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note
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In the mathematical literature such eigenfunction expansions have been shown to exist for sound speed profiles which are asymptotically constant (for example in Ref. 13). To this author's knowledge the extension to sound speed profiles which are asymptotically decreasing (indefinitely) has not been considered. A proof of the existence of such an extension will not be attempted here. One may view the expansion used here either as a hypothetical extension of the asymptotically constant case or as an approximate form for the case in which the sound speed profile eventually becomes constant, but at a height well above the domain of interest, and at a value well below the sound speed on the ground.
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