Which earth pressure theory takes into account the roughness of the back of the wall?

Coulomb's earth pressure theory is the correct choice because it incorporates the effects of wall friction and allows for variations such as the roughness of the back of the wall. This theory considers both the active and passive earth pressures that develop when a retaining wall moves, which can be influenced by the texture and roughness of the wall surface.

By accounting for wall friction, Coulomb’s theory provides a more realistic estimation of the forces acting on a retaining structure, particularly in scenarios where the wall does not have a smooth finish. This makes it particularly useful in practical applications where the construction materials and finishing can vary, impacting the interaction between soil and wall.

Other theories, such as Rankine’s, assume a smooth wall and do not consider the effects of wall friction, which limits their applicability in certain real-world situations. Terzaghi-Wegman theory focuses on conditions such as the difference in wall and soil weights, while Perrin's theory introduces a more complex approach without directly addressing wall roughness in its fundamental assumptions. Therefore, Coulomb's theory stands out for its ability to account for wall roughness and friction, making it a preferred choice in engineering practice.