Equations For Sound Propagation Research Articles

Higher-order Padé approximations for accurate and stable elastic parabolic equations with application to interface wave propagation

Higher-order Padé approximations are applied to derive accurate and stable parabolic equations for sound propagation in oceans bounded below by an elastic bottom or bounded above by ice cover. Accuracy is achieved by placing constraints on the derivatives of the Padé approximations at the point corresponding to the reference wave number. Stability is achieved by requiring that the Padé approximations map part of the lower-left quadrant of the complex plane into the upper half of the complex plane. Elastic parabolic equations based on these Padé series can handle problems involving compressional, shear, and interface waves, very wide propagation angles, and large depth variations and weak range variations in the seismoacoustic parameters. A finite-difference spectral solution is developed for generating reference solutions and starting fields. The rotated elastic parabolic equation is used to investigate the accuracy of the elastic parabolic equation for range-dependent problems.

Parabolic decomposition of the separable Helmholtz equation

The solution of the separable Helmholtz equation is represented as the convolution of solutions of two related (formally) parabolic partial differential equations. Separable means that the square of the index of refraction is a function of depth plus a function of range and that the depth of the ocean is assumed to be constant. This representation decomposes the separable Helmholtz equation into two parabolic partial differential equations. One equation is the usual Fock-Tappert parabolic equation for sound propagation. The solutions of the second parabolic equation may be regarded as candidate kernels of special integral transformations, or transmutations. All of these equations have many solutions, of course, so initial and/or boundary conditions are presented that will determine solutions of the parabolic equations so that their convolution will give the Green's function for the Helmholz equation. Examples of such transmutation representations will be presented.

Theoretical study of the photoacoustic signal produced in a relaxing gas

The photoacoustic signal generated in a photoacoustic gas cell with rigid wall boundaries is calculated by solving the coupled equations for sound propagation and thermal diffusion in a viscous and relaxing gas. For a cylindrical geometry, monochromatic radiation, and sinusoidal modulation, when a condition of zero heat flow through the lateral walls of the cell is verified, a general expression is found that can be applied whatever the deactivation scheme is for collisional relaxation in the gas. Both resonant and nonresonant modes of operation are considered.

Parabolic decomposition of the separable Helmholtz equation

The solution of the separable Helmholtz equation is represented as the convolution of solutions of two related (formally) parabolic partial differential equations. Separable means that the square of the index of refraction is a function of depth plus a function of range and that the depth of the ocean is assumed to be constant. This representation decomposes the separable Helmholtz equation into two parabolic partial differential equations. One equation is the usual Fock‐Tappert parabolic equation for sound propagation. The solutions of the second parabolic equation may be regarded as candidate kernels of special integral transformations, or transmutations. All of these equations have many solutions, of course, so we present initial and/or boundary conditions that will determine solutions of the parabolic equations so that their convolution will give the Green's function for the Helmholtz equation. Examples of such transmutation representations will be presented.

Diffraction tomography I: The wave-equation

A general wave equation for sound propagation in a viscoelastic medium is obtained. From this general equation an approximate inhomogeneous wave equation is derived by perturbation methods. Born's and Rytov's approximations are considered. The equation is finally brought into a form which provides transformation properties under rotation of the test object required for diffraction tomography.

Diffraction tomography I: The wave-equation.

A general wave equation for sound propagation in a viscoelastic medium is obtained. From this general equation an approximate inhomogeneous wave equation is derived by perturbation methods. Born's and Rytov's approximations are considered. The equation is finally brought into a form which provides transformation properties under rotation of the test object required for diffraction tomography.

Propagation of sound in a curved bend containing a curved axial partition

Sound transmission in a 180° bend containing a curved partition positioned on the axial center line is investigated theoretically and experimentally using equations for sound propagation in straight and curved ducts. Good agreement is obtained and small discrepancies are discussed. The partition is found to significantly alter the sound propagation through the bend and reasons for the different acoustic behavior are given.

A reactive acoustic attenuator

A reactive acoustic attenuator that combines high reflection of low frequency sound with low pressure drop coefficient is investigated experimentally and theoretically by using equations for sound propagation in straight and curved ducts. Good agreement is obtained and the theory is used to redesign the device to give a minimum transmission loss of ten decibels over a frequency range of three-quarters of an octave. Small discrepancies between theoretical and experimental results are discussed.

Relation between the solutions of the Helmhotz and parabolic equations for sound propagation

In this paper we establish an integral transform relation between the solutions of the Helmholtz and parabolic equations for sound propagation in an arbitrary two-dimensional waveguide. The sound speed is a function of both depth and range. For range-independent sound speeds, the integral transform is proved to be exact by using a normal-mode expansion. For range-dependent sound speeds the stationary phase approximation of the transform is, in lowest order, equivalent to the usual parabolic approximation. Corrections to the parabolic approximation are also calculated using the stationary phase method.

