Question

4.1 For a monochromatic (a single wavelength) traveling wave on a deep layer of fluid, the potential and the elevation are given by $(x,t) = -aceky cos k(x – ct), n(x,t) = a sin k(x - ct). The kinetic and potential energies contained in one wavelength are KE=[[[@)+C)] |d(x– ci)dy and PE = 3P8 /*r*d(x–ce). Show that the two energies are equal. This is called the principle of equi-partition of energy.

Answer 1

Ø (xit) = -acety cost(x-ct). n (x,t) = a sink (x-ct) ****lleder 16-02) 90.- aze"36-sina 62-6). akceky (sink (x-ct)) (2 axe cany (sin? a (x-ce) o ackeky cosk (x-ct) ag acek zky cos 2 k(x-Ct). ad 29 6 + = a ² k?ce sky CO) Cena K.E IS dlx-ct dy 12 a ²k?c? /e exy 2k - Relate ()

g=ke from dimensional analysis + Sl kale 4 P.E = 2 sg * 2 a sinek (x-et) d (x-ct) x-ct - е d(x-ct) de a sinakl dl. PE=359 deton's so Cosao = 1-2 sino 2 sino sino 1- cos 20 P.E = 2 gg a? i- cosak l de. 2 esg als - sga? [e- sinsel] K = 25/2 => akl= 2x25 l = 25 l sino + > P. E = 7 7 9ga? [2_singh ак Iggan? P.E = KE
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