Elastic waves in solids 2

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Introduction to Acoustics

Describing the journey of a sound signal deals with many different branches of science, including engineering, earth sciences, life sciences, and the arts. For example, a musician reads notes and plays the piano music. Engineers worked on the microphone that picks up the sound and other engineers optimized the reproduction of the sound by a loudspeaker electroacoustics. Architects and civil engineers ensured that the sound is reproduced correctly in the concert hall room acoustics.

The ear of a listener picks up the sound physiology , the sound is processed by the auditory system, and the listener perceives it as music psychoacoustics. Acoustics is clearly multidisciplinary and multiphysics in its nature. Here, we are primarily concerned with the physical principles of acoustics in relation to engineering and earth sciences.

The first known speculations about the wave nature of sound dates back to the ancient Greek philosophers at the time of Aristotle. They were inspired by how small waves propagate at the surface of water and how they interact with obstacles as well as the generation of sound by vibrating bodies.

Marin Mersenne — is often referred to as the father of acoustics , in particular because he gave the first correct published mathematical description of a vibrating string Harmonicorum Libri in This was two years before Galileo Galilei's — seminal work on mechanics. During the time of Mersenne and Galileo, other physicists were not convinced by the wave nature of sound and some thought of it as a stream of matter see Ref. The first formal mathematical description of the propagation of sound in a fluid was given by Isaac Newton — in his work Principia the second book from He described sound as the propagation of pulses as neighboring fluid particles interact and push each other.

Progress toward a complete theory for the propagation of sound was made in the eighteenth century with the development of continuum mechanics and fluid dynamics through the work of Leonhard Euler — , Joseph-Louis Lagrange — , and Jean le Rond d'Alembert — The modern theory for sound propagation is, for most, based on the work of these scientists and their contemporaries. Lord Rayleigh's — treatise The Theory of Sound from is thought to complete this long line of work with, for example, a detailed description of scattering. The work of Rayleigh is often said to mark the change from the classical to modern era in acoustics Ref.

To derive the governing equations for all wave phenomena, you need to start with the conservation equations in their most general form ; that is, conservation of mass, momentum, and energy. To close the system, these equations need to be supplemented by constitutive relations as well as thermodynamic equations of state. Many different types of waves exist depending on the medium in which they propagate and their interactions.

In the following sections, we derive the governing equations for waves in fluid liquids and gases , including details about loss models and assumptions. Then, we present the governing equations for elastic waves in solids as well as the combined propagation of elastic and pressure waves in porous materials.

The conservation equations describing the motion of fluids are the continuity equation mass conservation , Navier-Stokes equation momentum conservation , and general heat transfer equation energy conservation. They are given by. The conserved quantities here are the density , momentum , and total entropy. In principle, we will consider situations where mass is conserved and so, in general,. The acoustic perturbation to the mass source term see below can, however, be used as a representation for a complex process that we do not want to describe in detail.

There are many ways to write the conservation equations and select the dependent variables; above is just one of them. See, for example, Ref. The terms on the left-hand side of the equation represent the conserved quantities. These terms are also sometimes after manipulations written in a nonconservative form as. The operator is known as the material derivative, and is defined as. Some thermodynamic relations are necessary to reformulate the energy conservation equation in terms of the temperature and pressure variables. There are various ways to derive the relation; here, we present one.

One of Maxwell's relations is also necessary. It is given as. Constitutive relations are the expressions that define or approximate the properties of a material and how it responds to external stimuli. The bulk viscosity term models compression and expansion viscosity effects that, in effect, describe the difference between the mechanical and thermodynamic pressures. These are not always in equilibrium. Moreover, all of the material properties may, in general, depend on both temperature and pressure. This implies that the material properties should be treated as space-dependent quantities.

Using the above expressions, we arrive, after several manipulations, at the full set of coupled equations of motion for an isotropic, compressible, viscous, and thermally conducting fluid. Here, it is expressed in terms of the dependent variables for pressure , velocity , and temperature. In almost all applications, these terms are not included because of mass conservation resulting in. Acoustics is concerned with the transport and propagation of small perturbations.

