  We perform a three dimensional lattice simulation of the electroweak symmetry
breaking process through a two-step phase transition, where one of the two
steps is a first order phase transition. Our results show that: 1) when the
electroweak symmetry breaking is driven by the beyond Standard Model sector
around $\sim \mathcal{O}(10^{2-3})$ GeV, the gravitational wave spectra
produced from the phase transitions are of broken power-law double-peak shapes;
2) when the electroweak symmetry breaking is induced by a first-order phase
transition of a high-scale global U(1) theory, cosmic strings can form and then
disappear through particle radiation, and the yielded gravitational wave
spectra are of plateau shapes. The two scenarios can be distinguished through
probing gravitational wave spectra. Our study suggests that the stochastic
gravitational waves provide an alternative way to probe the beyond Standard
Model sector relevant to the electroweak symmetry breaking pattern in the early
Universe.
