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The article announces that Peter J. Westervelt, Professor of Physics Emeritus at Brown University, was awarded the Silver Medal in Physical Acoustics by the Acoustical Society of America.

Earlier studies [J. Acoust. Soc. Am. 29, 934 (1957)] of the mutual nonlinear interaction of two plane waves of sound with each other exhibited a singularity when the waves became collinear. The singularity is removed in this study, which leads to a method for calculating the pressure absorption coefficient α of a wave with wave vector k1 resulting from interaction with isotropic acoustic waves having the energy density spectrum u (k). Thus, α= (4ρ0c20)-1(1+1/2Λ)2π Fk1 0 ku (k) dk+k1F∞d1 2u (k) idk+k21Fk11k-1u (k) dk], in which ρ0 and c0 represent the density and velocity of sound of the fluid which is assumed to be dispersionless. The constant 1/2Λ =ρ0c d0-1(dc/dρ)ρ0. Subject Classification: [43]25.22, [43]25.35. [ABSTRACT FROM AUTHOR]

In addition to serving as an ideal discipline for introducing the concepts of quantum field theory and general relativity, nonlinear acoustics provides a new means for absorbing and generating sound. The many applications of this exciting subject are based in part on the fact that, in common with vector (electromagnetic and shear) and tensor (gravitational) waves, the scalar sound wave transports energy, angular momentum, linear momentum, and, incidentally, information. The methods of nonlinear acoustics can account for such disparate phenomena as the parametric array and the radar time delay in a gravitational field; the absorption of sound in superfluid helium and shock-wave attenuation in a gas; radiation pressure and sonic boom; the scattering of sound by sound; and the generation of gravitational waves. Based on the ability of nonlinear acoustics to subsume within its methodology, and hence to unify, different physical processes, as well as its demonstrated ability to contribute solutions to practical problems, it semms reasonable to expect that this subject will become a part of the curriculum in more colleges and universities than is the case at present. Subject Classification: 10.60; 25.10. [ABSTRACT FROM AUTHOR]

The effect on the generation of jet noise by the secondary air induction of a modified jet nozzle is analyzed. It is shown that the combination of the secondary air with the primary jet air creates a new jet stream of larger area, lower velocity, and lower noise generation. The decrease in noise radiation is found in terms of the area of the combined jet stream. Detailed check of the theory is not possible at present because measurements of the areas of the combined jet streams have not been generally made along with measurements of the noise. However, it is possible to estimate the upper limit of the appropriate jet stream area in order to obtain the upper limit on noise reduction. The upper limit on noise reduction obtained from the theory is consistent with existing measurements. Qualitative conclusions from the theory with respect to the spectrum and directivity of the noise radiated by jets with modified nozzles are in accord with measurements. [ABSTRACT FROM AUTHOR]

Owing to the inherent nonlinearity of the equations of motion for a perfect fluid, two or more sound waves passing through a common region will, in general, interact with one another and give rise to scattered waves. In this paper, a source function is obtained for the lowest order scattering process which is quadratic in the primary field variables. This function is rewritten in a form that demonstrates that no scattered waves exist outside the region of interaction of two sound beams intersecting each other at right angles. Furthermore, it is demonstrated that no such scattered waves exist when the two beams intersect at the angle θ = -cos-11/2ρ0c0-2d2p/dρ2ρ = ρ0, where ρ, c, and p stand for the density, sound velocity and pressure, respectively, and the subscript 0 means ambient values of these quantities. The interfering effect of pseudo-sound, induced by radiation pressure, is suggested as an explanation for the results of Ingard and Pridmore-Brown, who have reported recently what they believe to be scattered waves from the interaction region of two sound beams intersecting each other at right angles. [ABSTRACT FROM AUTHOR]

An expression obtained (1951) by the author for the radiation force on a scattering obstacle with arbitrary normal impedance is now shown to be valid for any scattering obstacle. The derivation of the final expression for the force in terms of the asymptotic scattering function for the obstacle in the field of an incident plane wave is accomplished by taking into account the interaction of the incident wave with the scattered wave. Thus the former assumption of a perfectly collimated beam (1951) is avoided by considering the incident plane wave to be of infinite extent. The result for the force in the direction of the incident wave is F∥ = c0-1power scattered+power absorbed-∫Γs cosθdA where c0 is the velocity of sound, Γs is the magnitude of the mean scattered intensity, θ is the angle formed between the incident and scattered waves. This is the same result as that obtained in 1951. The expression given in 1951 for the force perpendicular to the incident wave is correct only for an object which scatters in a restricted way. The correct general expression for the perpendicular component of the force is F⊥ = - c0-1∫Γs cosβdA in which β is the angle formed between F⊥ and Γs. [ABSTRACT FROM AUTHOR]

