Do all waves of any kind satisfy the principle of superposition?

Is it an inherent portion of defining something as a wave?

Say if I had something that was modeled as a wave. When this thing encounters something else, will it obey the principle of superposition. Will they pass through each other?

Answer

If a wave $f(x,t)$ is something that satisfies the wave equation $Lf=0$ where $L$ is the differential operator $\partial_t^2-c^2\nabla^2$ then, because $L$ is linear, any linear combination $\lambda f+\mu g$ of solutions $f$ and $g$ is again a solution: $L(\lambda f + \mu g)=\lambda Lf+\mu Lg=0$.

In general, there might be things that propagate (not exactly waves, but since the question is for waves of any kind) determined by other differential equations. If the equation is of the form $Lf=0$ with $L$ a linear operator, the same argument applies and the superposition principle holds.