I'm calling you out on that one Mike. The mass of the bat is the same, regardless of size. If it has a mass of 2lb9oz, the only way that the low density bat will have more mass than the other is if you glue more wood on it.
Stiffness is a function of the willow structure. If you have low density willow then the cells that make up the willow are larger/the material making them up is further apart, so the air that has replaced the liquid in the drying process makes up more of the willow.
Is it possible that the reason a low density willow performs better for the weight is actually down to the additional air in the structure?
Don't know about the rest of you, but my head is spinning with all these terms being bandied around - density, mass, volume, weight.
Let's do a quick recap of the basic physics:
1. Density = Mass / Volume
2. Weight = Mass x gravity i.e m*g
When we say a bat *weighs* 2lb 9 oz, that's really the mass (not weight)
Anywho, gravity being a constant on planet earth, mass and weight could be used interchangeably for the purposes of this discussion (since greater mass means greater weight, g being constant)
If mass (or weight) of two bats is the same, say 2lb 9oz, a low density bat must have greater volume.
From #1
Volume = Mass / Density -> lower density means higher volume, mass remaining constant.
Therefore a low density bat will look bigger (edges, spine, everything) for two bats with the same *shape* i.e. profile
I agree with Tim - the lower density bat just has bigger volume, mass of wood doesn't change. The difference in volume between the bats is made up by air. As an extreme analogy, think of low density willow as a block of sponge vs. a solid block of cheese. If they both weighed the same, the sponge would have more volume due to all the air pockets in it.
I partially agree with Mike. If both bats are pressed the same, the lower density one will have more stiffness and more rebound. For the same thickness of the hard top surface, there's probably more "layers" of cells comprising the pressed surface in the low density bat.
My theory (and this is just a theory) is that the airpockets create a spring effect. At a microscopic level they are able to compress when the ball hits and then spring back to their original size. This elastic effect probably wears off after repeated use, i.e. the cell walls of wood lose their ability to spring back after repeated compression, the air pockets eventually collapse, and the bat loses its rebound.
Thought I'd be one of the few brave souls to stick out their necks on this topic. Would love to hear what forum experts think about this. Feel free to debunk my theory

although I don't think it is completely implausible. I'm not a batmaker, just a guy crazy about bats and cricket in general. I do have a graduate degree in Mechanical Engg, and have a technical background in the mechanical behavior of materials...