The first Standard Sonar built puts one unit in your bases's "Short Range Detection" meter, and helps with UFO Detection. Further such modules also add to the meter, but do not increase the detection rate.
Every thirty minutes of game time, each base with a Standard Sonar has a 10% chance of detecting a USO within an approximate 1300-1500 mile radius (this may also vary based on USO size), and can track existing contacts up 2000 mile radius. If you advance the time an hour or day at a time, it will make multiple checks and interrupt if a USO is detected, setting the time to be at whatever point the detection happened (even if it is only five seconds after the last update).
Whether these chances are affected by the size of the USO is unknown, the UFOpedia suggest that it does play a part, but the description text is wrong about all the other details.
You cannot gain increased detection by building more Standard Sonar facilities at the same base, but you can add a single Wide Array Sonar to boost your total detection rate from the base to 30% at short range and 20% at long range. Any more sonars of either type have no game effect, except on the detection meters on the Base Information page. Both types of Sonar become redundant once a Transmission Resolver has been installed at a base.
Construction Time: 12 days
Construction Cost: $500,000
Maintenance Cost: $10,000/month
The above detection abilities are one-time-bonuses per base. The base detection values are recalculated every time a new building is completed on a base.