We have found the parametric equations for the circular raypath that starts at the origin and makes an angle ${\theta }_{0}$ with the positive yaxis (Figure 13). This raypath represents a diving wave that reaches a maximum depth at point $D{\rm {=}}\left(x_{D}{\rm {,\ }}y_{D}\right)$ and then returns to the surface. The tangent to the raypath is horizontal at D. Thus, the angle from the vertical is ${\rm {9}}0^{\rm {o}}$, that is, $\theta {\rm {=}}{\theta }_{D}{\rm {=}}\pi {\rm {/2}}$ at the depth point. Inserting ${\rm {\ cos\ }}\ \theta {\ =\ cos\ }{\theta }_{D}{\ =\ 0}$ and ${\rm {\ sin\ }}\theta {\ =\ sin\ }{\theta }_{D}{\ =}{\ 1}$ in the above parametric equations, we obtain the coordinates of the depth point as
${\begin{aligned}x_{D}&{\rm {=}}{\frac {v_{0}}{a{\rm {\ tan\ }}{\theta }_{0}}}\end{aligned}}$
${\begin{aligned}y_{D}&{\rm {=}}{\frac {v_{0}}{a{\rm {\ sin\ }}{\theta }_{0}}}{\frac {v_{0}}{a}}{\rm {=}}{\frac {v_{0}}{a}}\left({\frac {\rm {l}}{{\rm {\ sin\ }}{\theta }_{0}}}{\rm {1}}\right).\end{aligned}}$


(56)

Let $t_{D}$ denote the traveltime along the raypath from the origin to the point of maximum depth. At the point of maximum depth, the angle is $\theta {\rm {=}}{\theta }_{D}{\rm {=}}\pi {\rm {/2}}$. Thus, the implicit equation for the traveltime $t_{D}$ is
${\begin{aligned}e^{at_{D}}&{\rm {=}}{\frac {{\rm {\ tan\ }}{\frac {{\theta }_{D}}{\rm {2}}}}{{\rm {\ tan\ }}{\frac {{\theta }_{0}}{\rm {2}}}}}{\rm {=}}{\frac {{\rm {\ tan\ }}{\frac {\pi }{\rm {4}}}}{{\rm {\ tan\ }}{\frac {{\theta }_{0}}{\rm {2}}}}}{\rm {=}}{\frac {\rm {l}}{{\rm {\ tan\ }}{\frac {{\theta }_{0}}{\rm {2}}}}}{\rm {=}}{\frac {\rm {l}}{\frac {{\rm {1}}{\rm {\ cos\ }}{\theta }_{0}}{{\rm {\ sin\ }}{\theta }_{0}}}}{\rm {=}}{\frac {{\rm {\ sin\ }}{\theta }_{0}}{{\rm {l}}{\rm {\ cos\ }}{\theta }_{0}}},\end{aligned}}$


(57)

Figure 13. A diving wave from origin
O to maximum depth
D and back to the horizontal axis. It is called a diving wave because it plunges into the ground and comes up to the surface, not by reflection but by making a Uturn. Because depth is plotted upward, the diving wave appears inverted. The wavefront though the maximum depth point is also shown.
which also can be written as
${\begin{aligned}e^{at_{D}}&{\rm {=\ tan\ }}{\frac {{\theta }_{0}}{\rm {2}}}{\rm {=}}{\frac {{\rm {l}}{\rm {\ cos\ }}{\theta }_{0}}{{\rm {\ sin\ }}{\theta }_{0}}}.\end{aligned}}$


(58)

These implicit equations involve exponentials. There are two common ways to get rid of exponentials: by logarithms and by hyperbolic functions. We choose the latter. The hyperbolic cosine is defined as ${\rm {\ cosh\ }}u{\rm {=}}\left(e^{\mathbf {u} }{\rm {+}}e^{\iota \iota }\right){\rm {/2}}$. Similarly, the hyperbolic sine is defined as ${\rm {\ sinh\ }}u{\rm {=}}\left(e^{\mathbf {u} }e^{\mathbf {u} }\right){\rm {/2}}$. We thus have
${\begin{aligned}{\rm {\ cosh\ }}at_{D}{\rm {=}}{\frac {e^{at_{D}}{\rm {+}}e^{at_{D}}}{\rm {2}}}{\rm {=}}{\frac {{\frac {{\rm {\ sin\ }}{\theta }_{0}}{{\rm {1}}{\rm {\ cos\ }}{\theta }_{0}}}{\rm {+}}{\frac {{\rm {1}}{\rm {\ cos\ }}{\theta }_{0}}{{\rm {\ sin\ }}{\theta }_{0}}}}{\rm {2}}}{\rm {=}}{\frac {{\rm {sin}}^{\rm {2}}{\theta }_{0}{\rm {+}}{\left({\rm {1}}{\rm {\ cos\ }}{\theta }_{0}\right)}^{\rm {2}}}{{\rm {2}}\left({\rm {l}}{\rm {\ cos\ }}{\theta }_{0}\right){\rm {\ sin\ }}{\theta }_{0}}}{\rm {=}}{\frac {\rm {l}}{{\rm {\ sin\ }}{\theta }_{0}}}\end{aligned}}$


(59)

and, similarly,
${\begin{aligned}{\rm {\ sinh\ }}at_{D}&{\rm {=}}{\frac {e^{at_{D}}e^{a{\rm {t}}_{O}}}{\rm {2}}}{\rm {=}}{\frac {\rm {l}}{\rm {2}}}\left({\frac {{\rm {\ sin\ }}{\theta }_{0}}{{\rm {1}}{\rm {\ cos\ }}{\theta }_{0}}}{\frac {{\rm {1}}{\rm {\ cos\ }}{\theta }_{0}}{{\rm {\ sin\ }}{\theta }_{0}}}\right){\rm {=}}{\frac {\rm {l}}{{\rm {\ tan\ }}{\theta }_{0}}}.\end{aligned}}$


(60)

In terms of traveltime to the point of maximum depth, the point of maximum depth is given by the coordinates
${\begin{aligned}x_{D}{\rm {=}}{\frac {v_{0}}{a{\rm {\ tan\ }}{\theta }_{0}}}{\rm {=}}{\frac {v_{0}}{a}}{\rm {\ sinh\ }}at_{D}\end{aligned}}$
${\begin{aligned}y_{D}&{\rm {=}}{\frac {v_{0}}{a{\rm {\ sin\ }}{\theta }_{0}}}{\frac {v_{0}}{a}}{\rm {=}}{\frac {v_{0}}{a}}\left({\rm {\ cosh\ }}at_{D}{\rm {1}}\right).\end{aligned}}$


(61)

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