Starting from the Navier-Stokes equations, general equation governing the generation of vorticity R is obtained: ∇2R = μ0-1∇ × ∇ · <ρ0uu>, wherein R is the time-independent vorticity in the Eulerian frame and <ρ0uu> is the average value of Reynolds' stress dyadic. The solution to this vorticity equation, when properly transformed to the particle coordinates, is shown to be divergence free. A specialization of the vorticity equation to the case of solenoidal first-order motion is shown to lead to the generating term employed by Rayleigh and by Schlichting; a specialization to the case of irrotational first-order motion is shown to lead to the generating term employed by Eckart. The sum of the two specialized driving terms does not equal the general term, indicating that the contributions to vorticity from rotational and compression effects are not independent of one another. The theory is applied to find the streaming generated by a well-defined beam of sound giving results agreeing with Eckart and with Markham. However, when it is applied to a two-dimensional standing wave problem, the configuration of the streaming velocity in the boundary layer is found to differ from the results obtained by Rayleigh. Finally, the effect on streaming of a time-dependent viscosity coefficient is examined. [ABSTRACT FROM AUTHOR]

The one-dimensional wave equation is discussed to the second order of approximation by means of a transformation that carries the equation from the Eulerian to the Lagrangean form. Airy's solution to this equation in Lagrangean form has been shown by Fubini to be an excellent approximation to an exact solution of Earnshaw's equation of motion; therefore, Airy's solution is chosen as the basis for much of this discussion. An expression for the local mean hydrostatic pressure in a plane progressive wave is obtained by transforming Airy's solution from particle to local coordinates. In a similar way, the particle velocity in fixed coordinates is shown to possess a time-independent component proportional to, and in a direction opposite to, the intensity vector This d.c. counter-velocity is predicted without recourse to viscous forces and is compatible with zero average mass velocity. For a sound pressure level of 151 decibels in air there should exist a steady particle velocity of 1 cm/sec. Small particles suspended in the field can, under certain circumstances, acquire this velocity. An approximate treatment is suggested for handling second-effects arising from stationary field configurations. The influence of viscosity is discussed qualitatively. One purpose of this paper is to correlate work in the field which has not so far been published in this country. [ABSTRACT FROM AUTHOR]

The generation of transient signals by the propagation of large-amplitude pulses has been investigated. Measurements of the transient signal along the axis of propagation of 10-MHz pulses in carbon tetrachloride confirm the validity of the theoretical treatment of Berktay [J. Sound Vibration 2, 435-461 (1965)]. Data are also presented for the angular dependence of the transient signals generated by 20-MHz pulses in water. [ABSTRACT FROM AUTHOR]

This paper presents the theory of highly directional receivers and transmitters that may be 'constructed' with the nonlinearity of the equations of fluid motion. [ABSTRACT FROM AUTHOR]

It is proved that a linearization of the hydrodynamic equations in the manner of Oseen does not lead to a description of the steady flow phenomena which occur in connection with oscillatory flow past solid obstacles. [ABSTRACT FROM AUTHOR]

When high energy neutrinos interact with nucleons in the ocean a jet of hadrons is produced which deposits thermal energy. This thermal energy is expected to produce a sonic pulse which hopefully will be sufficiently intense and directional to enable the energy and direction of incidence of the primary neutrino to be determined [Antares Parvulescu, J. Acoust. Soc. Am. 61, 580(A) (1977)]. This paper discusses the physical mechanism whereby the energy of the neutrino is converted into a sound pulse. A simple model will be exploited to account for the signature expected from such an event. [Work supported in part by the U. S. Naval Ocean Research and Development Activity and by the U. S. Department of Energy.] [ABSTRACT FROM AUTHOR]

The following topics are discussed from a unified point of view: acoustic streaming; radiation pressure and other second-order forces; scattering of sound by sound and self-scattering of sound; interaction of sound with orifices. [ABSTRACT FROM AUTHOR]

A high-power modulated sound beam acts like an end-fire directional array for the modulation frequency. This occurs because the nonlinear terms in the equations of motion cause such a beam to act like a distribution of sources for the modulation frequency. If the sound beam is unmodulated, it behaves precisely like a parametric amplifier for any sound traveling in the direction of beam and thus can be used as a highly directional receiver. In order to test this concept, two carrier beams having approximately the same frequency ω1 ≈ ω2 ≈ ω were superposed on each other. The modulation frequency is then the difference ω8 = /ω1 - ω2/. If the pressure of each carrier is the same and equal to P0 the modulation percentage will be 100. The expression for the radiated intensity Is at a distance R0 far from this source is Is=ωs4P04S02[1 + 12ρ0c0-1(d2p/dρ2)]22(8π)2ρ03C09R02 × 1α2 + k2sin4(θ/2) in which S0 is the cross-sectional area of the carrier beam, α is the pressure attenuation coefficient for the carrier beam, θ is the angle measured from the axis of the carrier beam, and k = ω8/c0. Maximum radiation occurs at θ = 0 in the forward direction of the carrier. Experiments, described in an accompanying paper, agree well with the theoretical expression which, incidentally, shows the same angular dependence as Rutherford scattering in atomic theory. (Supported by the Office of Naval Research.) [ABSTRACT FROM AUTHOR]

The suggestion is put forward that the effect of sound waves on heat transfer is dominantly result of the modification of the inner streaming boundary layer which is known to occur when the sound particle displacement amplitude s exceeds in magnitude the acoustic boundary layer thickness δac. [ABSTRACT FROM AUTHOR]

A formula for the force on an arbitrary scatterer is derived without making any assumptions about the scatterer. [ABSTRACT FROM AUTHOR]

