. The collected papers of Sir Thomas Havelock on hydrodynamics. Ship resistance; Water waves; Hydrodynamics. Cases of Three-dimensional Fluid Motion. 361 The terms R]2 and E21 represent the interference effects. Let Bi and B2 be the centres of the two systems pi and p2. To calculate E21, consider a constituent plane wave-front through Bi; take this line as a new axis O'y', and a perpendicular line through B2 for the axis O'x', as in fig. JT Then, corresponding to the expression (15) in § 4, we have as an element of E21 the quantity ^ J-« -^ J-»{(a;'-l-2^cos(^)2 + 2/'2+/^}3/2' ^ ' The simil
. The collected papers of Sir Thomas Havelock on hydrodynamics. Ship resistance; Water waves; Hydrodynamics. Cases of Three-dimensional Fluid Motion. 361 The terms R]2 and E21 represent the interference effects. Let Bi and B2 be the centres of the two systems pi and p2. To calculate E21, consider a constituent plane wave-front through Bi; take this line as a new axis O'y', and a perpendicular line through B2 for the axis O'x', as in fig. JT Then, corresponding to the expression (15) in § 4, we have as an element of E21 the quantity ^ J-« -^ J-»{(a;'-l-2^cos(^)2 + 2/'2+/^}3/2' ^ ' The similar element in the value of E12 is the expression (30) when we have replaced a;' + 2Acos(/) by ,r' —2Acos^. Adding the two elements and carrying out the integration with respect to y', we have, as an element of E12 + E21, "^ ^ ]-^{x'+ 2h con ^f+f = SttWA^ sec^ ^(;-2«»/sec=* pos (2Koh sec <^). (31) Eeplacing from (11) the proper factor and integrating with respect to (f), we have E12 + E12 = (87r/5rp)«;o=*AM sec5^e-2«(2«o7i-see) d^. (33) We can express E in series of known functions by expanding cos(2«oA sec <j)) either in powers of Kah, or in Bessel functions J„(2/t they are of no use for numerical calculations. It is not difficult to calculate numerical values directly from the integral (33) for given values of the constants. To obtain a graph showing the I See note by Editor on page 150. 153. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Havelock, Thomas, Sir, 1877-. Washington, Office of Naval Research, Dept. of the Navy; for sale by the Superintendent of Documents, U. S. Govt. Print. Off
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