The geometry of a weld pool affects its solidification behavior. Here we show that the welding parameters and material properties responsible for the formation of unusual, wavy fusion boundaries can be identified from heat and fluid flow calculations and confirmed by independent experiments. These parameters and properties can be expressed by dimensionless numbers using the Buckingham π-theorem. The wavy boundary originates from the interaction of counter-rotating liquid metal loops at high Marangoni numbers. The Prandtl number affects the type of inflection.