These perturbations can be many orders of magnitude smaller than the background conditions; for example, normal speech signals with an amplitude of compared to the atmospheric pressure of about , Pa. It is very often not practical to solve the full governing equations presented above.

They are nonlinear in nature and also multiscale in both space and time when comparing:. In numerical applications, it would require very high numerical precision to resolve the different physical scales and timescales simultaneously. In most practical cases, the acoustic problem can be assumed linear. Perturbation theory is used to simplify and analyze the acoustics separately from the background properties. Depending on the order of the perturbation expansion, the order of retained terms in the equations of the physical mechanisms modeled, and other approximations, the reformulation of the governing equations will lead to different acoustic equations.

These equations describe everything from general linear acoustics and the Helmholtz equation to advanced nonlinear acoustic models and equations for shocks. For details about perturbation theory, see Ref. For most acoustics applications, the expansion of the dependent variables will be done to first order, such that. Typically, the background fields zeroth order only depend on space or they can, at most, vary slowly in time when compared to the acoustic timescale.

In certain applications, such as the analysis of acoustic streaming, perturbation is carried out to the second order to separate timescales see, for example, Ref. In many textbooks, the acoustic perturbation in pressure is represented by. In linear acoustics, the perturbation is carried out to the first order, and only the first-order terms are retained in the governing equations as shown below.

The equation of state is expanded, using a Taylor series, to first order in the dependent variables about the zeroth-order solution. In most nonlinear acoustics when boundary layer effects are neglected , the dependent variables are perturbed to the first order but terms up to the second order are retained in the governing equations see the Nonlinear Acoustics section below.

In this case, the equation of state is expanded to second order with a Taylor series see, for example, Ref. The perturbation approach does not solely depend on a mathematical scheme, but relies heavily on the physical effects that need to be captured by the governing equations. In the linear theory, the governing equations are linearized and expanded to first order in the small parameters around the stationary background solution.

The small parameter variables first order represent the acoustic variations on top of the stationary background mean or average flow zeroth-order solution. Beginning from equation 8 , the dependent variables are defined as. In the frequency domain, the time dependency is harmonic and the dependent variables and sources can be expressed in their Fourier components.

For example, for the pressure, this reads. The perturbation to the density is expressed in terms of the pressure and temperature using a first-order Taylor expansion or generally, the density differential given above. These thermodynamic quantities are related to the speed of sound, the speed of the propagating wave, by the important relations. Here, we have explicitly stated that the speed of sound is the isentropic speed of sound. This is the value normally tabulated and referred to as the speed of sound.

The isothermal speed of sound is given by. Inserting the above equations into the governing equations and retaining only the terms that are linear in the perturbed quantities will yield the full linearized Navier-Stokes equations. After reordering the terms, they are. Recall that the background fields are assumed stationary or at least slowly varying in time compared to the perturbations. We have, however, retained the perturbation mass source term. This term is seldom included, but it can be used to model complex processes that we do not want to describe in detail.

For example, the action of a pulsating sphere or heat injection may be well approximated by such a mass source term in acoustics. A mass-like source term also appears when using second-order perturbation approaches. In the equations presented above, no perturbation has been performed in the material properties, like the viscosity or the thermal conductivity. The linearized Euler and linearized Navier—Stokes equations form the basis for the field of [aeroacoustics] Summary of Acoustics Pillar aeroacoustics.

In the presence of a background flow , these equations sustain the propagation of nonacoustic waves like entropy and vorticity waves. These are convected at the background flow speed.

Elastic Waves in Solids II

The equations show great numerical challenges to solve. In the quiescent case when the background flow is assumed to be zero, , the full linearized equations simplify significantly to. These equations describe the propagation of acoustic waves, including thermal and viscous losses, explicitly. When modeling or studying miniature acoustic devices, like mobile devices and microphones, it is essential to include the physics described by the thermoviscous equations in order to predict correct behavior and response.

