The transformation of surface waves over varying obstacles is far from a new problem, but it has become an increasing challenge in our world due to growing environmental, societal and technological needs. Many stakeholders, from scientists and engineers through to governments and private companies, are searching for superior models to describe the influences of changes in depth of the ocean or cross-section of canals, and how the speed of the current affects the wave transformation. This research can have many impacts, from protecting coastlines or natural treasures like the Great Barrier Reef and bringing greater safety to marine vessels and offshore structures, to answering complex new theories and concepts, including the application of the Hawking effect in relation to black holes. Whenever a wave passes over a surface, the size of the transmitted wave may be significantly increased compared to the amplitude and speed of the incident wave. The impact of a transmitted wave on offshore or coastal structures is impossible to calculate without closely calculating the transmitted waves. It is also impossible to trust the numerical results without deep analytical modelling of these problems due to the complex nature of the effects. This research looks at four important considerations: the transformation of long linear waves in an ocean with a variable depth, long-wave scattering in a canal with a rapidly varying cross-section, long linear surface waves on stationary currents in a canal of constant depth and variable width, and the re-estimation of the forced Korteweg–de Vries equation to describe a resonant interaction of a solitary wave with external pulse-type perturbations.