A simple analysis is given of the effect on jet noise by mixing with an induced secondary air flow. The combination of the secondary air with the primary air of the jet creates a new jet stream of larger area and lower velocity. Power level, spectrum, and directivity changes are presented. The mixing mechanism provides a possible explanation of the noise reduction of such devices as ejectors, corrugated nozzles, and vaned nozzles. The noise reduction is given as a function of several ratios involving the mixing area, the jet temperature, and the forward speed. Also, jet noise is examined as a function of efficiency of conversion of jet stream power to acoustic power. An upper limit appears to exist on the efficiency, and this upper limit is estimated by a simple argument. The resulting limit is compared with experimental values. [ABSTRACT FROM AUTHOR]

The expansions and contractions of an absorbing medium resulting from the passage of an intensity-modulated laser are utilized to form a thermoacoustic array producing a highly directional acoustic wave propagating perpendicular to the axis of the laser beam. [ABSTRACT FROM AUTHOR]

An intensity modulated laser beam is incident on a conical reflector which produces a spreading fan of light. The fan acts like an acoustic directional array, producing an acoustic beam along the incident direction of the laser beam. The radiated acoustic intensity in the farfield is given by I = 1/2ρ0c0aωβP0/4πrcp2 1/a2+k2 sin2θ, where P0 is the average power of the laser, a is the intensity attenuation coefficient for the laser beam, θ is the angle measured from the laser beam direction, k = ω/c0, ω is the modulation frequency, c0 is the speed of light, cp is the specific heat per unit mass, β is the logarithmic coefficient of thermal expansion, and ρ0 is the density. Maximum radiation occurs at θ = 0. [This work was supported by the U. S. Office of Naval Research.] [ABSTRACT FROM AUTHOR]

The average intensity of a finite-amplitude wave satisfies a simply expressed energy conservation law [P. J. Westervelt, in L. Cremer, Ed., Proc. Int. Cong. Acoust., 3rd (Elsevier, Amsterdam, 1960), p. 316] in terms of the coefficient of small-amplitude attenuation, the first-order pressure, and the virtual source density. This equation when applied to a plane wave yields an equation for the attenuation of the pressure amplitude, in terms not only of the small amplitude attenuation coefficient, but also the shock formation distance. This latter equation exhibits a smooth transition through the region of shock formation. An approximation applicable to the end-fire array can be developed, and from this a compatible end-fire virtual source distribution. This source distribution results in a directional pattern modified from the low-amplitude Rutherford law [P. J. Westervelt, J. Acoust. Soc. Amer. 35, 535 (1963)] by a multiplicative factor, which is expressible as a hypergeometric series. This series can be summed exactly on axis, and a simple closed-form expression giving the reduction in sound pressure level due to saturation is found. These results, obtained independently by each of us, are to be published, flint in a Letter to the Editor by J. F. B. accepted by J. Acoust. Soc. Amer. and later in greater detail in a joint paper by both of us. [ABSTRACT FROM AUTHOR]

An experimental demonstration of transient-signal generation during the propagation of a large-amplitude ultrasonic pulse has been made. This phenomenon was treated theoretically by Berktay [J. Sound Vibration 2, 435-461 (1965)]. Data are presented for a pulsed 10-MHz carrier propagating in CCl4. [ABSTRACT FROM AUTHOR]

Recently, Westervelt [J. Acoust. Soc. Am. Suppl. 1 88, S167 (1990)] has developed an analytical treatment of the transient parametric array that is valid for both far field and near field, for arbitrary taper function and arbitrary modulation of the primary wave. The result was previously shown to be consistent in the far field with results of P. Stepanishen [J. Acoust. Soc. Am. 82 (1987)]. The analysis is based on earlier work in general relativity by Westervelt [Acta Phys. Pol. 27, 831-841 (1965)]. Here, a useful numerical treatment of the same problem is presented. A computer model for this numerical formulation has been implemented, and results have been obtained to compare with Westervelt's analytical results. Comparison is also made between numerical results and relevant experimental data found in the literature. [ABSTRACT FROM AUTHOR]

The pressure impulse response of an array with arbitrary taper τ(y′) is shown to be P = βE(4πa)-1[ψeτ(x0 - R0)].00 in which ψc = a(x0 - x1)-1θ(x0 - R) [see Eq. (6), in P. J. Westervelt, Acta Phys. Pol. 27, 831-841 (1965)], β is the coefficient of nonlinearity, and E is the impulse energy. Applied to the truncated array of length L with a square wave input of duration T, it can be shown, by methods similar to those used by Westervelt [P. J. Westervelt, JETP Lett. 4, 225-228 (1965)], that the far-field pressure is given by P = βE(8πR0cT sin2 [lowercase_phi_synonym]){δ(x0 - R0) - δ(x0 - R0 - cT) - δ[x0 - R0 - L(1 - cos [lowercase_phi_synonym])] + δ[x0 - R0 - L(1 - cos [lowercase_phi_synonym]) - cT]}, in which R0 is the length of the vector from the transducer to the field point, L is the length of the vector from the transducer to the absorber, [lowercase_phi_synonym] is the angle between R0 and L, and E is the energy of the square wave. [ABSTRACT FROM AUTHOR]