If the thermodynamic processes in the system are assumed to be adiabatic and reversible isentropic , and viscosity and thermal conductivity can be neglected, the thermoviscous acoustic equations reduce to. The latter energy equation is used to express the time differential of the pressure in terms of the density. Using the density differential from Eq. This equation is essential in the general case, when the density is not a constant. For constant material data and no heat source, the equation reduces to the usual.

Note that similar formulations can also be expressed in terms of a velocity potential or the density. We can now recover the scalar wave equation. With the usual trick of taking the time derivative of the continuity and the divergence of the momentum equation, combined with the time derivative of the pressure, we get. This is the most general form of the scalar wave equation with source terms that are valid for nonconstant density and nonconstant speed of sound.

In the frequency domain, the Helmholtz equation is recovered by replacing the time derivative with multiplication by , which gives. This general form of the governing equations is essential in, for example, underwater acoustics , where material data is depth dependent. In a multiphysics context, one example is the temperature distribution in a muffler system, , influencing the material properties, and.

Here, the correct formulation is also essential. The concept of impedance is important in mechanical engineering and acoustics in particular. Impedance is a frequency domain concept and is defined as the ratio between the force and the flow variables at a boundary or in a point. An impedance boundary condition can be used to impose the properties of the boundary without modeling it explicitly. Impedance boundary conditions thus generalize the sound-hard and sound-soft boundary conditions to address a large number of cases between these two extremes. A frequency-dependent impedance can be used to model or simplify a complex mechanical system or the properties of an absorbing boundary.

One example is the impedance of the human eardrum tympanic membrane , which is well described and studied. It is common to operate with three different impedance concepts in acoustics: acoustic, specific, and mechanical impedance. This quantity is also sometimes known as the surface normal impedance and is often used to specify or define boundary conditions. Different wave types can be defined by their characteristic specific acoustic impedance, relating the pressure and particle velocity at every point. The characteristic specific acoustic impedance of a plane-traveling lossless wave is given by the well-known quantity.

There is an energy transport, or flow of energy, associated with the propagation of acoustic waves. The magnitude of the sound intensity or simply intensity magnitude is defined as the time-averaged energy per unit time through a unit area, where the normal of the area is in the wave propagation direction.


In general, the intensity is a vector describing the energy transport magnitude and direction. The intensity is the time average of the instantaneous intensity. The expression for follows from the so-called acoustic-energy corollary this is a statement of energy conservation for acoustic waves; see Ref. The integration time depends on the type of acoustic signal. For noise, it should be a long time, while for a harmonic signal, it is the signal period. For plane-propagating waves in a lossless quiescent fluid, the intensity is given by.

In the more general case for wave problems solved in the frequency domain in a quiescent fluid solving the Helmholtz equation , the instantaneous intensity is and the intensity is given by. A very general form of the acoustic-energy corollary valid for all linear acoustics is given by Myers in Ref. Knowing the intensity vector field distribution in a system is useful when analyzing the energy transport and dissipation. The figure above shows the intensity through an absorptive muffler system. Specifically, knowing the combination of incident and reflected waves at a boundary can be used to compute a transmission loss; that is, the ratio between the transmitted power and the incident power.

For loudspeaker systems, the integral of the normal intensity over a closed surface defines the radiated power. The calculation and prediction of absorbed acoustic energy in, for example, human tissue or porous materials also involves the intensity. These conditions describe how the acoustic field is generated, how it behaves at boundaries, and how it behaves as it radiates in an infinite domain.

In situations with no mean background flow, the multiphysics nature of acoustics appears at boundaries where the sound field interacts with its surroundings. At the boundary to an elastic structure, the acoustic field will experience the acceleration of the structure, while the structure experiences the pressure load. This is the typical vibroacoustic coupling or acoustic-structure interaction.

