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General Integral Equation for the With-Height Reproduction of a Focused Source Inside

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A general integral formula to reproduce sound field from a virtual sound source located inside of secondary sources is proposed. The proposed formula extends the theory on the focused sound source of Wave Field Synthesis, for the reproduction of three-dimensional sound field. To resolve the non-existence problem involved with the reproduction of a source inside, an alternative sound field satisfying homogeneous wave equation is derived. The Kirchhoff-Helmholtz integral is reformulated in such a way that the alternative field is reproduced in terms of the secondary sources distributed on a surface. Then the general equation is reduced to simpler forms using various approximations, such as single layer formula reproducing the sound field only by monopole sound sources.

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