= Right-triangle geometry of internal-wave velocities
{title2=$\mathbf c_p\cdot\mathbf c_g=0,\quad\mathbf c_p+\mathbf c_g=(N/K,0)$}
Choose $k,m>0$, $K=(k^2+m^2)^{1/2}$ and the positive <angular frequency> $\omega=Nk/K$. Then the <phase velocity> is $Nk(k,m)/K^3$ and the <group velocity> is $N(m^2,-km)/K^3$. Their <dot product> vanishes and their sum is horizontal with magnitude $N/K$. Draw the <group velocity> from the tip of the <phase velocity>: the two form the perpendicular sides of a <right triangle>. The horizontal hypotenuse is their sum, not the <horizontal phase velocity> $\omega/k$.
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