Nonhomogeneous Wave Equation Research Articles

One-Way Waves

One representation of the solution of the one-dimensional nonhomogeneous wave equation $u_{tt} - c^2 u_{xx} = g(x,t)$, with $u = u_t = 0$ at $t = 0$, is $u = ({1/2}c)\iint_\Delta gdxdt$, where the integral is taken over a certain triangle in the $x - t$ plane. This form of solution, known as the d’Alembert solution, is illustrated by an example of a $g(x, t)$ which produces a wave of finite length traveling only in one direction. For more general $g(x, t)$ it is shown how to decompose $g(x, t)$ into four components each of which produces a component wave with a simple character.

The moving load on a string as free boundary problem

A free boundary problem for the nonhomogeneous wave equation is studied. Such a problem arises when the motion of a load on a string is analyzed without supposing that the load velocity is known and fixed. Preliminarily the complementary equations which characterize the free boundary are determined. The mechanical problem is then solved by proving a uniqueness and existence theorem.

A Theoretical Prediction of the Acoustic Pressure Generated by Turbulence-Flame Front Interactions

The equations of momentum and continuity are combined and linearized yielding the one-dimensional nonhomogeneous acoustic wave equation. Three terms in the nonhomogeneous equation act as acoustic sources and are taken to be forcing functions acting on the homogeneous wave equation. The three source terms are: fluctuating entropy, turbulence gradients, and turbulence-flame interactions. Each source term is discussed. The turbulence-flame interaction source is used as the basis for computing the source acoustic pressure from the Fourier transformed wave equation. Pressure fluctuations created in turbopump gas generators and turbines may act as a forcing function for turbine and propellant tube vibrations in earth-to-orbit space propulsion systems and could reduce their life expectancy. A preliminary assessment of the acoustic pressure fluctuations in such system is presented.

Asymptotic determination of the formation process of nonlinear distortion of one-dimensional pulses in a layered medium: PMM vol.40, n≗ 6, 1976, pp. 1093–1103

Nonlinear effects in the propagation, reflection, and refraction of one-dimensional pulses in a medium consisting of two layers lying on a half-space are considered and analyzed. Properties of layers and of the half-space are different, and stresses are defined by an expansion in powers of strains. The initial pulse of finite duration is specified in the form of boundary condition at the surface of the external layer either for the deformation or for the dislocation rate, and the problem of wave pattern when the initial pulse amplitude tends to zero,i.e. in the case of small nonlinear effects, is solved. Problem is solved by the method of successive integration of nonhomogeneous linear wave equations, in which the solution of the linear problem is taken as the first approximation and the subsequent approximations are derived by approximating the nonlinear terms with the use of the preceding approximation.

ASPECTS OF THE DYNAMICS OF A RUBBERLIKE MATERIAL

The first part of this paper, consisting of sections 1 and 2, presents a general analysis of continuous isentropic motions of a particular class of isotropic elastic solids. The materials in question have been shown to simulate the essential thermo-mechanical properties of solid polymers exhibiting rubberlike behaviour and, in the present context, their most important characteristic is near-incompressibility. The actual motion considered is accordingly viewed as an isochoric motion, generally involving finite deformations, on which is superimposed a dilatational disturbance of small amplitude. The main objectives of the analysis are a rationally conceived definition of the isochoric motion and an approximation procedure enabling the dilatational motion to be calculated to leading order, and for these purposes the constitution of the moving body is fully specified by a single response function representing the influence of the polymer network on the thermoelastic behaviour of the material. The isochoric motion is characterized by a strain-energy function formed by appropriately specializing the network response function, and the perturbation displacement is shown to be governed by a non-homogeneous wave equation in which the source term is derived from the isochoric motion. In the second part of the paper (sections 3 and 4) the general theory is applied to spherically symmetric motions of an infinite body excited by the application of a uniform time-dependent pressure to the surface of a cavity. The isochoric approximant to the actual motion is studied in section 3 and detailed results are worked out for an empirical network response function which, in a variety of situations, has been found to yield theoretical predictions in good agreement with experimental findings. The perturbation wave field is determined in section 4. A discussion of its properties shows that the combined motion conforms to an expected pattern and that certain peculiarities associated with the isochoric motion are annulled by superposition of the dilatational disturbance.

Nonexistence of Global Solutions to the Nonhomogeneous Wave Equation, Regardless of Boundary Conditions

The fact that the ordinary derivative y′ is known to be a Darboux function implies that the right-hand side of the differential equation y′ = f(x, y) must also be such a function. Since it is now known that the mixed hyperbolic derivative uxy is also a Darboux function, this implies that under certain mild conditions the equation uxx − utt = f may not have a classical global solution unless f has the proper Darboux structure. As with the ordinary differential equation, this nonexistence result does not depend on boundary conditions.

Spherical wave propagation in elastic-plastic work-hardening materials

Equations governing the propagation of finite amplitude waves with polar symmetry through an isotropic elastic-plastic work-hardening material are formulated. Formal characteristics solutions for elastically and plastically deforming regions are given, and the respective validity conditions obtained. Matching conditions across a moving interface between elastic and plastic regions at which the normal stress is continuous are determined from the balance laws and constitutive response. Analogous to the situation for waves in uni-axial strain, six types of interaction can occur corresponding to interface propagation speeds in the six ranges limited by the elastic and plastic wave propagation speeds. Given the initial motion, the required number of interface conditions to determine the interface path and wave motion in the two regions is obtained for each type. The role of vanishing plastic work rate is shown precisely. Linearized equations for infinitesimal deformation and a parabolic work-hardening law are recovered, and give rise to displacement potentials which satisfy a non-homogeneous wave equation in both regions. Global solutions can then be formally expressed in terms of wave functions.