Finite element (FE) simulations are popular for studying propagation and scattering of ultrasonic waves in nondestructive evaluation. For a large number of degrees of freedom, time domain FE simulations are much more efficient than the equivalent frequency domain solution. However, unlike frequency domain simulations, time domain simulations are often poor at representing the speed and the attenuation of waves if the material is strongly damping or highly dispersive. Here, the authors demonstrate efficient and accurate representation of propagated and scattered waves, achieved by combining a set of time domain solutions that are obtained for a set of frequency ranges known as bands, such that, in combination, the authors' multiband solution accurately represents the whole wave spectrum. Consequently, high accuracy is achieved, at minor computational cost, using a modest number of bands. The multiband technique is implemented for ultrasonic wave propagation in highly attenuating polyethylene material, using three frequency bands, and can yield a reduction in empirical acoustic properties fractional error compared with respective time domain simulations, in propagation duration, of a factor of 1.4, and in full-width-half-maximum, of a factor of 10. Last, the accuracy of this approach is further exemplified in a wave scattering simulation.