Connection between the solutions of the Helmholtz and parabolic equations for sound propagation

Using a conformal mapping technique in a rectangular waveguide, we present an exact integral relation between the solutions of the Helmholtz equation ψ whose sound speed c(x,y) is a function of depth y and range x, and the solution p of a parabolic equation whose sound speed varies in the mapped depth coordinate and which satisfies a related boundary-value problem. The integral relation is a Fourier-Mellin-inversion transformation of p. The conformal transformation interrelates the sound speeds of the two equations. Examples presented include the special case of Polyanskii when c(x,y)=c(y), and profiles which are sinusoidal in depth and exponentially decrease to a constant in range. The exact relation between ψ and p also clarifies the use of p as an approximation to the asymptotic value of ψ. For example, in multimode propagation problems in cylinderical coordinates, p preserves at most one modal amplitude of the asymptotic value of ψ, the others being scaled by an eigenfrequency dependent factor. Also, because of the exact relation between ψ and p, their respective incident fields cannot be independently chosen.

Numerical Solution of the Transport Equation for Sound Propagation in He II

Assuming an anomalous phonon dispersion, the velocity shift and the attenuation of ultrasound in liquid ${\mathrm{He}}^{4}$ below 0.5 K are calculated by a direct numerical solution of the Boltzmann equation for the phonon distribution function. In order to account for the hydrodynamic regime the collision invariants are treated separately. The results are in qualitative agreement with the experiments and show a nonmonotonic increase of the velocity shift as a function of the applied frequency, but they are different from an approximate solution using relaxation times. The method developed to solve the Boltzmann equation can also be applied directly to calculate transport properties of phonons in solids.

Derivation of the Stochastic Helmholtz Equation for Sound Propagation in a Turbulent Fluid

The basic stochastic wave equation for sound propagation through a turbulent field is derived from first principles as represented by the continuity equation, the Navier-Stokes equations, and an appropriate equation of state. It is shown that, for very low turbulent Mach numbers and monochromatic transmissions sound propagation in a turbulent field can be adequately represented by the stochastic Helmholtz equation, ∇2p+k02μ2p = 0, if the acoustic wavelength does not exceed the Taylor microscale λg divided by [(σκ)12(u/c)]12, where σκ is the Prandtl number and u/c is the turbulent Mach number. In addition, the comparison of similar turbulent flows in air and in water, with respect to estimating their acoustic frequency limitations, is illustrated by contrasting: (1) the Baerg and Schwartz experiments with the Stone and Mintzer experiments, and (2) the turbulent lower atmosphere with the turbulent upper ocean.

A Fifty Horsepower Siren

This paper contains a description of the theory, construction, and testing of a siren with an acoustic output of fifty horsepower, at an efficiency of seventy percent. The siren is of the traditional type in which the air stream from a compressor passes through an orifice which is opened and closed at an audiofrequency. Part I contains a development of the theory on which the design was based. The theory is developed with the aid of an equivalent circuit diagram (Fig. 1) in which the various acoustic impedances are represented as lumped circuit elements. The theory is for the most part based on the customary linear equations for sound propagation, but the non-linear nature of the impedance of the port is taken into account. Mere inspection of the equivalent circuit diagram is sufficient for the understanding of the considerations which are important for the obtaining of a high efficiency. According to this theory the best efficiency can be realized in a siren when the orifice is provided with a suitable acoustic horn, when the excess pressure of the air supplied by the compressor is small compared with the absolute pressure of the atmosphere, and when the operations of opening and closing the port occupy a small fraction of the period of one cycle. Expressions are obtained for the theoretical efficiencies of several types of sirens, and are found to be near one hundred percent for a good design. Part II contains a description of the siren which was constructed to operate with a compressor that supplied 2500 cubic feet of air per minute at a pressure of five pounds per square inch. The siren contains six ports with a total area of 22 square inches, which are opened and closed by a rotary chopper at a frequency of about 500 cycles per second. Each of the ports is provided with an exponential horn whose cut-off frequency is 125 cycles per second. The theoretical efficiency of this siren is computed to be between 70 and 90 percent. Part III is devoted to a description of the testing of the siren and a discussion of the results obtained. The tests indicated that at full output the siren produces an average level of 138 db at a distance of 100 feet. The measured level at the mouth of the horns is 174 db, and the calculated level in the throat of the horns is 184 db above 10−16 watt per square centimeter. The level in the throat of the horns corresponds to a sound intensity of 260 watts per square centimeter. The most reliable method of determining the efficiency, based on thermodynamic principles, indicates that the efficiency of the utilization of the available energy in the air from the compressor is about 72 percent.

The reflexion of sound pulses by convex parabolic reflectors

The reflexion of a train of simple harmonic waves by a convex paraboloid of revolution, and by a parabolic cylinder, has been discussed by Lamb. In the present paper these results are extended to the reflexion of plane waves of arbitrary form. It is found that on the introduction of suitable variables the equation of sound propagation transforms (in each case) into a simpler equation whose general integral can be obtained by quadratures. Two unknown functions are introduced during the integration, which have to be determined from the boundary conditions. This involves in both cases the solution of a Volterra integral equation, which is effected numerically by calculation of the first terms in the series development of the resolving kernel. An interesting feature of the solutions obtained is that when a suitable time scale is introduced (for a sharp-fronted pulse the time must be counted from the onset of the wave), the reflected wave experienced is the same at all points on any paraboloid (or parabolic cylinder) confocal with the reflector.