probably one of the best videos on wave equation i have seen up till now , clear concise and expertly taught in a way that even an amateur with basic knowledge of the concept (i.e. me) can understand
10:40 I think the left hand side has dimension L/T^2 (acceleration) and the second order derivative on the right side has dimension 1/L. Therefore T/p has dimension L^2/T^2 as you correctly state.