A teatcup liner configured to be mounted in a shell of a teatcup and to be applied to a teat of an animal, with a pulsation chamber between the liner and an inner side of the shell being connected to vacuum and atmospheric pressure, or a sub-pressure less than the milking vacuum, in a cyclic manner during a milking operation, the liner having a longitudinal shape extending along a longitudinal axis (x), the teatcup liner comprising: a barrel having a top that extends downward, wherein the barrel is comprised of i) an upper barrel of a first length, and ii) a lower barrel of a second length, where the first and second lengths (L1, L2) together equal the overall length of the barrel, and a top part of the lower barrel is connected to a bottom part of the upper barrel, the barrel defines an inner space with an interior surface within the upper barrel for receiving the teat, wherein, during the milking operation, the teat does not extend or extend to a minor degree into the lower barrel, wherein the barrel, in the relaxed state, has transversely to the longitudinal axis (x), a polygonal cross-sectional shape comprised of corner portions and side walls, each corner portion connecting two of said side walls, each corner portion and each side wall having an exterior surface and an opposite, inner surface, the inner surface of each corner portion being an arcuate first concave curvature open to the inner space, wherein, for the upper barrel, each corner portion has a first flexibility for closing such that during a liner collapse phase of a milking operation the corner portions close to the inner space and the side walls collapse towards the longitudinal axis (x) into the inner space of the upper barrel, and wherein, for the lower barrel, each corner portion has a second flexibility for closing such that during the liner collapse phase of the milking operation the corner portions close to the inner space and the side walls collapse towards the longitudinal axis (x) into the inner space of the lower barrel, characterized in that the second flexibility is greater than the first flexibility such that the corner portions of the lower barrel are weaker than the corner portions of the upper barrel, thereby assuring the lower barrel collapses prior to the upper barrel to create an upward wave movement of the lower barrel towards the upper barrel that collapses the upper barrel onto the teat. a barrel having a top that extends downward, wherein the barrel is comprised of i) an upper barrel of a first length, and ii) a lower barrel of a second length, where the first and second lengths (L1, L2) together equal the overall length of the barrel, and a top part of the lower barrel is connected to a bottom part of the upper barrel, the barrel defines an inner space with an interior surface within the upper barrel for receiving the teat, wherein, during the milking operation, the teat does not extend or extend to a minor degree into the lower barrel, wherein the barrel, in the relaxed state, has transversely to the longitudinal axis (x), a polygonal cross-sectional shape comprised of corner portions and side walls, each corner portion connecting two of said side walls, each corner portion and each side wall having an exterior surface and an opposite, inner surface, the inner surface of each corner portion being an arcuate first concave curvature open to the inner space, wherein, for the upper barrel, each corner portion has a first flexibility for closing such that during a liner collapse phase of a milking operation the corner portions close to the inner space and the side walls collapse towards the longitudinal axis (x) into the inner space of the upper barrel, and wherein, for the lower barrel, each corner portion has a second flexibility for closing such that during the liner collapse phase of the milking operation the corner portions close to the inner space and the side walls collapse towards the longitudinal axis (x) into the inner space of the lower barrel, characterized in that the second flexibility is greater than the first flexibility such that the corner portions of the lower barrel are weaker than the corner portions of the upper barrel, thereby assuring the lower barrel collapses prior to the upper barrel to create an upward wave movement of the lower barrel towards the upper barrel that collapses the upper barrel onto the teat.