The virtual source of combination frequencies is well known to be Q = β(∂/∂t)(p1p2), where β = ρ0-2c0-4 [2 + (d2p/d ρ2)ρ0] and the pressure of each of the two primary waves may be expressed as p1 = a1 cos ψ1, and p2 = a2 cos ψ2, where the a's and the eikonals, the ψ's, depend on the four-radius vector xi (refer to 'eikonal' in the index of any late edition of Landau and Lifshitz, The Classical Theory of Fields). The restriction to plane primary waves implicit in my derivation of Q [J. Acoust. Soc. Am. 35, 535-537 (1963)] was later removed by me [Proc. Univ. of Texas Nonlinear Acoustics Symposium, edited by T. G. Muir, pp. 167-181 (1970)]. The present paper reports a function R whose d'Alembertian equals Q when the region of interaction common to both primary waves resides many primary wavelengths from each of the two sources. It turns out that R = 12β(∂/∂t) [a1a2[(∂ψ1/∂xi)(∂ψ2/∂xi)]-1sin ψ1 sin ψ2] in which (∂ψ1/∂xi)(∂ψ2/∂xi) means ∇ψ1·∇ψ2 - c0-2ψ1ψ2. Thus ∇2R - c0-2 R = Q plus terms that are negligible if the primaries are not collinear since these terms are of order E-1 and E-2 smaller than Q where E = ψ + iωt is the space-only dependent eikonal. E1 and E2 have magnitudes of the order r1 λ1-1 and r2λ2-1, respectively, where r1 and r2 are the distances from the interaction region to the primary sources of waves having dominant wave-lengths λ1 and λ2, respectively. The conclusion is that no scattering occurs outside the interaction region of two waves, neither one of which needs to be strictly plane, nor for that matter steady state. [ABSTRACT FROM AUTHOR]

A recent claim [D. H. Trivett and P. H. Rogers, J. Acoust. Soc. Am. 71, 1114-1117 (1982)] that a cw plane wave is scattered by a pulse is subject to the same criticism [P. J. Westervelt, J. Acoust. Soc. Am. 29, 199-203, 934-935 (1957) and M. L. Lighthill, Math. Revs. 19, 915-916 (1958)] leveled at an earlier claim [U. Ingard and D. C. Pridmore-Brown, J. Acoust. Soc. Am. 28, 367-369 (1956)] that scattering occurs from the interaction region of two cw sound beams. The pulse wavefunction employed by Trivett and Rogers does not satisfy the homogeneous wave equation, a fact which gives rise to spurious sources responsible for their nonzero result. The correct result is approximated by Esq. (2t) or Eq. (22) [P. J. Westervelt, J. Acoust. Soc. Am. 59, 760-764 (1976)] multiplied by the longitudinal and transverse taper factors of the pulse. At θ = 0 when the pulse and the cw wave are travelling in the same direction, this solution has an infinite directivity index indicating that the scattered wave never levels the interaction region; this situation, unachievable in practice, should not be called a parametric array and contradicts recent results [D. H. Trivett and P. H. Rogers, J. Acoust. Soc. Am. Suppl. 1 72, S40 (1982)]. About thirty years ago I repeated Ingard and Pridmore-Brown's experiment using their apparatus and convinced myself that nonlinearities in the electrostatic transducer were responsible for their nonzero results. [ABSTRACT FROM AUTHOR]

Introducing the variable π=ρsc20+L+ΛV into Eq. (14) of Westervelt [P. J. Westervelt, 200 (1957)], two equivalent dissipationless wave equations are obtained for the scattered variable ρsc20: Eq. (1) is [Laplacian_variant]2[π+2(ψ2),00]=ΓV,00 and Eq. (2) is [Laplacian_variant]2[π-Λ(ψ2),00]=ΓT,00 in which [Laplacian_variant]2ψ2=L and Γ=-(2+Λ). Outside the region of interaction π=ρsc20 so the source for ρsc20 can be either ΓV,00 or ΓT,00 and since, in general, T≠V, ρsc20 must be zero (except for unidirectional waves when T=V in which case the well-known solution is still restricted to the interaction zone). As an example, consider two colliding waves for which the interaction energies satisfy T12=-V12 requiring ρsc20 from Eq. (1) to be the negative of that from Eq. (2) which demands that ρsc20=0. Within the interaction region the solution to either equation is ρsc20=(Λ-2)([lowercase_phi_synonym]+[lowercase_phi_synonym]-,00+[lowercase_phi_synonym]-[lowercase_phi_synonym]+ 00-2[lowercase_phi_synonym]+,0[lowercase_phi_synonym]-,0), where ψ=[lowercase_phi_synonym]++[lowercase_phi_synonym]- the sum of right and left traveling waves. The scattered pressure ps=ρsc20+ΛV. For scattering to occur outside the interaction region, there must exist losses, as witnessed in the parametric array, or dispersion [see, e.g., P. J. Westervelt,''Scattering of Sound by Sound,'' in Finite Amplitude Wave Effects in Fluids, edited by L. Bjo\rno\, Proc. 1973 Symp. Copenhagen (IPC Science and Technology, London, 1974), or interaction with real sources [P. J. Westervelt, Virtual Sources in the Presence of Real Sources in Nonlinear Acoustics, edited by T. G. Muir (Proc. of Univ. of Texas at Austin Conference, Nov. 1969)]. [ABSTRACT FROM AUTHOR]

