### Pulses and Other Wave Processes in Fluids: An Asymptotical Approach to Initial Problems

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Description Table of Contents Product Details Click on the cover image above to read some pages of this book! Our Awards Booktopia's Charities. Physiologically, such a representation is motivated by the fact that the human ear itself filters via the auditory filters and perceives sound on a logarithmic frequency axis. This is also a fundamental concept used in music. In the same way, the human ear also has a logarithmic sensitivity, hence the use of a decibel scale for levels, as already mentioned.

Mathematically, the propagation of a sound wave along one dimension in space and time is described by the scalar wave equation. This is a partial differential equation PDE where t denotes time, x the spatial coordinate, c the speed of sound, and p denotes pressure the dependent variable. The equation is derived from the conservation laws of continuum mechanics — conservation of mass, momentum, and energy — under several assumptions that are discussed in more detail below.

The general solution to the 1D wave equation is given by. Actually, the most general solution is a linear combination of such functions, which is known as d'Alembert's formula when initial conditions are considered.

## Multiphysics Cyclopedia

One such solution is a propagating sine wave. The wave number is usually defined as a vector, such that it also contains information about the direction of propagation of the wave. An example of a propagating wave is depicted in the animation above. The period of the oscillation is defined by. The equivalent of the wave equation formulated in the frequency domain is the all-important Helmholtz equation. The Helmholtz equation can be derived in several ways: by expanding the pressure into its Fourier components or equivalently using separation of variables time and space.

The simplest method is to assume that pressure is a time-harmonic signal of the type. Inserting this expression into the wave equation and rearranging it yields the Helmholtz equation, with constant material parameters, for a time-harmonic signal. The Helmholtz equation very often forms the basis of the numerical or analytical analysis of acoustic problems. In order to solve the wave equation or the Helmholtz equation, they should be combined with material parameters, boundary conditions, and initial conditions that describe the physical problem at hand.

For more details on fundamental acoustics, see Ref. In acoustics, sound is produced, transmitted, and affected by the medium in which it is propagating and finally detected, perceived, and analyzed. 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.

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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.

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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.

Velocity of Sound Wave in a Fluid

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.

### Introduction to Acoustics

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.