With the strong mobility and autonomy, autonomous underwater vehicles(AUVs) have become necessary equipment in maritime search and rescue. Generally, the existing researches realize target detection in a given water area by randomly deploying AUVs, and they do not theoretically analyze the detection trajectory and formation shape. Therefore, we first prove that the linear tracking mode is the optimal trajectory to achieve the maximum coverage and the maximum target detection probability for AUVs with randomly selected starting point and mobile direction. Based on the above detection trajectory, a detection model for underwater search and rescue is established using the measure theory. Furthermore, we consider the communication loss of the AUVs, and hence, it is proved that the sensing area of the AUVs is the optimal detection formation when it is tangent. Meanwhile, the effects of the angle, distance, moving speed and sampling period of the AUV formation on the target detection probability are strictly analyzed. Finally, final simulation results verify that the constructed model can effectively improve the detection probability of targets.