The exact second-order transient solution to the interaction of an arbitrary wave with a plane wave is given by Westervelt [P. J. Westervelt, 3320 (1994)]. Let the arbitrary wave be χ(x0-m·r), a plane wave traveling in the m direction. In this case σ=(1-&ncirc;·m)-1χ and the solution to Eckart's equation becomes ρsc20=E12 +(1-&ncirc;·m)-1(Λ+2&ncirc;·m)=cos θ ∇2ψ2, which is identical to Eq. (10) of Westervelt [P. J. Westervelt, 934 (1957)] provided the substitutions ψ2=1/2(w1w2)-1W12 and &ncirc;·m=cos θ are made. It is asserted that this exact solution serves as an approximate solution to the far-field interaction of arbitrary sources. This is done by allowing m and &ncirc; to be space dependent. As an example, the exact solution for the cardioid wave χ=(4πr)-1[&ncirc;·m(G,0+Gr-1)-G ,0] interactions with a plane wave is obtained from σ=-(4πr)-1G,0, where G=G(x0-r) and m=rr-1. In the far field of the cardioid source, σ=(1-&ncirc;·m)-1χ, as in the plane wave-plane wave interaction, thus demonstrating the assertion. [ABSTRACT FROM AUTHOR]

Two arbitrary pulses p2(t-c-10x) and p1(t+c-10x) each of width c0τ pass through one another at x=0, generating a differential surface mass rate source density dσs(t,x)=qs dx where qs=Ad(p1p2)/dt and A= (ρ0c40)-1[2 + ρ0c0-2(d2p/dρ2)ρ0]. The equation for the scattered pressure ps is d2ps/dx2-c-20 d2ps/dt2= -qs and its solution is ps(x',t)=∫dps=(c0/2)∫dσs= (c0/2)∫qs(x,t')dx, t' being the retarded time; thus, (ps)± = (c0/2)A∫±c0τ/2x'[d(p1p2) t=t'±/dt]dx, where + is chosen when x'=x, and t±' = t ± c0-1(x'- x). Suppose x'>=c0τ/2 then x' lies outside the interaction zone, it is found that (ps)-= (c20A/4)(p2p1+p2∫t-∞p1dt)=0 unless p1 has a nonzero average which is possible for plane waves but it is assumed that this is not the case [L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, MA, 1959), p. 267]. In the event x'<-c0τ/2x' again lies outside the interaction zone and (ps)+= (c20A/4)(p1p2+p1∫t-∞p2 dt)= 0. Within the interaction region -c20/2

Starting with Eckart's equation for ρs the scattered density [P. J. Westervelt, J. Acoust. Soc. Am. 29, 934 (1957)], [Laplacian_variant]2ρsc20=[Laplacian_variant]2E12 -∇2(2T12+ΛV12), the variables x0=c0t and ψ,0=-(4ρ0c20)-1/2p are introduced for which [Laplacian_variant]2ψ=0, [Laplacian_variant]2ψ2=T12-V12, and ∇2V12=[Laplacian_variant]2V12+(V12),00 to obtain [Laplacian_variant]2[ρsc20+T12+(Λ-1)V12 +2(ψ2),00]=-2(2+Λ)[(ψ,0)2],00. Next it is assumed that ψ=[lowercase_phi_synonym]+χ, where [lowercase_phi_synonym](x0-n·r) is a plane wave and χ,0=σ,0+n·∇σ, where σ(x0,r) is an arbitrary wave. Terms bilinear in [lowercase_phi_synonym] and χ are retained; thus (ψ,0)2=2χ,0[lowercase_phi_synonym],0, and since ∇[lowercase_phi_synonym]=-n[lowercase_phi_synonym],0, it is found [Laplacian_variant]2(σ[lowercase_phi_synonym])=2∇σ·∇[lowercase_phi_synonym]-2σ,0[lowercase_phi_synonym],0=-2[lowercase_phi_synonym],0 χ,0, leading to the solution of Eckart's equation, ρsc20=(2-Λ)V12-E12 +2[(2+Λ)σ[lowercase_phi_synonym]-ψ2],00, valid within the interaction zone, but vanishing outside where V12=E12=σ=[lowercase_phi_synonym]=ψ=0. The feasibility of making optical measurements of ρs is being investigated. [ABSTRACT FROM AUTHOR]

Following a suggestion by Professor M. Korman, the transient solution of an undamped oscillator as a simple analogy to ρs=-W being the solution of [Laplacian_variant]2(ρs+W)=0 has been used. Berntson et al. [J. Acoust. Soc. Am. 86, 1968 (1989)] criticized the solution stating that solutions to the homogeneous equation need to be appended in order to satisfy boundary conditions. This is not the case here. Say the forcing function F(t) for the oscillator mx+kx=F(t) is derivable from F(t)=mG(t)+kG(t) then x(t)=G(t). As an example, if the arbitrary transient starts at t0 and ends T later then G(t)=[U(t-t0)-U(t-t0-T)]H(t) and F(t)= m[δ(t-t0)-δ(t-t0-T)]H(t)+2m[δ(t-t0) -δ(t-t0-T)]H(t)+[U(t-t0)-U(t-t0-T)] [mH(t)+kH(t)]. Let us obtain the solution for the case H(t)=b, a constant. In this case F(t)=f1(t)+f2(t), where f1(t)=bk[U(t-t0)-U(t-t0-T)] and f2(t)=bm[δ(t-t0) -δ(t-t0-T)]. f1(t) is a square wave and the response to it is x1(t)=bU(t-t0)(1-cos ω0t) for t0=t0+T. In f2 the δ generate displacement boundary conditions and x2(t)=bU(t-t0)cos ω0(t-t0) for t0=t0+T. The complete solution x=x1+x2 is x(t)=0 for t=t0+T confirming the fact that x(t)=G(t). This one-dimensional analogy confirms the fact that there is no response outside the interaction region t0