Mathematically speaking, two conditions apply at the interface: a kinematic condition stating continuity in displacement or velocity and a dynamic condition stating continuity in stress. At the surface of a sound source, like a speaker, sound is not only generated but also impacts the speaker with an added mass and couples back all the way to the electrical source. As already discussed, impedance models can be used as boundary conditions to model the behavior of a complex system in a simplified manner; for example, specifying the impedance at the surface of a porous absorber instead of modeling it in detail Ref.

Impedance conditions are widely used in computational acoustics to reduce the complexity of a model. The impedance models can be seen as engineering relations, as they approximate the true behavior of a complex system but simplify its description. Using an impedance condition can often give a good first approximation to a complex problem. The behavior of the acoustic waves as they enter or leave a system is described by radiation conditions or port conditions. This class of conditions is especially important in numerical simulations where idealized conditions cannot always be used.

At the inlet of a system, like a waveguide, waves should be allowed to enter the guide, while reflected waves should be allowed to exit. This is one example of a radiation or nonreflecting boundary condition NRBC. At the inlet and outlet of ducts, this type of condition can be implemented as a so-called port condition. In an open system where waves can freely propagate toward infinity, they have to fulfill the Sommerfeld radiation condition, as the distance from sources and scatterers tends to infinity.

Formats and Editions of Elastic waves in solids / 1, Free and guided propagation. [abymedoxuhav.tk]

In numerical simulations, the proper implementation and formulation of this condition is important in order to avoid spurious numerical reflections that can pollute the solution. A classical example of this type of condition is given by Bayliss, Gunzburger, and Turkel Ref. Using a so-called sponge layer to mimic an infinite open problem can be an alternative to a boundary condition. This is a domain where outgoing waves are killed with little or no reflections. One such layer method is the well-known perfectly matched layer PML.

Other formulations also exist for time domain problems where real coordinate stretching is combined with artificial numerical diffusion. In general, these are referred to as absorbing layers. When acoustic waves propagate in, for example, porous material or a narrow duct, the loss mechanics involved can be complex to describe and model from first principles. In other situations, where a detailed description of the loss mechanism exists, it can be difficult to find analytical or numerical solutions to the problem.

In these cases, it can be advantageous to use a so-called equivalent fluid model to describe the loss process in a homogenized way. The typical application is when solving the Helmholtz equation, where the losses are included by defining a complex-valued density and speed of sound both parameters are typically frequency dependent. Many equivalent fluid models exist to describe losses in porous materials; for example, the Delany—Bazley—Miki model Ref.

The validity of the models are restricted to certain parameter ranges representing the rigid or limp porous matrix approximations. In numerical simulations, an equivalent fluid model is also computationally less demanding than a full poroelastic wave model solving Biot's equations.

When acoustic waves propagate in narrow regions, the losses viscous and thermal associated with the acoustic boundary layer need to be taken into account. For narrow slits or waveguides of constant cross sections, homogenized models exist. They are often referred to as low-reduced-frequency models see the Thermoviscous Acoustics section. Homogenized equivalent fluid models are also used in underwater acoustics , where losses are defined by an attenuation coefficient. Values are typically frequency dependent and given in nepers per meter, dB per kilometer, or dB per wavelength Ref.

The propagation of sound in a solid happens through small-amplitude elastic oscillations of the solid's shape and structure. Example of the vibration analysis of a loudspeaker cabinet the deformation amplitude is exaggerated for visualization. Simulations can be used to identify vibration modes that radiate sound in an unwanted manner from a loudspeaker system.

The equations that govern the propagation of linear elastic waves in solids are derived from the general governing equations of structural mechanics. The equations are linearized and formulated in the limit of small perturbations in the stress and strain. The most general linear relation between the stress and strain tensors in solid materials can be written as. This is Hooke's law for a linear elastic material.

For small perturbations, the strain tensor on vector form is given by. The elastic wave equation is obtained from Newton's second law conservation of momentum and is given by. In the frequency domain, the dependent variable and sources are decomposed into their Fourier components. The resulting Helmholtz-like equation for the elastic waves in a solid is given by.

