This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1895 edition. Excerpt: …reader will have no difficulty in examining these points, by means of the general value of Q given in the Appendix. Wave-Motion in a Canal of Variable Section. 181. When the section (8, say) of the canal is not uniform, but varies gradually from point to point, the equation of continuity is, as in Art. 166 (iv), o–liTM where b denotes the breadth at the surface. If h denote the mean depth over the width b, we have S = bh, and therefore–iaB (2). where h, b are now functions of x. The dynamical equation has the same form as before, viz. (3) Between (2) and (3) we may eliminate either 17 or f; the equation in t) is S-f(S The laws of propagation of waves in a rectangular canal of gradually varying section were investigated by Green. His results, freed from the restriction to a special form of section, may be obtained as follows. If we introduce a variable 6 defined by dx/d0=(gh) (i), in place of x, the equation (4) transforms into where the accents denote differentiations with respect to 8. If 6 and h were constants, the equation would be satisfied by t)=F(6-t), as in Art. 167; in the present case we assume, for trial, n = e.F(6-t) (iii), where e is a function of 6 only. Substituting in (ii), we find e’ F’ e fb’ 1 h! (F’ eA…. 2e-F+e+(b+2h)F + e)=0(lv) The terms of this which involve F will cancel provided
e’. V. 1 A’ C being a constant. Hence, provided the remaining terms in (iv) may be neglected, the equation (i) will be satisfied by (iii) and (v). The above approximation is justified, provided we can neglect e /©‘ and e’/e in comparison with F’/F. As regards &/&, it appears from (v) and (iii) that this is equivalent to neglecting 6_1. dbjdx and h'1. dhjdx in comparison with Ij-1….
