Which material sounds better for speakers construction? Wood, Ply or MDF?

I’ve oft wondered why no one has come out with speaked cabinets made of Smart Metal. Head tennis racquets used “smart” materials in their tennis racquets many years ago, that change stiffness characteristics in proportion to how hard the tennis ball is hit. Seems like it might be appropriate for cabinets. 

All speaker manufactures attempt to raise the resonance of their cabinets since low frequencies are extremely difficult to dissipate. Those bass busters deal more with the cabinet resonance from poorly designed speakers then from the waves off the front of the driver. 

Energy can not be destroyed. There is the challenge. 

All speaker manufactures attempt to raise the resonance of their cabinets since low frequencies are extremely difficult to dissipate.

False. The thin wall, BBC-derived approach is to lower the resonance frequencies to the bass region. Since you like to reference Harbeth, read up on their philosophy.

Energy can not be destroyed. There is the challenge

True, but it can be altered, as into heat.

Lively Harbeth cabinets color the sound. Not the way I would build. 

Best materials are composite polymers. And this is not debatable. 

As far as I'm concerned, Wilson has this nailed. Goldmund used to use composite methacrylate in the Dialogues. 

Anyone building DIY cabinets can use Corian at the very least. It's not too expensive. Or anything from McMaster's catalogue. 

Well the Harbeth " model " is the exception. The resonance of MDF is low but their cabinets are thin and " loose " which raises the resonance, so they have chosen to lower it.
In most speakers  low resonance excite the cabinet creating a very large driver. We can all relate to the example of a car with the stereo turned up load. The bass gets amplified buy the body of the car, the doors which essential are the back of the cabinet become a huge " driver ". The area of the door, perhaps 20 times larger then the driver in the door when excited will sound 20 times louder the the driver itself.