Poroelastic waves describe the combined propagation of pressure waves and elastic waves in porous materials. The pressure waves propagate in the saturating fluid in the pores and the elastic waves propagate in the porous matrix frame. In the limit where the porous matrix frame is almost motionless rigid , homogenized equivalent fluid models exist to describe the acoustic behavior. The same is true in the limp limit, where the frame follows the movement of the fluid. This description is not generally true for all frequencies or material parameters.

Moreover, when the porous material is in contact with a vibrating solid surface, the elastic waves also need to be included. The model that describes the combined propagation solving for both the displacement and the pressure is given by the Biot theory.

In his seminal work from , Biot extended the classical theory of linear elasticity to porous media saturated with fluids see Ref. The porous matrix is described by linear elasticity. Damping is introduced by considering the frequency-dependent losses due to viscosity and thermal conduction of the saturating fluid in the pores. The formulation of the Biot equations that include the thermal effects is sometimes referred to as the Biot—Allard theory see Ref. One form of the governing equations is. The two frequency-dependent quantities, the viscosity function and the fluid compressibility , include information about the viscous and thermal losses, respectively.

The equations only have a limited number of analytical solutions that are not applicable to realistic geometries, complex boundary conditions, or multiphysics applications. There are many techniques available to solve the governing equations of acoustics, and each has its weaknesses and strengths.

The material tensors in Eqs. It is seen that the electric subspaces are the same as magnetic ones for polarized ceramics.

Elastic waves in solids 2

Thus, the physical quantities, the coupled coefficients and the corresponding operators of polarized ceramics are calculated as follows. It is seen that for quasi-static electromagnetic approximation, there exist four elastic waves in polarized ceramics solids. Only two waves v2 and v4 are affected by the piezoelectric coefficients which speed up the propagation of the second and fourth waves. For quasi-static mechanical approximation, there exist two electromagnetic waves in polarized ceramics solids, both are affected by the.

The Maxwell's equations, coupled to the mechanical equations of equilibrium, and the mechanical equations of motion, coupled to the equation of static electric field, are studied here in the standard spaces of the physical presentation. The complete set of uncoupled electromagnetic waves equations and elastic waves for anisotropic piezoelectric solids are deduced. The results show that the number of electromagnetic waves and elastic waves in piezoelectric solids is determined by both the subspaces of electromagnetically anisotropic media and ones of mechanically anisotropic media.

For the piezoelectric material of class 6 mm, it is seen that there exist four elastic waves, but only two waves are affected by the piezoelectric coefficients, two electromagnetic waves are affected by the piezoelectric coefficients. Eringen, A. Springer, New York Mindlin, R. Solids Struct. Lee, P. Sedov, A. Piezoelectric halfspace. Dieulesaint, E. Ting, T. Wave Motion 44 1 , Li, S. Yang, J. Guo, S. Acta Mech. Solida Sinica 14 1 , An eigen theory of waves in piezoelectric solids Academic research paper on " Materials engineering ". Effect of negative permeability on elastic wave propagation in magnetoelastic multilayered composites.

Guo School of Civil Engineering and Architecture, Central South University, Changsha, China Electromagnetic waves generated by mechanical fields need to be studied in the calculation of radiated electromagnetic power from a vibrating piezoelectric device [4,5], and are also relevant in acoustic delay lines [6] and wireless acoustic wave sensors [7], where acoustic fields produce electromagnetic waves.

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Jk sik 4. Transposing Eq. Substituting Eq. For quasi-static mechanical approximation, there exist two electromagnetic waves in polarized ceramics solids, both are affected by the piezoelectric coefficients which slow down the propagation of electromagnetic waves. References 1.

Elastic waves in solids 2 Elastic waves in solids 2
Elastic waves in solids 2 Elastic waves in solids 2
Elastic waves in solids 2 Elastic waves in solids 2
Elastic waves in solids 2 Elastic waves in solids 2
Elastic waves in solids 2 Elastic waves in solids 2
Elastic waves in solids 2 Elastic waves in solids 2

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