Scattering occurs as elastic waves propagate through a random polycrystalline medium, exhibiting scattering-induced attenuation and velocity dispersion. These behaviours carry bulk information about microstructure and can therefore be used for microstructure characterisation. The purpose of this thesis is to gain knowledge about the behaviour of elastic waves in polycrystals in order to facilitate the characterisation of microstructure. This thesis contributes mainly in six aspects.

First, a theoretical second-order approximation (SOA) model is developed to calculate the scattering-induced attenuation and velocity dispersion of plane elastic waves in random polycrystals. This model provides solutions of second-order accuracy in material inhomogeneity that are valid across all scattering regimes and partially account for multiple scattering. It applies to statistically equiaxed and elongated grains of arbitrary crystal symmetries, with decoupled geometric and elastic statistics represented respectively by the two-point correlation (TPC) function and the elastic covariance. A simple Born approximation, with a reduced accuracy considering only single scattering, is formulated based on the SOA model, and analytical asymptotes are derived for the low-frequency Rayleigh and high-frequency stochastic regimes.

Second, a three-dimensional (3D) finite element (FE) method is advanced to solve the wave propagation problem in the time domain. This method uses grain-scale spatial representation, in significant sample volumes of large numbers of grains, to describe polycrystalline materials. It captures the exact interactions of waves with grains without low-order scattering approximations. The numerical errors and statistical uncertainties of the FE method are minimized to deliver very accurate calculations of attenuation and phase velocity. The TPC function of the FE model is accurately determined and incorporated into the SOA model to enable a direct comparison of both models.

Then, the SOA and FE models are used to study the propagation of plane longitudinal waves in polycrystals with statistically equiaxed grains and greatly differing inhomogeneities. Attenuation exhibits fourth- and second-power dependences on frequency in the Rayleigh and stochastic regimes, while phase velocity is nondispersive in both regimes. Attenuation and phase velocity also show proportionalities to material inhomogeneity, and in the Rayleigh regime, the difference between the SOA and FE models is quadratically related to inhomogeneity for both attenuation and velocity.

The fourth contribution relates to using the SOA and FE models to study plane longitudinal wave propagation in polycrystals with statistically elongated grains. The models are found to agree very well with each other for the studied polycrystals over a wide frequency range. In the Rayleigh regime, attenuation and phase velocity exhibit dependencies on the fourth- and zeroth-power of frequency, show respective proportionalities to the effective volume of the grains and the mean grain radius in the direction of propagation, and both manifest a proportionality to the mode-converted elastic scattering factor. In the stochastic regime, attenuation and phase velocity show dependencies on the second- and zeroth-power of frequency, demonstrate positive and negative proportionalities to the mean grain radius in the direction of propagation, and both are proportional to the same-mode elastic scattering factor.

Subsequently, a practical problem is addressed to represent the actual TPC statistics of polycrystals by a single exponential. A variety of potential parameters are identified for the single exponential and their goodness is evaluated by using the SOA and FE models. It is found that the effective grain radius is an optimal choice for the single exponential to represent the microstructure of a range of polycrystals with greatly differing grain uniformities and to achieve a fairly accurate calculation of attenuation and velocity.

The last contribution concerns the theoretical discovery of three modes for elastic waves in polycrystals. For either longitudinal or transverse propagating waves, three solutions are found in the far-field approximation and the SOA model, indicating that three modes may co-exist. A further study by the spectral function approach reveals that the two non-dominant modes mostly have negligible energy in comparison to the dominant mode.