Uniform asymptotic behaviour of integrals of Bessel functions with a large parameter in the argument

In this paper, we deal with integrals whose integrand has a rapidly oscillating zero-order Bessel function of the first kind with real parameters in its argument, which can become large. We introduce and tabulate model integrals depending on a single parameter, which can determine the behaviour of the original integral near the zeros of the argument of the Bessel function. As an example of the uniform asymptotic analysis, we evaluate the multi-parameter integral, which arises in the solution of the transition problem for an accelerating moving load on an elastically supported infinite string. Asymptotic predictions are compared with the results obtained by direct numerical integration.