The authors use a model elastodynamic theory to study the propagation of a crack along a plane in an unbounded solid. Let the crack front motion \(x= v_ 0 t+ \varepsilon f(z,t)\), \(y=0\) correspond at \(\varepsilon=0\) to a straight crack propagating along the \(x\)-axis in the space \((x,y,z)\) with a uniform velocity \(v_ 0>0\). According to the method of perturbations, the authors write the displacement field as \(u(x,y,z, t;\varepsilon)= u_ 0 (x,y,t)+ \varepsilon\varphi (x,y,z,t)+ O(\varepsilon^ 2)\) when \(\varepsilon\to 0\). The function \(u_ 0 (x,y,t)\) is the well-known solution of the plane problem. The first order perturbation term \(\varphi\) satisfies the wave equation \(c^ 2 \nabla^ 2 \varphi=\partial^ 2 \varphi/\partial t^ 2\) in the half-space \(y>0\) and the boundary conditions \(\partial\varphi/\partial y(x,0,z,t) =0\), \(x<v_ 0 t\); \(\varphi(x,0,z,t) =0\), \(x>v_ 0 t\). Let the polar coordinates be understood to denote \(r\exp (i\theta)= x-v_ 0 t+i (1- (v_ 0/ c)^ 2)^{1/2} y\). Then it is proved that the function \(\varphi\) satisfies the asymptotic relation \(\lim_{r\to 0} \{\varphi (x,y,z,t) r^{1/2}\}= F(z,t)\sin (\theta/2)\), where \(F(z,t)\) is a given continuous function. Moreover, the function \(\varphi\) has been found in closed form by the Fourier transformation.