On page 1982 of their article, Berntson et al. [J. Acoust. Soc. Am. 86, 1968 (1989)] criticized virtually all experimental results on the scattering of sound by sound. In addition they gave their necessary conditions for this scattering to be observed. These conditions were fulfilled in a recent experiment [R. Roy and J. Wu, in Proceedings of the 13th ISNA, edited by H. Hobæk (Elsevier Science, London, 1993)] where no scattered sound was obtained in spite of the fact that the minimum detectable signal level was of the order of 50 dB below Ingard and Pridmore-Brown's theoretical predictions. The one-dimensional treatment given in the previous abstract to two space-time dimensions will be discussed further. [ABSTRACT FROM AUTHOR]

The bilinear interaction of two primary waves p1 and p2 generates a scattered wave ps satisfying [Laplacian_variant]2ps=-qs=-A(p1p2+p1p2), where A= (ρ0c04)-1[2+ρ0c0-2 (d2p/dρ2)ρ0]. Here, p1=pd+pm was chosen in which pd=-∇·[(4πr)-1ιG(t-c0-1r)] and pm=-(4πc0r)-1G(t-c0-1r). This combination of dipole and monopole sources located at r=0 has a far-field cardioid pattern with a null in the ι direction. Also chosen was p2= c0U(ι·r+r0)σ[t-c0-1(ι·r+r0)], a plane wave originating at r=-r0ι and traveling in the ι direction. G(t) and σ(t) may each have arbitrary time dependence, however, in order to avoid the bilinear interaction of primary waves with their sources, the sources were activated at t=0 and terminated at 0@xf k (Elsevier Science, London, 1993)] this theory is easily extended to apply to transducers of arbitrary configuration. [ABSTRACT FROM AUTHOR]

A general operator solution to the nonlinear acoustic wave equation for a viscous medium, V21 + ρ0-1c0-2η∂/∂t - c0-2∂2/∂t2P = Q = -βρ0-1c0-9∂2/∂t2 (P2) is obtained from perturbation series, P = N=1∞ εNP(N); Q = N=2∞ εNQ(N). The exact solution for P(N) ∼ 0(εN) is shown to be P(N) = 1 - exp∂/∂t|c0∇|-11 - 1/2ηρ0-1c0-2∂/∂t - 1r·∇ × exp[-Re(r·∇)]∇21 + ρ0-1c0-2η∂/∂t - c0-2∂2/∂t2-1·Q(N). This solution is a formal extension of Westervelt's 1972 nonsingular operator solution [J. Acoust. Soc. Am. 59, 760 (1976)] to the case of a medium with viscosity. The evaluation of the operator expression above is shown to be straightforward, provided the primary field P(1) is first expanded into a Fourier super-position of plane waves. Parametric array applications are discussed. [ABSTRACT FROM AUTHOR]

Recent experiments by L. R. Sulak et al. [J. Acoust. Soc. Am. 61, S580(A) (1977)] demonstrate that an acoustic signal is produced by proton beams deposited in water. If the unmodified theory of the thermoacoustic array were applicable the signal would be expected to vanish at about 4 °C, the temperature for which the static thermal expansion of water vanishes. The signal vanishes at about 5.7 °C [L. R. Sulak et al., Nucl. Instrum. Methods, to be published]. This discrepancy in temperature can be explained by two assumptions which will be justified: In the static case it is assumed that about 10% of the thermal energy resides in Debye-like modes and these modes contribute to the steady-state thermal expansion were the Debye modes to be excited in the transient case they would generate sound by a mechanism equivalent to the parametric array: however, I assume that the Debye modes fail to be excited for short-period processes. Thus the time-dependent thermal expansion is diminished with respect to the static expansion by an amount which shifts the zero expansion to a 2 °C higher temperature. The consequence of this analysis is that the theory of thermoacoustic sources should be modified by subtracting out the parametric source contribution. Experiments should be performed with laser arrays to determine the frequency dependence of the Debye amplitudes. The close connection between thermoacoustic and parametric sources has already been mentioned [P. J. Westervelt, McGraw-Hill Yearbook of Science and Technology (McGraw-Hill, New York, 1977), p. 389]. [This work has been supported by the U.S. Naval Ocean Research and Development Activity.] [ABSTRACT FROM AUTHOR]

