The main effect is the opposite of what you say--- the metals have a high speed of sound and low dissipation compared to wood. So wood attenuates faster, and has lower frequencies, and this makes a dull thump, while metals ring like a crystal and don't decay for longer, at higher frequencies because of the higher speed of sound (which is ultimately because of the stiffer Young's/Bulk modulus, as you say)
The presence of disorder is also important in diffusing the sound. Wood and other nonmetals scatter the sound into a complicated waveform, where attenuation is enhanced.
Firstly, although it is a bit of a 'how long is a piece of string?' kind of argument I would not really accept:
Even gases are assumed mostly as incompressible at velocities much lower than Speed of sound.
...as very accurate. Gases are easily compressed to half of their initial volume (and less) with something as basic as a bicycle pump.
To understand why liquids and solids are far less compressible than gases we need to look at the micro-structure of matter. Within a reasonable temperature window of $0\:\mathrm{K}$ to about $5000\:\mathrm{K}$, matter is made up of atoms or molecules. These quantum structures are comprised of positively charged nuclei and surrounding negatively charged electron clouds (bound to the nuclei by electrostatic forces and explained by Quantum Physics).
When molecules and/or atoms collide, the electron clouds repel each other, resulting in quasi-elastic collisions.
Gases are far more compressible than liquids and solids because the inter-molecular distances are far greater than in liquids and solids (hence also the lower densities of gases).
In the case of liquids, imagine these atoms or molecules to practically be 'crawling' over each other, much like in an agitated bath full of ping pong balls. The inter-molecular distances are very small and any attempt in reducing the volume the liquid takes up results in increased electronic repulsion and thus an increase in pressure. Liquids thus resist compression much more than gases.
In the case of solids the scenario is similar, except that the atoms/molecules are now fixated in a lattice and their inter-molecular distances even smaller than in the case of liquids. So they generally resist compression even more.
Best Answer
All you need is the Navier's equation of motion (you can consult it in any book of elasticity)
Naturally, you can decompose a wave $ w$ in a transversal and a longitudinal part:
$$ w = w_L + w_T $$
with the following properties :
$$ \nabla \times w_L = 0$$ $$ \nabla .w_T = 0 $$
If we focused only in the transversal part, retaking the Navier equation we have:
$$ \rho \frac{\partial^2 w_T}{ \partial t^2 } = \mu \nabla^2 w_T $$
Now you can remember that the general wave equation is
where c is the phase velocity of the wave $ u $
So, that's it
$$ v_T = \sqrt \frac \mu\rho$$
I hope you help
J.