Acoustic Analogues

Abstract

The previous chapters illustrate with simple models the emergence of a new science of sound that accounts not only for spectral and refractive characteristics of waves but also addresses the amplitude and phase of these waves. In this chapter, these concepts are taken further and related to the development of acoustic analogues of other physical phenomena ranging from quantum phenomena to general relativity. These analogues may offer perspectives for applications and technological developments of this new science of sound. In this chapter, we first introduce the concept of phase bit (φ-bit) based on the fermion-like behavior of phonons in some elastic structures as an analogue of a quantum bit (qubit). The analogy with the notion of spin suggests the possibility of developing quantum information technologies based on superposition of elastic waves as well the capability of achieving exponentially parallel algorithms utilizing the non-separability of elastic waves. We then consider the analogy between fermion-like elastic waves (waves with spinor characteristics) supported by a structure subjected to a spatio-temporal modulation of its properties. We show the analogy between the equation that drives the dynamics of this elastic system and the Dirac equation for a charged particle including an electromagnetic field. The modulation can be used to tune the spinor part of the elastic waves suggesting that it can work as a gauge field analogue. Staying within the realm of analogies with quantum phenomena, we explore the acousto-hydrodynamics of bubbles in a fluid irradiated with an acoustic standing wave. We argue that the secondary sound field emitted by bubbles may lead to self-interaction that can modify the translational motion of bubbles. This phenomenon is reminiscent of the pilot wave model of quantum mechanics and suggests the possibility of acoustic bubbles to exhibit particle-wave duality. Finally, we review within the context of simple models the analogy between the propagation of acoustic waves in moving fluids and general relativity. We relate these concepts to the model of a one-dimensional harmonic crystal subjected to a directed spatio-temporal modulation of its stiffness. The dynamics of this system is described within the context of interpretations based on differential geometry.

Appendices

Appendix 1: Laplacian and d’Alembertian in a General Coordinate System

Let us consider a 3-vector x ^μ = x , y , z for μ = 1 , 2 , 3 in a general coordinate system. The Laplacian is given by [28]:

$$ \varDelta u=\frac{1}{\sqrt{\left|g\right|}}{\partial}_{\mu}\left(\sqrt{\left|g\right|}{g}^{\mu \nu }{\partial}_{\nu }u\right) $$

(6.113)

Let us apply (6.113) to the well know spherical coordinate system, (r, θ, φ). The covariant metric tensor is given by:

$$ \left[{g}_{\mu \nu}\right]=\left(\begin{array}{ccc}{g}_{rr}& 0& 0\\ {}0& {g}_{\theta \theta }& 0\\ {}0& 0& {g}_{\varphi \varphi}\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ {}0& {r}^2& 0\\ {}0& 0& {r}^2{sin}^2\theta \end{array}\right) $$

(6.114)

The contravariant metric tensor is the inverse of [g _μν ]:

$$ \left[{g}^{\mu \nu}\right]=\frac{1}{r^4{sin}^2\theta}\left(\begin{array}{ccc}{r}^4{sin}^2\theta & 0& 0\\ {}0& {r}^2{sin}^2\theta & 0\\ {}0& 0& {r}^2\end{array}\right) $$

(6.115)

The determinant of the covariant metric tensor is g = r ⁴sin² θ. Inserting this determinant and (6.115) into (6.113), leads to the expression for the Laplacian in spherical coordinates:

$$ \varDelta u=\frac{\partial^2u}{{\partial r}^2}+\frac{2}{r}\frac{\partial u}{\partial r}+2\frac{\mathit{\cos}\theta }{r^2\mathit{\sin}\theta}\frac{\partial u}{\partial \theta }+\frac{\partial^2u}{{\partial \theta}^2} $$

(6.116)

If we now consider the 4-vector x ^μ = t , x , y , z for μ = 0 , 1 , 2 , 3 in a general coordinate system. The d’Alembertian is given by [28]:

$$ \square u=\frac{1}{\sqrt{\left|g\right|}}{\partial}_{\mu}\left(\sqrt{\left|g\right|}{g}^{\mu \nu }{\partial}_{\nu }u\right) $$

(6.117)

The condition □u = 0 leads to a wave equation in any general system of coordinate.

Appendix 2: Geometrical Representation of the Dynamics of One-Dimensional Harmonic Systems

The Lagrangian density , $\mathcal{L}\left(x,t,u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\right),$ for the continuous one dimensional elastic string (i.e., long wavelength limit of the one dimensional harmonic crystal) is given by:

$$ \mathcal{L}=\frac{1}{2}\rho {\left(\frac{\partial u}{\partial t}\right)}^2-\frac{1}{2}{\beta}^2{\left(\frac{\partial u}{\partial x}\right)}^2 $$

(6.118)

u(x, t) is the displacement field. ρ and β ² are the mass density and stiffness, respectively.

