$1.$ An "ideal" waveguide is given with a soft top $\backslash (($p$(0)=0)\backslash )$ and hard bottom $\backslash ((\backslash$mathrm$\{$dp$\}/\backslash$mathrm$\{$dz$\}(\backslash$mathrm$\{$D$\})=0)\backslash )$ of depth $\backslash (\backslash$mathrm$\{$D$\}=40\backslash$mathrm$\{$~m$\}\backslash )$ with a sound speed of $\backslash (1500\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ and density of $\backslash (1025\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ Assume no absorption in the seawater. A CW source of unit strength of frequency $100$ Hz is at a depth of $22\mathrm{.}8$ m $.$

a$.$ What is the eigenvalue $($horizontal wavenumber$)$ of the first mode? What is the vertical wavenumber of the first mode? Please provide the answers with six significant digits.

$(5$ points$)$

b$.$ What is the mode number of the first evanescent mode and what is its eigenvalue $($horizontal wavenumber$)?$

$(5$ points$)$

c$.$ Which of the propagating modes $($ones with real eigenvalues$)$ has the least energy? $(5$ points$)$

d$.$ What is the modal grazing angle of the first mode?

$(5$ points$)$

e$.$ Sketch the complex wavenumber $\backslash (\backslash$mathrm$\{$k$\}\_\{\backslash$mathrm$\{$r$\}\}\backslash )$ plane with the locations of the first ten modal eigenvalues.

$(5$ points$)$