How many *lowest* modes in a solid of size **L**?

Density of lowest modes ~ (Number of lowest modes) / (frequency)(volume)

Trick to find lowest modes:

remove boundary constraints,

count *free *modes with 0 frequency

---3 translations.. eg

Re-apply boundary constraints: walls fixed

pure
translations are no longer possible

lowest modes
are *distorted* translations

Frequency w0(L) ~ 1/L

Density of lowest modes ~ (Number of lowest modes) /
(frequency)(volume)

~ **L****-2**

Since frequency ~ 1/L, density of lowest modes ~
frequency**2**

cf *jammed* solids: they must have
lots of lowest modes or

very small w0(L)