Wave equation with a concentrated moving source (Q1175705)

The purpose of this paper is to find a tempered distribution satisfying the Cauchy problem \[ (\Delta-\partial^ 2/\partial\tau^ 2)u=- \delta(x)\delta(y)\delta(\eta\tau-z)H(z) H(\tau),\quad u(x,y,z,0)=u_ \tau(x,y,z,0)=0. \] Here \(\Delta\) is the three dimensional Laplace operator, \(\delta\) is the Dirac delta function and \(H\) is the Heaviside step function, \(\eta\) is a parameter. The solution is explicitly constructed using complex variables and distribution theory techniques and is found for both the supersonic \((\eta>1)\) and the subsonic \((0<\eta<1)\) cases.