The conventional treatment [J. Weber, General Relativity and Gravitational Waves (Interscience, New York, 1961)] of an undamped mass quadrupole oscillator driven by a gravitational wave assumes the excitation force to be that exerted by the wave on the mass. The force exerted by the wave on the spring has heretofore been ignored. I have calculated this latter force Fs driving the spring and discovered that it is related to the force Fm driving the mass; thus, Fm = ω0-2 d2dt2 Fs, in which ω0 is the resonant circular frequency of the oscillator. The total driving force is the sum Fm + Fs. This total force is incapable of exciting the normal mode of the oscillator, thus supporting my earlier conjecture [P. J. Westervelt, Soviet Phys. JETP Letters 4, 225 (1966), Footnote 4] that passive detectors of gravitational waves cannot be realized in principle. Active detectors, i.e., generators of gravitational waves, can measure radiation from another source. [ABSTRACT FROM AUTHOR]

The gravitational intensity radiated by a 100% modulated well-collimated and truncated laser beam has been shown by me [JETP Letters 4, 225 (1966)] to be S = (kc-5)P2E(4πR02)-1 {1 - cos[kmL (1 - cos[lowercase_phi_synonym])]}[(1 - cos[lowercase_phi_synonym])-2sin4[lowercase_phi_synonym]]n. The crucial nature of this expression is the fact that S vanishes in the forward direction as a consequence of the sin4[lowercase_phi_synonym] term characteristic of tensor waves. For vector waves and for scalar waves the sin4[lowercase_phi_synonym] term is replaced by sin2[lowercase_phi_synonym] and by 1, respectively, and then resonant interaction leads to a Rutherford-like peaking in the forward direction which corresponds, in the scalar case, to the parmetric acoustic array. It is clear that gravitational radiation from ultrarelativistic particles will not exhibit the forward peaking characteristic of electromagnetic radiation from ultrarelativistic charged particles which corresponds to the vector case. [This work was supported by the U. S. Navy Office of Naval Research.] [ABSTRACT FROM AUTHOR]

The steady-state response of a rectangular cavity rotating about an axis passing through the center of the floor and ceiling of the cavity is computed under the assumption that one of the walls parallel to the axis of rotation executes sinusoidal pistonlike motion. Losses in the medium and at the walls are included in the analysis. When the floor and ceiling are square, the twofold degeneracy of the lowest two modes of the stationary room is removed by the Coriolis force that couples the two otherwise orthogonal modes. In this case, the splitting is ±8ωR/π2 with ωR being the angular velocity of rotation. The same amount of splitting was found by Lamb in his investigation of tidal oscillations in a rotating rectangular basin [see H. Lamb, Hydrodynamics (Dover Publications, Inc., New York, 1945), p. 330, Eq. 20.] [This work was supported by the U. S. Air Force under contract AF 33(616)-7045 with the Aeronautical Systems Division at Wright-Patterson Air Force Base, Ohio.] [ABSTRACT FROM AUTHOR]

A criterion recently derived by the author predicts the sound-pressure level at which sound begins to increase the rate of heat transfer. The application of this criterion to recent experiments is discussed, and it is concluded that, while theory and experiment are in essential agreement, more experiments covering a wider range of variables are needed in order to test the full range of applicability of the theory. [ABSTRACT FROM AUTHOR]

N. W. McLachlan and A. L. Meyers [Proc. Phys. Soc. (London) 47, 644 (1935)] and K. A. Naugol'nykh [Soviet Phys.-Acoustics 5, 79 (1959)] have obtained solutions valid to second order in the wave amplitude for spherically symmetric waves. We have generalized their results and obtained solutions for nonspherically symmetric waves. If the first order density wave satisfying the linear wave equation is taken to be ρ1 = (A/r)e-i(ωt-kr) × f(θ,φ), we find the second-order correction to the density to be ρ2 = -1/2 iρ0-1kA2/rlnr/r02+c0-2ρ0d2p/dρ2ρ=ρ0×e-2t(ωt-kr)f2(θ,φ). This result is subject to the restrictions that kr>1 and that |∇f|

The observed configuration of sonically induced flow in the vicinity of solid objects may depart significantly from predictions based on first-order perturbation theory whenever the acoustic particle displacement amplitude s exceeds in magnitude the acoustic boundary-layer thickness δAC. This change in the flow pattern seems to be intimately connected with two related physical effects: The observed increase in heat transfer from heated objects subjected to intense sound; the observed increase in drag experienced by an object subjected simultaneously to a steady unidirectional flow and to an intense sound field. An attempt is being made to understand these effects by extending the conventional perturbation theory to higher orders, a formidable task unless analytic solutions are available for the first-order perturbations. In connection with an attempt to find analytic solutions for the streaming in the vicinity of a right-circular cylinder, a formula has been obtained for the streaming boundary layer thickness δDC divided by the cylinder radius a, δDC/a = 1 - 13[square_root_of_2]/3 δAC/a-1/2 -1 + 2 - 13[square_root_of_2]/6δAC/a. (This work has been supported by the Air Research and Development Command, U. S. Air Force.) [ABSTRACT FROM AUTHOR]

As part of a fundamental study on the nonlinear behavior of acoustical materials at high sound intensities, we have examined a fine-wire screen. The theoretical analysis has been based on considering the screen to be an assemblage of noninteracting orifices. On this basis, a formula for the specific nonlinear resistance, valid at low frequencies, has been found which agrees quite well with static flow measurements. Criteria have been established for the onset of nonlinear behavior. These criteria are valid over a wide range of frequencies. (This effort has been supported by Nobs 77075.) [ABSTRACT FROM AUTHOR]

