In the case of incompressible uids the density is

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Unformatted text preview: ouble Fourier sine series of h. As in the case of ordinary Fourier sine series, the bmn ’s can be determined by an explicit formula. To determine it, we multiply both sides of (5.12) by (2/a) sin(pπx/a), where p is a positive integer, and integrate with respect to x to obtain a 2 a h(x, y ) sin(pπx/a)dx 0 ∞ = bmn m,n=1 a 2 a sin(pπx/a) sin(mπx/a)dx sin(nπy/b). 0 The expression within brackets is one if p = m and otherwise zero, so ∞ a 2 a h(x, y ) sin(pπx/a)dx = 0 bpn sin(ny/b). n=1 Next we multiply by (2/b) sin(qπy/b), where q is a positive integer, and integrate with respect to y to obtain 22 ab a b h(x, y ) sin(pπx/a) sin(qπy/b)dxdy 0 0 ∞ = bpn n=1 2 b b sin(nπy/b) sin(qπy/b)dy . 0 The expression within brackets is one if q = n and otherwise zero, so we finally obtain bpq = 22 ab a b h(x, y ) sin(pπx/a) sin(qπy/b)dxdy. 0 0 Suppose, for example, that c = 1, a = b = π and h(x, y ) = sin x sin y + 3 sin 2x sin y + 7 sin 3x sin 2y. 126 (5.13) In this case, we do not need to carry out the integration indicated in (5.13) because compar...
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This document was uploaded on 01/12/2014.

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