Question

Problem 10. It can be shown that a wave in water of depth h has dispersion relation dº = gk tanh (kh), where g is the gravitational acceleration. 1. In shallow water (kh < 1), show that the phase and group velocities are identical, and Op = vg = Vgh. 2. In deep water (kh > 1), show that the group velocity is half the phase velocity. Hint: work with approximate formulas for tanh (x) at x < 1 and x > 1.

Answer 1

W z gk tanh (kh) o aw dla - z tanklag) + 9kh sech čah) Kama ok n[ tach (kk) + kh sech ke? ] PW gtanksh + kh seeich) 2V gk tanh (kh) emek akh te зу • ev4 - e-uk ekhte-kh ikht. Ta kh... = lig 1+Xhen- +- Kh tr.. (1+kk+ -- +1-kh) 4kh to nx+kha... 1+kk tn..+1- Х

w z gk tankh . th zgk. sk. lik kh + Tikh ....) en Cienhon. 1- who = gk. Kh - shk phone relocity is Up z w z ſ sh Graperetary & Vg z V x = V sh

. 2. In depth water khine, tanh kh 1 v 23k 如歌 | 2-7 | s蹤 3 ok