assourdir

To deafen — a loud noise assourdit — or, just as commonly, to muffle or deaden a sound (des rideaux épais assourdissent le bruit de la rue). Figuratively, it can describe overwhelming or numbing someone with noise or information.

• L'explosion a assourdi tout le quartier. — The explosion deafened the whole neighborhood.
• Ces murs épais assourdissent les bruits extérieurs. — These thick walls muffle outside noise.
• Le vacarme du concert m'a assourdi. — The concert's roar deafened me.

Assourdir covers both extremes — deafening with loud noise and muffling/dampening noise — the direction depends entirely on context: thick walls assourdissent by blocking sound, an explosion assourdit by overwhelming the ears.