This paper is concerned with a method for solving response problems of an elastic solid of arbitrary shape with a circular cylindrical cavity subjected to a transient pressure along the cavity. In the analysis, the exact solutions which satisfy the boundary conditions along the cavity are obtained by applying the Laplace transformation to the equations of motion of elastic solids based on the two‐dimensional theory of elasticity. The boundary conditions along the outer surface of arbitrary shape are satisfied by means of the Fourier expansion collocation method. The Laplace transform inversion integral is obtained from the residue theorem. Reasonably accurate results are obtained when the integration for finding the coefficients of the Fourier series and the numerical differentiation are carried out appropriately. To discuss the accuracy of the present analysis, results are compared with those from an exact method for circular rings. Numerical calculations are carried out for sample cases.