The equations of motion are derived from Euler-Lagrange equation :

$$ \frac{\partial }{\partial t}\left(\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial u}{\partial t}\right)}\right)+\frac{\partial }{\partial x}\left(\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial u}{\partial x}\right)}\right)-\frac{\partial \mathcal{L}}{\partial u}=0 $$

(6.119)

ℒ is independent of the displacement u so the last term in (6.119) is zero. The elastic wave equation of motion is obtained in the usual form:

$$ \rho \frac{\partial^2u}{{\partial t}^2}-{\beta}^2\frac{\partial^2u}{{\partial x}^2}=0 $$

(6.120)

If we define u(x, t) =  ∫ dk u(k, t)e ^ikx, then the wave equation reduces to:

$$ \rho \frac{\partial^2u\left(k,t\right)}{{\partial t}^2}+{\left(\beta k\right)}^2u\left(k,t\right)=0 $$

(6.121)

This equation can be obtained from a Lagrangian density of the form:

$$ \mathcal{L}=\frac{1}{2}\rho {\left(\frac{\partial u\left(k,t\right)}{\partial t}\right)}^2-V(u) $$

(6.122)

with the k-dependent quadratic potential $V(u)=\frac{1}{2}{\left(\beta k\right)}^2{u}^2$. We now note that with the form (6.122) the Lagrangian density is now only an explicit function of u and $\frac{\partial u}{\partial t}$.

Inserting (6.122) into (6.119) yields the wave equation:

$$ \rho \frac{\partial^2u\left(k,t\right)}{{\partial t}^2}+\frac{\partial V(u)}{\partial u}=0 $$

(6.123)

The one dimensional harmonic system is conservative and the constant energy is given by the Hamiltonian :

$$ E=T+V=\frac{1}{2}\rho {\left(\frac{\partial u\left(k,t\right)}{\partial t}\right)}^2+V(u) $$

(6.124)

In (6.124), T is the kinetic energy. During harmonic motion energy passes from kinetic form into potential form and vice versa maintaining the overall energy constant. The crux of this appendix lies in the possibility of deriving the wave equation (6.123) from another Lagrangian density form, namely:

$$ \mathcal{L}\left(u,\frac{\partial u}{\partial t}\right)=2\left(E-V(u)\right){\left(\frac{\partial u}{\partial t}\right)}^2 $$

(6.125)

For pedagogical reason we apply Euler-Lagrange equation step-by-step. That equation reads: $\frac{\partial }{\partial t}\left(\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial u}{\partial t}\right)}\right)+-\frac{\partial \mathcal{L}}{\partial u}=0$. The first terms is obtained in the form:

$$ \frac{\partial }{\partial t}\left(\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial u}{\partial t}\right)}\right)=\frac{\partial }{\partial t}\left(2\left(E-V(u)\right)2\frac{\partial u}{\partial t}\right)=2\left(E-V(u)\right)2\frac{\partial^2u}{{\partial t}^2} $$

The second term is simply: $\frac{\partial \mathcal{L}}{\partial u}=-2\frac{\partial V(u)}{\partial u}{\left(\frac{\partial u}{\partial t}\right)}^2$. The equation of motion then becomes:

$$ 2\left(E-V(u)\right)2\frac{\partial^2u}{{\partial t}^2}+2\frac{\partial V(u)}{\partial u}{\left(\frac{\partial u}{\partial t}\right)}^2=0 $$

(6.126)

Using (6.124) (energy conservation), we can replace $2\left(E-V(u)\right)=\rho {\left(\frac{\partial u}{\partial t}\right)}^2$and simplify (6.126) by ${\left(\frac{\partial u}{\partial t}\right)}^2$. This yields again the wave equation (6.123).

It is worth noting that the Lagrangian density (6.122) in absence of a potential reduces to:

$$ \mathcal{L}=\frac{1}{2}\rho {\left(\frac{\partial u}{\partial t}\right)}^2 $$

(6.127)

Equation (6.127) implies uniform motion at constant velocity. This is free motion in a Euclidian space . The importance of (6.125) resides in its similarity to (6.127). Indeed, the right hand side of (6.125) can be defined as the kinetic energy:

$$ T=\frac{1}{2}{\left(\frac{\partial {u}^{\prime }}{\partial t}\right)}^2=2\left(E-V(u)\right){\left(\frac{\partial u}{\partial t}\right)}^2 $$

(6.128)

In the previous equation, we have defined the square of a new segment length along the axis of displacements by:

$$ du{\prime}^2=2\left(E-V(u)\right){du}^2 $$

(6.129)

The quantity g = 2(E − V(u)) is the metric that relates the square of the length element du ² to du′². The length du, varies as a function of u (or t) for fixed time intervals dt. This is illustrated in Fig. 6.3. For a solution u = sin t of (6.121) using ρ = βk = 1, the displacement increases toward its maximum in a non-uniform way. The intervals du shortens to zero for fixed time intervals dt as one approaches the maximum displacement.

