An extension of the analytical results of Kaup & Newell (1978) concerning the effect of a perturbation on a solitary wave of the Korteweg–de Vries equation is given and numerical studies are conducted to verify the conclusions. In all cases, the numerical results agree with the results predicted by the theory. The most striking feature of the perturbed flow is the presence of a shelf in the lee of the solitary wave whose role is to absorb (provide) the extra mass which is created (depleted) by the perturbation.

Resistivity measurement is a weighted averaging of local resistivities. We develop a formalism to calculate the weighting function, applying it to square van der Pauw samples and to linear and square four-point probe arrays. In each case, some regions of the sample are negatively weighted, but these regions can be reduced or eliminated by van der Pauw averaging. We discuss negative weighting, which we feel is responsible for spurious reports of superconductivity above room temperature. We show how a square four-point array can be more effective at measuring local resistivity than a linear one. Finally, we show how to apply our formalism to anisotropic materials.

We have found that the reflected wave that is created by a right-going solitary wave as it travels in a region of slowly changing depth does not satisfy Green's law. The amplitude of the reflected wave is constant along left-going characteristics rather than proportional to the negative fourth root of depth. This new finding allows us to satisfy the mass-flux conservation laws to leading order and establishes that the perturbed Korteweg–de Vries equation is a consistent approximation for the right-going profile.

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A formalism for calculating the sensitivity of Hall measurements to local inhomogeneities of the sample material or the magnetic field is developed. This Hall weighting function g(x,y) is calculated for various placements of current and voltage probes on square and circular laminar samples. Unlike the resistivity weighting function, it is nonnegative throughout the entire sample, provided all probes lie at the edge of the sample. Singularities arise in the Hall weighting function near the current and voltage probes except in the case where these probes are located at the corners of a square. Implications of the results for cross, clover, and bridge samples, and the implications of our results for metal–insulator transition and quantum Hall studies are discussed.