation we will neglect the dependence on $z,y$ of the displacement vector which becomes:

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

• Within this approximation write the elastic energy in term of $\partial _{x}u_{x},\partial _{x}u_{y},\partial _{x}u_{z}$

The action associated to this energy 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 ${\mathcal {L}}(x,t)$.
• Using the variational methods write the 3 equations of motion for $u_{x}(x,t),u_{y}(x,t),u_{z}(x,t)$.
• Compute the longitudinal mode velocity and the transverse mode velocity.

Sound waves in fluids

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.

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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.

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.