  We define a tower of injections of $\tilde{B}$-type (resp. $\tilde{D}$-type)
Coxeter groups $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$) for $n\geq 3$. Let
$W^c(\tilde B_{n})$ (resp. $W^c(\tilde D_{n})$) be the set of fully commutative
elements in $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$), we classify the
elements of this set by giving a normal form for them. We define a
$\tilde{B}$-type tower of Hecke algebras and we use the faithfulness at the
Coxeter level to show that this last tower is a tower of injections.
  We use this normal form to define two injections from $W^c(\tilde B_{n-1})$
into $W^c(\tilde B_{n})$. We then define the tower of affine Temperley-Lieb
algebras of type $\tilde{B }$ and use the injections above to prove the
faithfulness of this tower. We follow the same track for $\tilde{D}$-type
objects
