The equations of fluid mechanics and sound are derived here based on a molecular model of an ideal diatomic gas. Physics books [e.g. Feynman, and Landau and Lifshitz] typically discuss the molecular nature of sound, but then derive the sound wave equations by modeling air as a continuous elastic medium (i.e. no lumpy molecules). Mathematically this is just fine, but I find it much more satisfying to derive the wave equations directly from the molecular point of view. I also think this is a more straightforward derivation, since it completely avoids any need to deal with "specific heat ratios" or "adiabatic processes." The effects associated with these terms arise quite naturally directly from the molecular model.

This is a rather technical section, and assumes familiarity with calculus, differential equations, and statistics. There is an equation-free description of the physics of sound in the section on music and sound, which includes animations, etc. The focus here is on sound waves in air. Sound also propagates in other media, and the behavior can be quite different. One example is sound propagation in crystal structures, which is considered in a short separate section (see lattice waves).

The basic part of the derivation is contained in the numbered sections below. You can follow them sequentially, go directly to any section by clicking on the title following the numbers, or click on a highlighted topic. There are also a number of sections on related subjects listed at the bottom of this page, which can be accessed directly by clicking on the subject title. Within the physics section links generally lead to other places in the physics section, and not to the Glossary. Note that some browser versions garble the greek letters in the text; see Greek Letters and Special Characters: Microsoft vs. the World.

Comments welcome. E-mail to: aludwig@silcom.com

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**Derivation Outline**

The first step is to derive basic differential equations of fluid mechanics from the molecular model. This part of the derivation is essentially the same as the one presented by Vincenti and Kruger Chapter IX (who, however, develop the acoustic equations using the continuous-elastic-medium model). The sections are:

1. Molecular Model and Basic Physical Variables.

The basic molecular model is defined. Temperature and pressure are related to the molecular velocity statistics. The manner in which molecular motion transports mass, momentum, and energy is quantified in terms of fluxes across a small area in space.

2. Basic Differential Equations.

The fluxes are used to derive the basic differential equations governing the dynamic molecular model: the equation of continuity, Euler's equation, and a differential equation for pressure. No approximations or assumptions are made beyond those made in defining the model itself.

Small-signal approximations are made to derive the acoustic and wave equations from the above differential equations. These are the basic equations for linear acoustics. A separate section on nonlinear acoustics deals directly with the basic differential equations without the small-signal approximations.

4. Plane Waves.

The general plane wave solution to the sound wave equations are analyzed. Sound created by a piston vibrating harmonically in a tube is analyzed. The small-signal approximations are validated for very loud sound levels (validation meaning the results are accurate to within a fraction of a percent), but a case where a solution of the non-linear equations is important is also noted. Sound wave momentum is evaluated, leading to some interesting questions.

5. Coupling Power out of a Plane Wave.

The reception of power from a sound wave is analyzed. Conditions for total power absorption by a piston in a tube are determined. An analysis of coupling to an open-ended tube, and the relative response to pressure and velocity is presented.

An elementary spherical wave solution is presented. A more detailed description is given in the section on plane and spherical wave spectra.

7. Sound Waveguide and Cavity Modes

The theory for computing room resonances is developed. Resonant build-up, or room gain, is analyzed, and resonances in odd-shaped rooms are briefly discussed.

8. Image Theory

Image theory is developed, as a technique for a more accurate evaluation of room acoustics. The reverberation time calculated using image theory is compared to a standard handbook formula. This is a case where sound acts as if it propagates along rays, and a separate section provides a derivation of geometrical optics dealing with such ray behavior.

**Separate Sections on Related Subjects**

General integral solution of the sound equations

Derivation of the Green's function boundary integral solution of the wave equations.

Plane and spherical wave spectra

General representations of sound waves in terms of wave spectra.

Cone and flat piston radiating in an infinite baffle

Exact and numerical solutions, including the radiation impedance.

Scattering from Disks and Apertures

Generalized multipole technique

Numerical solution method based on multipoles.

Point source radiating in a moving medium

General solution for Doppler shift, for both transmitter and receiver moving with arbitrary velocities relative to the medium.

Molecular noise and "blackbody" radiation.

Calculation of the background noise level due to random molecular motion, and relation to the threshold of human hearing.

An analysis of the collision mechanics, combined with a Monte-Carlo simulation, demonstrating that the static pressure exerted by diatomic and monatomic molecules is the same, at a given temperature.

The nonlinear acoustic equations, the equation of continuity, Euler's equation, and the pressure equation, are solved for the case of a plane wave. The particle velocity is initially a pure sinusoid at a single frequency. The density and variance are solved as a quadratic function of the velocity. Harmonics grow as the wave propagates, eventually creating a shock wave.

The basic plane wave solutions propagate without loss. The loss mechanism at the molecular level is determined to be due to a small deviation from the Gaussian velocity statistics assumed in the derivation.

Physical Constants, Units, and Conversion Factors

Important physical constants, and conversion factors.

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Bohdan Zograf informs me that he has created a Belourussian translation of this section and posted it here: Belorussian translation