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A plane wave in the rest frame of the medium has phase $\mathbf{k}\cdot \mathbf{x}-{\Omega }_0(\mathbf{k})t$. Under the Galilean transformation ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{U}t$, this becomes $\mathbf{k}\cdot {\mathbf{x}}^{\prime }-[{\Omega }_0(\mathbf{k})-\mathbf{U}\cdot \mathbf{k}]t$. ThereforeHere ${\mathbf{c}}_g$ is the group velocity in the moving frame.

For acoustic waves, ${\Omega }_0=c_0|\mathbf{k}|$ and take $\mathbf{U}=M{c}_0{\mathbf{e}}_x$. A steady wave has $\omega =0$, requiring $\cos \beta =1/M$, where $\beta$ is the angle of $\mathbf{k}$ to the flight direction. Its ray velocity is $c_0\hat{\mathbf{k}}-M{c}_0{\mathbf{e}}_x$, with backward component $-c_0(M^2-1)/M$ and transverse magnitude $c_0\sqrt{M^2-1}/M$. Thus the angle $\theta$ of the ray to the backward flight direction obeysThe axial symmetry gives the Mach cone behind the aircraft.

For unsteady emitted disturbances, all ray velocities lie on the sphere ${\mathbf{c}}_g=-\mathbf{U}+c_0\hat{\mathbf{k}}$. Since $M>1$, the origin lies outside this velocity sphere and its tangent cone has exactly the same semi-angle. All other rays are strictly inside the cone. Equivalently a signal emitted $\tau$ time units ago has a wavefront sphere centered at $-\mathbf{U}\tau$ with radius $c_0\tau$; these spheres are enclosed by their Mach-cone envelope. Consequently the unsteady wavefield is confined to the cone's interior, with the steady envelope on its boundary.

For capillary waves put $a=\sqrt{T/\rho }$, so ${\Omega }_0=a{k}^{3/2}$. The positive-frequency steady condition is $a\sqrt{k}=U\cos \beta$, requiring $\cos \beta >0$. ThenandIts endpoint traces the circlesince $c_{gx}/U=-1/4+(3/4)\cos 2\beta$ and $c_{gy}/U=(3/4)\sin 2\beta$. As $\beta$ runs through $(-\pi /2,\pi /2)$ this traces the circle except its leftmost limiting point, corresponding to the zero-wavenumber limit. The circle surrounds the origin, so every direction except the precisely backward axis is attained by a nonzero-wavenumber steady ray; the backward direction is approached as $k\downarrow 0$. Thus the closure of the steady ray field extends all around the insect, with no excluded angular sector such as the acoustic Mach-cone exterior.