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Equations of Linear Acoustics

In this chapter, we establish the equations that govern the propagation of small-amplitude acoustic waves in fluids under the simplest possible conditions. The fluid is considered as perfect, that is non-viscous and non-heat-conductive. Acoustic disturbances are regarded as small-amplitude perturbations of the ambient state of a time-independent homogeneous fluid at rest. External forces, such as gravity, are not taken into account.

These assumptions may seem very strong, and we will have to relax some of them in later chapters. However, they offer considerable advantages in terms of simplicity, while often remaining unrestrictive in practice. This simplicity makes it possible to emphasize the fundamental properties of acoustic waves, which are generally only slightly modified in more complex situations where, for example, the inhomogeneous nature of the medium will have to be taken into account. They also allow the construction of analytical solutions with very wide applications, which also serve as references for the numerical simulations that become necessary when the geometries of the problems under consideration become complex.

1.1. Validity of the assumptions of linear acoustics and a perfect fluid

It is important to form a first qualitative idea on the legitimacy of the assumptions of a perfect fluid and linearization of fluid dynamics equations by performing order-of-magnitude analyses in a simplified situation. We thus consider a wave of frequency f chosen within the range of audible sounds, for instance, f = 1 kHz, and propagating through the air in the x1 direction only.

To evaluate the validity conditions of the linearization process, we consider the two components of acceleration in the Navier-Stokes equations (these equations are recalled in Appendix 1):

where u1 is the component in the x1 direction of the velocity associated with the propagation of an acoustic wave. The partial derivative of velocity with respect to time ?u1/?t is linear with respect to fluctuations, while its convective derivative u1?u1/?x1 exhibits a quadratic nonlinearity. The order of magnitude of the linear term is ?u1, where ? = 2pf is the angular frequency of the wave. That of the nonlinear term is , where ? is the spatial scale of the wave, called the wavelength. The wavelength is related to the angular frequency by ? = 2pc0/?, where c0 denotes the speed of propagation of acoustic waves in the medium under consideration; in air, the speed of sound is about 340 m.s-1. The ratio of the orders of magnitude of the nonlinear term to the linear term is therefore u1/c0 = Ma; this ratio defines the acoustic Mach number. We will see in Chapter 3 that a sound wave with a 94 dB level, which is perceived by humans as very intense and painful, is only associated with a root-mean-square (rms) value of pressure fluctuations of only 1 Pa. For this wave, the acoustic Mach number is extremely low, being about 10-5. In the most unfavorable situations, it will hardly exceed 10-3, which fully justifies why terms composed of products of fluctuations in the equations of motion can be neglected, and thus validates the linearization process of the equations.

To evaluate the validity of the perfect fluid assumption, we simply analyze the influence of viscous terms, since the thermal effects are typically of the same order of magnitude as them. The viscous term in the Navier-Stokes equations:

has an order of magnitude ?u1/?2, where ? is the kinematic viscosity of the fluid. The ratio of the linear acceleration term to the viscous term is therefore of the order , or c0?/?, some sort of local acoustic Reynolds number. For air, ? ~ 15×10-6 m2.s-1 and for a 1 kHz frequency, this number is about 106. This very high value of the Reynolds number ensures that viscous effects are negligible for the usual range of frequencies.

It is important to note, however, that the above reasoning is qualitative and only has a local value, that is to say on the scale of a wavelength. Viscous, thermal and nonlinear effects are cumulative, and their consequences can be significant if the distances traveled by waves correspond to a large number of wavelengths. As a case in point, very high frequency acoustic waves (greater than a hundred kilohertz) propagating in the air are very quickly attenuated by visco-thermal effects over distances of only a few meters. It will therefore sometimes be necessary to take into account dissipative effects, often in an approximate way by an a posteriori correction of a calculation made for a perfect fluid. This is, for example, the case in room acoustics, as we shall see in Chapter 10.

1.2. Linearized equations of fluid dynamics

Since the fluid is now considered perfect and external forces are neglected, the equations of fluid dynamics are reduced to the system of Euler's equations (see Appendix A1.4):

where p, ?, s and Ui denote fluid pressure, density, entropy and velocity, respectively1.

The propagation medium is assumed to be time independent, homogeneous and at rest. The variables describing the ambient state (i.e. when there is no acoustic disturbance) are therefore uniform; they will be expressed using a "0" subscript. Since the medium is at rest, the ambient velocity is zero, U0 = 0. Acoustic disturbances, which are time and space dependent, will be indicated by a "prime" symbol. During the propagation of an acoustic wave, the different variables will thus be decomposed according to:

The disturbances are assumed to be small enough so that the products of fluctuations that will appear when the above decompositions are introduced into the equations of fluid dynamics can be neglected. This constitutes the fundamental assumption of linear acoustics.

We now outline the linearization procedure on the conservation of mass equation (or "continuity equation"):

which reduces to:

The last term of the left hand side of [1.10] is of second order as the divergence of a product of fluctuations. It is therefore neglected in the linear approximation, and the linearized continuity equation is finally written as:

To linearize the equations in which the total time-derivative appears, it can be observed that the total derivative of a fluctuation can be reduced to its partial derivative with respect to time, because:

by neglecting terms of order greater than or equal to 2. To obtain the linearized equations, thanks to the assumptions of homogeneity of the medium and the absence of ambient flow, any total derivative can therefore be replaced by a simple partial derivative with respect to time.

The system of linearized Euler's equations is thus written as:

The last equation shows that the entropy fluctuation associated with an acoustic disturbance is identically zero, which leads to an important simplification. The pressure then depends only on the density and the equation of state takes the simple form p = p(?, s0)= p(?).

For the equation of state [1.4], the process of linearization must be carried out differently insofar as there is no explicit form for it, except for the special but important case of ideal gases to which we will return later. The idea is to consider that the presence of acoustic waves only slightly modifies the ambient state; the perturbed variables can then be obtained through a Taylor-series expansion around the ambient state that is limited to the first order. It thus follows that:

Partial derivatives are taken at constant entropy (as recalled by the subscript "s") and evaluated at the ambient state. By definition, p(?0,s0) = p0 and only considering the first order in the linearization assumption, a simple relation of proportionality between pressure and density fluctuations is therefore obtained.

It is important to understand that the coefficient (?p/??)s,0 is a thermodynamic quantity characteristic of the medium in which acoustic waves propagate (but that it is independent of them, since it is evaluated at the ambient state). In thermodynamics, it is shown that this coefficient is strictly positive and it will later be seen that it is equal to the square of the propagation speed of acoustic waves (the speed of sound), denoted by c0:

1.3. The wave equation

We therefore have a linear system of three first-order equations for three variables, p', ?' and . These equations are generally rearranged to eliminate fluctuations in...