Separable Coordinate Systems Research Articles

Solid Torsional Horns

The propagation of torsional waves in tapered solid elastic rods is examined both theoretically and experimentally from the viewpoint of acoustic horn theory. Such tapered rods in torsional vibration are dubbed torsional horns. Two differential equations are derived that describe the propagation of torsional waves. One of these is an “exact” wave equation that can be readily solved only when the horn boundaries fit a separable coordinate system. The other is an approximate wave equation based on the assumption that the wavefronts are plane cross sections of the horn. This equation is very similar to Webster's plane-wave equation for compressional waves in an acoustic horn. Experimentally determined standing-wave patterns and resonance frequencies of torsional horns are compared with the solutions of the two wave equations for selected horn contours. A quantitative estimate of the error introduced by the plane-wave approximation is obtained for the exponential horn.

An Integral Transform Associated with Boundary Conditions Containing an Eigenvalue Parameter

It has long been known that certain integral transforms and Fourier-type series can be used to solve many classical boundary and initial value problems in separable coordinate systems. More recently, it has been shown that these classical transforms and series are spectral representations associated with an ordinary differential system which results on applying separation of variables to the given boundary value problem. This has been the basis for recent work concerned with systematically generating the proper spectral representation needed to solve a given problem. See [1]-[4] for a list of references. We shall consider the problem of finding the associated spectral representation when the resulting ordinary differential system has the eigenvalue parameter occurring in both the equation and one boundary condition. Moreover, the differential equation is to be satisfied on a semi-infinite interval, thus leading to a singular problem which does not seem to have been studied before. In §2 by using a transformation due to B. Friedman (which we modify appropriately for our singular case), we first give a formal derivation of the spectral representation, and then we rigorously prove the result. In §3 our representation is applied to solve an initial-boundary value problem arising in the theory of diffusion and heat flow in one dimension. We should bear in mind that even in cases where solutions are already known, our method systematically yields alternative representations which are often more rapidly convergent and from which asymptotic expansions of solutions with respect to parameters can often be found. In the problem to be considered a representation of the solution can also be found easily by a straight-forward application of the Laplace transform in time t, while the new transform derived in §2 yields the solution when applied in space x. The new transform, however, can be applied when the coefficients in the boundary value problem are time dependent, a situation which, in general, precludes the use of the Laplace transform.. As a general rule [1] there should be a spectral representation associated with each ordinary differential system resulting from applying separation of variables to the original boundary value problem, and each spectral representation should lead to a different representation of the solution of the original problem.

Scattering of a plane scalar wave by fundamental surfaces with mixed boundary conditions

The problem of scattering of an arbitrarily directed scalar incident plane wave by a fundamental surfacE, of orthogonality with mixed linear boundary conditions is solved for arbitrary separable coordinate system. From the Fredholm integral equation of the second kind, derived from the Helmholtz solution, a simple linear relationship is found between the unknown coefficients in the expansions of the wave potential function and its normal derivative on the fundamental surface of orthogonality, in terms of the eigenfunctions of the separable coordinate system. By applying the mixed linear boundary conditions on the fundamental surface of orthogonality another set of simple linear relationships between the unknown coefficients is found. From the two sets of linear equations, the unknown coefficients may be determined and the scattered field calculated. The above method may be applied to both Laplace and Helmholtz equations. Since we require for the solution only the free space Green’s function expansion, the problem of finding explicit representations of the various specialized Green’s functions is avoided.