A wave equation riddle

We continue to work on the quantization of the exterior derivative. We were motivated by the wave equation on a Riemannian manifold of dimension q with Dirac operator D. This equation u_tt = - D^2 u is solved explicitly by u(t) = cos(Dt) u(0) + t sinc(Dt) u'(0) which involves two Bessel functions, The cos function solves the q=1 Bessel equation and the sinc function solves the q=3 Bessel equation f''(r)+(q-1) f'/r +f = 0 with f(0)=1,f'(0)=0. The strong Huygens principle fails for even dimensional manifolds. Update 2/22/2026: the computation works only for the initial position and not for the initial velocity. u(t) = phi(tD) u(0) satisfies the PDE u_{tt} + D^2 u + (q-1) u_t/t =0 if phi is the Bessel function for q and u'(0)=0. We had hoped that also for the velocity, we would have t psi(tD) u'(0) satisfying this PDE if psi is the Bessel function for q+2 ,but not everything cancels. The Bessel case q+2 is interesting to us, because d_t f = psi(tD) t df is an exterior derivative that has the property that its value at p only depends on f evaluated on the wave front W_t(p). Update 2/24/2026: in order for the puzzle to work for for the initial velocity we need to change the wave equation a bit more. Indeed, u_{tt} + D^2 u + (q-1) u_t/t - (q-1) u/t^2, u(0)=0, u'(0) = D f, is solved by u(t) = D_t f = psi(tD) D f. The modified wave equation pushes the initial exterior derivative away and renders it a bounded operator. The solution u(t,x) satisfies now in all dimensions a strong Huygens principle. u(t,x) only depends on u_t(t,y) where y is on the wave front W_t(x). See the blog https://www.quantumcalculus.org/3d-dirac-operator where the computation can be verified by running 3 lines of Mathematica code for any q. This picture now makes it more transparent why the q+1 Bessel function comes in. In the case of the usual wave equation we have in arbitrary dimensions the solution cos(Dt) u(0) + sinc(Dt) u'(0) which involves Bessel functions for q=1 and q=3 but unlike we are in one dimension, this does not give us the exterior derivative deformation we want. Pictures were shot on Thursday during a 18 K run to the Cummings park.