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From Rosenthal's 3D equation

(2.20)

At a fixed value of x, x = c, where c is a constant. For a given material under a given welding condition, T _o , k, Q, V, and α are all constant. Furthermore, T = T _m (the melting point) at the fusion boundary. Therefore, everything in Eq. (2.20) is constant, and the radial distance R between the origin and a point at the fusion boundary must, therefore, also be constant.

(2.21)

Since R and x are both constant, y ² + z ² = constant. Thus, the transverse cross‐section of the weld pool is round. In reality, however, the transverse cross‐section of a bead‐on‐plate weld is often not round, even when the plate is very thick.