  The spin excitations in the spin-liquid phase of the anisotropic triangular
lattice quantum antiferromagnet Cs$_2$CuCl$_4$ have been shown to propagate
dominantly along the crystallographic $b$ axis. To test this dimensional
reduction scenario, we have performed ultrasound experiments in the spin-liquid
phase of Cs$_2$CuCl$_4$ probing the elastic constant $c_{22}$ and the sound
attenuation along the $b$ axis as a function of an external magnetic field
along the $a$ axis. We show that our data can be quantitatively explained
within the framework of a nearest-neighbor spin-$1/2$ Heisenberg chain, where
fermions are introduced via the Jordan-Wigner transformation and the
spin-phonon interaction arises from the usual exchange-striction mechanism.
