What is the $3$D equation for a damped sine wave? This is the equation for a $2$D damped sine wave, what is the $3$D equivalent?
For reference, a $2$D sine wave would be of form $y=\sin x$, and a 3D sine wave would be of form $z=\sin(\sqrt{x^2 + y^2})$.
As in the comments, the equation
$$z=e^{-\sqrt{x^2+y^2}}\sin(\sqrt{x^2+y^2})$$
or else phase-shifted and scaled as
$$z=e^{-\sqrt{x^2+y^2}}\cos(2\pi\sqrt{x^2+y^2})$$
expresses damping along any ray $ax+by=0$ extending from the origin, and this was achieved by substitution of $\sqrt{x^2+y^2}$ in place of $t$ in the Wikipedia listing at https://en.m.wikipedia.org/wiki/Damped_sine_wave.
As in the comments, the equation
$$z=e^{-\sqrt{x^2+y^2}}\sin(\sqrt{x^2+y^2})$$
or else phase-shifted and scaled as
$$z=e^{-\sqrt{x^2+y^2}}\cos(2\pi\sqrt{x^2+y^2})$$
expresses damping along any ray $ax+by=0$ extending from the origin, and this was achieved by substitution of $\sqrt{x^2+y^2}$ in place of $t$ in the Wikipedia listing at https://en.m.wikipedia.org/wiki/Damped_sine_wave.