(Solved): 3) The dispersion equation for linear waves has the expression below 2=gktanh(kh) where =2/Tp ...

3) The dispersion equation for linear waves has the expression below ${?}^2=gk\tanh (kh)$ where $?=2?/T_p$ is angular frequency, $T_p$ is wave period, $k=2?/L$ is wave number, $L$ is wave length, and $h$ is water depth. Note that wave (form) travel speed $c=$ $L/T_p=?/k$. Two approximation methods to solve $k$ (given frequency $?$ and depth $h$ ) by Guo (2002) and Beji (2013) are given below. (a) Guo's (2002) approximation $k=k_o/{\left[1?1/\exp \left(x^{1.25}\right)\right]}^{0.4},\text{ where }x=k_{o}h$ (b) Beji's (2013) approximation $k=k_o\left[1+\frac{{\left(k_{o}h\right)}^{1.3}}{\exp \left(1.1+2k_{o}h\right)}\right]/\sqrt{\tanh \left(k_{o}h\right)}$ (i) Given $T_p=5\sec ,h=10\mathrm{m}$, use MATLAB to calculate $L$ using Methods (a) and (b). The true $L$ (based Newton-Raphson Method) is $?36.563\mathrm{m}$. Is the estimate from (a) better than (b)? (ii) Given $T_p=5\sec ,h=3\mathrm{m}$, calculate $L$ using Methods (a) and (b). The true $L$ (based Newton-Raphson Method) is $?24.917\mathrm{m}$. Is the estimate from (a) better than $(b)$ ? (iii) Based on the linear wave dispersion equation, wave (form) travel speed can be expressed as $c=c_o\tanh (kh)$, where $c_o=g{T}_p/2?$. Does the wave in (i) travel faster than the wave in (ii)?