Fig. 6.3

Displacement u = sin t showing that equally spaced intervals of time Δt lead to a non-uniform set of intervals: Δu

Finally, we consider the optimization problem of minimizing the length along the u′ axis between two points A and B:

$$ L=\underset{A}{\overset{B}{\int }}{du}^{\prime }=\underset{A}{\overset{B}{\int }}\sqrt{2\left(E-V(u)\right){\left(\frac{\partial u}{\partial \lambda}\right)}^2}d\lambda =\underset{A}{\overset{B}{\int }}Fd\lambda $$

(6.130)

In (6.130), we have introduced a parameter λ which enables us to move along the axis u′. This optimization problem can also be solved using Euler-Lagrange equation : $\frac{\partial }{\partial \lambda}\left(\frac{\partial F}{\partial \left(\frac{\partial u}{\partial \lambda}\right)}\right)+-\frac{\partial F}{\partial u}=0$ which yields the equation:

$$ \frac{\partial^2u}{{\partial \lambda}^2}-\frac{1}{2}\frac{1}{g}\frac{\partial g}{\partial u}{\left(\frac{\partial u}{\partial \lambda}\right)}^2=\frac{1}{F}\frac{\partial F}{\partial \lambda}\frac{\partial u}{\partial \lambda } $$

(6.131)

This is the equation of a geodesic in a curved space which metric is g.

Let us choose, dλ = du′, then by (6.130), F = 1 and $\frac{\partial F}{\partial \lambda }=0$. Equation (6.131) reduces to

$$ \frac{\partial^2u}{{\partial u\prime}^2}-\frac{1}{2}\frac{1}{g}\frac{\partial g}{\partial u}{\left(\frac{\partial u}{\partial {u}^{\prime }}\right)}^2=0 $$

(6.132)

or

$$ \frac{\partial^2u}{{\partial u\prime}^2}+\frac{1}{2\left(E-V(u)\right)}\frac{\partial V}{\partial u}{\left(\frac{\partial u}{\partial {u}^{\prime }}\right)}^2=0 $$

(6.133)

Along the geodesic the velocity is a constant (free motion), v such that du′ = vdt. Inserting that relation in (6.133) and also using energy conservation (6.124) yields again the wave equation (6.123).

It is possible to construct curved spaces in some dimension by embedding them in higher dimensional spaces. The additional dimension add a curvature term to the usual metric of flat space. To illuminate this concept and the meaning of (6.130) we consider the two dimensional space ($u,v=\frac{dv}{dt}\Big)$ in which the one dimensional space describing the displacement of an oscillator is embedded, namely u. For the sake of simplicity, we take all physical constants equal to 1.

The conserved energy of the oscillator gives:

$$ T+V={v}^2+{u}^2=2E $$

(6.134)

We note the well-known result that the oscillator follows a circular trajectory in the (u, v) plane. A line element on the trajectory circle is defined as (Fig. 6.4):

$$ {ds}^2={dv}^2+{du}^2 $$

(6.135)

Fig. 6.4

Trajectory of oscillator in two-dimensional phase space (u, v). The vertical lines are drawn with equal values of ds. One notices the corresponding non-uniformity of the intervals du

Differentiating (6.134) results in:

$$ vdv+udu=0 $$

(6.136)

We express the line element ds ² in terms of u by inserting (6.136) into (6.135):

$$ {ds}^2=\left(\frac{2E}{2E-{u}^2}\right){du}^2 $$

(6.137)

This line element ds is uniform along the circular trajectory. The velocity $\frac{ds}{dt}$ is a constant along the circle. The circular trajectory is that of a free moving object which kinetic energy is equal to the total energy of the oscillator, namely

$$ \frac{1}{2}{\left(\frac{ds}{dt}\right)}^2=E $$

(6.138)

Combining (6.138) and (6.137) yields

$$ {dt}^2=\left(\frac{1}{2E-{u}^2}\right){du}^2 $$

(6.139)

One recovers the inverse of the metric, g = 2(E − V(u)). When plotting $t={\int}_0^u\left(\frac{1}{2E-{u}^2}\right)du$ with u = sin t one converts the plot of Fig. 6.3 into that of a straight line (Fig. 6.5).

Fig. 6.5

${\int}_0^u\left(\frac{1}{2E-{u}^2}\right)du$ versus time t. The straight line is the signature of a geodesic