It has been made great progress on single image deraining based on deep convolutional neural networks (CNNs). In most existing deep deraining methods, CNNs aim to learn a direct mapping from rainy images to clean rain-less images, and their architectures are becoming more and more complex. However, due to the limitation of mixing rain with object edges and background, it is difficult to separate rain and object/background, and the edge details of the image cannot be effectively recovered in the reconstruction process. To address this problem, we propose a novel wavelet approximation-aware residual network (WAAR), wherein rain is effectively removed from both low-frequency structures and high-frequency details at each level separately, especially in low-frequency sub-images at each level. After wavelet transform, we propose novel approximation aware (AAM) and approximation level blending (ALB) mechanisms to further aid the low-frequency networks at each level recover the structure and texture of low-frequency sub-images recursively, while the high frequency network can effectively eliminate rain streaks through block connection and achieve different degrees of edge detail enhancement by adjusting hyperparameters. In addition, we also introduce block connection to enrich the high-frequency details in the high-frequency network, which is favorable for obtaining potential interdependencies between high- and low-frequency features. Experimental results indicate that the proposed WAAR exhibits strong performance in reconstructing clean and rain-free images, recovering real and undistorted texture structures, and enhancing image edges in comparison with the state-of-the-art approaches on synthetic and real image datasets. It shows the effectiveness of our method, especially on image edges and texture details.