In many continuum-mechanical Problems it is assumed that liquid and solid substances cannot Change the total value of volume where it holds $\rho = const, \vec{\nabla}\cdot \vec{v} = 0$.
In the 1-dimensional problem of the vibrating string it is often assumed that the string cannot be deformed in longitudinal direction. Even gases are assumed mostly as incompressible at velocities much lower than Speed of sound.
The reason for this assumption lies in the high modulus of compression $K$ of the substance. Instead of an "elastic spring force" it is assumed that there acts a Deformation resistance. But is it really easier to lock degree of Deformational freedom?
And what is the reason for that many substances are almost non-deformable (statistical mechanics explaination)? Is it derivable by microscopic theories???
Best Answer
Firstly, although it is a bit of a 'how long is a piece of string?' kind of argument I would not really accept:
...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.