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Mapped wave-envelope elements of variable radial order are presented for the computation of time-harmonic, unbounded, three-dimensional acoustical fields. Their application to transient problems is described in a companion article (Part II). Accuracy is assessed by a comparison of computed and analytic solutions for multi-pole fields generated by a vibrating sphere. Solutions are also presented for plane wave scattering. Elements of radial order m+l are shown to be capable of modeling multi-pole components of order m, although the provision of adequate transverse resolution is shown to be a stringent requirement, particularly at high frequencies. Ill-conditioning of the coefficient matrix limits the practical implementation of the method to elements of radial order eleven or less. The utility of the method for more general geometries is demonstrated by the presentation of computed solutions for the sound field generated by the vibration of a cylindrical piston in a plane baffle and of an idealised engine casing under anechoic conditions. The computed results are shown to be in close agreement with the analytic solution in the case of the cylindrical piston, and with a boundary element solution in the case of the engine casing.

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A variable-order, infinite “wave-envelope” element scheme is formulated for transient, unbounded acoustical problems. The transient formulation which is local in space and time is obtained by applying an inverse Fourier transformation to a time-harmonic wave-envelope model whose formulation is described in a companion article. This procedure yields a coupled system of second-order differential equations which can be integrated in time to yield transient pressure histories at discrete nodal points. Far-field transient pressures can also be obtained at adjusted times. The method can be applied quite generally to two-dimensional and three-dimensional problems and is compatible with a conventional finite element model in the near field. The utility of the method is confirmed by the presentation of transient solutions for axisymmetric and fully three-dimensional test problems. An implicit time integration scheme is used and computed results are compared to analytic solutions and to solutions obtained from alternative numerical schemes. Close correspondence is demonstrated and the scheme is shown to be stable for the problems which are presented. CPU times for a large three-dimensional problem are shown to compare favorably with those required for an equivalent transient boundary element computation.

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