When a bullet travels at supersonic speeds, it generates a shock-wave. For a bullet traveling in a straight path, the shock-wave is oriented in the same direction as the bullet. But if a bullet is being deflected by the wind, the nose of the bullet will orient itself into the wind (the opposite of the direction of flight, which is counter-intuitive).
My question: is the shockwave oriented to the nose of the bullet, or is it oriented to the direction of flight?
A shock wave will obey the Rankine–Hugoniot relations, which means the normally incident flow direction is the relevant direction.
My question: is the shockwave oriented to the nose of the bullet, or is it oriented to the direction of flight?
What you are looking for is called aberration. So first find the angle between the velocity of the fluid incident on the bullet in the absence of wind and the velocity of the wind. That is, imagine you are sitting on the bullet traveling through a perfectly calm, uniform fluid background. The incident flow is perfectly anti-parallel to the velocity of the bullet in the lab frame. If you now add wind, then the incident flow is canted at an angle defined by: $$ \alpha = \tan^{-1}{\left( \frac{ v_{wind} }{ v_{b} } \right)} \tag{0} $$ where $v_{wind}$ is y-component of the velocity of the wind and $v_{b}$ is the x-component of the velocity of the bullet relative to some lab rest frame. You can then construct a transformation matrix, $\mathcal{R}$, to transform from the lab rest frame to the aberrated frame, where: $$ \mathcal{R} = \begin{pmatrix} \cos{\alpha} & -\sin{\alpha} & 0 \\ \sin{\alpha} & \cos{\alpha} & 0 \\ 0 & 0 & 1 \end{pmatrix} \tag{1} $$ You can then apply $\mathcal{R}$ to any of the lab frame vectors to get the aberrated coordinate basis vectors.
But if a bullet is being deflected by the wind, the nose of the bullet will orient itself into the wind (the opposite of the direction of flight, which is counter-intuitive).
Most bullets, early in their flight, will have their spin axis mostly aligned with their direction of flight unless the wind is really strong or the bullet is a low velocity type (e.g., the .22 caliber rounds like those used in biathlon are influenced by even ~5 mph winds). The more massive and fast the round, the smaller the influence of the wind.
