Archives of Acoustics,

**31**, 1, pp. 57-76, 2006### The saddle point method applied to selected problems of acoustics

The paper is aimed at the presentation of examples of applying the saddle point approximation method to some fundamental problems of acoustics and the discussion of some interesting couplings between applied mathematical methods and the physical interpretation of the results.

The saddle point method is shortly reminded in its basic form, from which more advanced versions, improving the results and widening the range of possible applications, are derived. Two problems, solved at first for electromagnetic waves and then applied to acoustics by means of ``the analogous method'', have been chosen as examples. The first one is the phenomenon of the reflection of a spherical wave at a plane interface between two media, the lower of which is characterised by a higher velocity (water/sand, air/water). In this case there is a critical angle above which a total reflection and a lateral wave occur. The second example is the far field radiated from the outlet of a semi-infinite circular duct. The physical insight into the understanding of the physical phenomena provided by the saddle point method is stressed.

The saddle point method is shortly reminded in its basic form, from which more advanced versions, improving the results and widening the range of possible applications, are derived. Two problems, solved at first for electromagnetic waves and then applied to acoustics by means of ``the analogous method'', have been chosen as examples. The first one is the phenomenon of the reflection of a spherical wave at a plane interface between two media, the lower of which is characterised by a higher velocity (water/sand, air/water). In this case there is a critical angle above which a total reflection and a lateral wave occur. The second example is the far field radiated from the outlet of a semi-infinite circular duct. The physical insight into the understanding of the physical phenomena provided by the saddle point method is stressed.

**Keywords**: saddle point method, waves propagation in layered media, lateral wave, radiation from cylindrical duct

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