A technique involving the higher Wronskians of a differential equation is presented for analysing the dispersion relation in a class of wave propagation problems. The technique shows that the complicated transcendental-function expressions which occur in series expansions of the dispersion function can, remarkably, be simplified to low-order polynomials exactly, with explicit coefficients which we determine. Hence simple but high-order expansions exist which apply beyond the frequency and wavenumber range of widely used approximations based on kinematic hypotheses. The new expansions are hypothesis-free, in that they are derived rigorously from the governing equations, without approximation. Full details are presented for axisymmetric elastic waves propagating along a tube, for which stretching and bending waves are coupled. New approximate dispersion relations are obtained, and their high accuracy confirmed by comparison with the results of numerical computations. The weak coupling limit is given particular attention, and shown to have a wide range of validity, extending well into the range of strong coupling.

提出了一种涉及微分方程的更高 Wronskian 的技术，用于分析一类波传播问题中的色散关系。该技术表明，在色散函数的级数展开中出现的复杂超越函数表达式可以非常精确地简化为低阶多项式，并具有我们确定的显式系数。因此，存在简单但高阶的扩展，其适用于基于运动学假设的广泛使用的近似值的频率和波数范围之外。新的扩展是无假设的，因为它们是从控制方程中严格推导出来的，没有近似。展示了沿管传播的轴对称弹性波的完整细节，其中拉伸波和弯曲波是耦合的。获得了新的近似色散关系，并通过与数值计算结果的比较证实了它们的高精度。弱耦合极限受到了特别的关注，并被证明具有广泛的有效性，可以很好地扩展到强耦合的范围。