To obtain experimental verification of the expression for an acoustic end-fire array derived by Westervelt a 13.4-Mc crystal, mounted inside a water-filled tank, was driven simultaneously by 13-Mc and 14-Mc transmitters. A small barium titanate probe, free to rotate in the horizontal plane, was used to detect the 1 Mc difference frequency radiated by the interacting carrier beams. This transducer was connected in turn with a sensitive calibrated receiver. The entire system permits a determination of the directivity pattern and the absolute magnitude of the radiated 1-Mc signal. The results of experiments using this arrangement, as well as of trials with both higher and lower difference frequencies, are in good agreement with the theoretical expression mentioned above. [ABSTRACT FROM AUTHOR]

The nonlinear absorption of waves has been calculated by a method which does not require the solution of the nonlinear wave equation. The method appears to be useful for waves of arbitrary configuration. (This work has been supported by the Bio-acoustics Branch of the Wright Air Development Center, U. S. Air Force.) [ABSTRACT FROM AUTHOR]

The power required to maintain the Eckart type of acoustic streaming is calculated and found to be of sufficient magnitude to contribute measureably to the attenuation of large amplitude sound waves. A second problem has also been solved. When sound interacts with objects, streaming occurs due to the vorticity sources localized near the surface of the objects. An energy theorem is proved which is valid for objects small compared with the wavelength of sound and which enables the energy dissipated by the streaming motion to be expressed entirely in terms of the first-order oscillatory motion thus obviating the need for calculating the second-order streaming motion. [Work supported jointly by the U. S. Air Force (Wright Air Development Command) and the Office of Naval Research.] [ABSTRACT FROM AUTHOR]

Pulse jet noise arises from three separate and distinct mechanisms. The pulsating air flow at the inlet and exhaust constitute simple monopole souces whose source strengths can be related to the instantaneous volume flow. The fact that the exhaust flow is directed causes a large periodic flux of momentum to be directed along the axis of the jet giving rise to a dipole or force source. The strength of this force source is clearly related to the thrust of the jet. Finally, the jet, mixing with the surrounding air, will produce noise of the Lighthill variety stemming from the convected quadripole sources. One problem consists in determing the proper admixture of the three fundamental sources necessary to reproduce the measured far-field directivity. Another problem consists in relating the three source strengths to the measured flow and thrust characteristics of the jet. Both of these problems are currently being studied. Preliminary analysis indicates that the fundamental frequency is generated by a monopole and that the higher harmonics contain monopole, dipole, and quadripole radiation. (Work supported by the Wright Air Development Command.) [ABSTRACT FROM AUTHOR]

A substantial simplification in the calculation of forces due to radiation pressure results from the application of the concepts of wave momentum [P. J. Westervelt, J. Acoust. Soc. Am. 23, 312 (1951); J. S. Pyett, ibid. 28, 156 (1956)]. It will be shown that a similar simplification occurs in calculating the torque on an object in a sound field in terms of the flux of angular momentum crossing a closed surface far removed from the scattering object. [This work has been supported by the U. S. Air Force, Wright Air Development Center.] [ABSTRACT FROM AUTHOR]

The evaluation of the stream function describing the acoustic streaming in the vicinity of a sphere has been completed. This stream function was described at the last meeting [R. W. Thrasher and P. J. Westervelt, J. Acoust. Soc. Am. 28, 155 (1956)], and it contained two definite integrals of the exponential type. These integrals have been evaluated on the Brown University I.B.M. C.P.C. computer with an accuracy adequate to enable a comparison of our results with those worked out for the cylinder [S. Skavlem and S. Tjötta, J. Acoust. Soc. Am. 27, 27 (1955)]. Steady flow velocity profiles and boundary layer thickness curves will be presented. The application of this work in connection with determining the interaction force between two spherical particles in a sound field will be discussed. Heretofore the influence of streaming on the two-body interaction force in viscous fluids has been ignored. [This work has been supported by the U. S. Air Force, Wright Air Development Center, and the National Science Foundation.] [ABSTRACT FROM AUTHOR]

The force owing to radiation pressure on an object of any shape and having an arbitrary normal boundary impedance can be expressed in terms of the asymptotic scattering functions for the object [J. Acoust. Soc. Am. 23, 312 (1951)]. It will be shown that this specification of the radiation force is valid for any obstacle irrespective of whether it is possible to specify a normal boundary impedance. Two typical obstacles having nonlocal boundary conditions are a metal disk and a lossy bubble. A simple relation between the radiation pressure in a plane standing acoustic wave and the acoustic impedance will be given. Equations for the generation of acoustic streaming have been derived in terms of Brillouin's radiation stress tensor [J. Acoust. Soc. Am. 25, 60 (1953)]. It is emphasized that these equations are valid for any nonreacting fluid irrespective of the loss mechanisms involved. For the special case of a barotropic fluid, W. P. Raney has shown that these equations reduce to those obtained by Medwin and Rudnick [J. Acoust. Soc. Am. 25, 538 (1953)]. [This work has been supported jointly by the Office of Naval Research and the U. S. Air Force, Wright Air Development Center.] [ABSTRACT FROM AUTHOR]