## 1 Introduction  
Water flooding technology is widely used to improve oil recovery efficiency in oil-fields [1, 2]. The accurate control of the water injection rate for stratified water injection is a key issue in water injection technology [3, 4, 5]. Intelligent stratified water injection technology without ground mechanical operations has been gradually carried out at home and abroad [6]. Reliable and efficient wireless intelligent measurement has become a core technology in the field of water injection wells. In particular, data transmission technology is the most important part in wireless intelligent measurement technology. Using acoustics to transmit data in tubular strings has been reported, but the severe water and energy lossesin these systems has indirectly resulted in the insufficiency and inefficiency of these existing techniques [7]. There have been many applications for using transient flow transmission signals to detect pipe blockages and leaks [8; 9; 10]. Using transient flow waves to transmit signals is an effective way to control water injection and provide real-time guidance and optimization for wireless intelligent measurement and regulation. Signal transmission techniques using transient flow have been developed and demonstrated in the engineering field [11; 12]. It is well known that the transient wave transmission signal method is simple to operate. Moreover, the technology is economically efficient and can provide timely intelligent measurement and adjustment optimization decisions [6]. The drilling fluid pulse in measurement with drilling (MWD) is a typical transient flow wave communication method. It was developed in the late 1960s, matured and began to be used commercially in the 1970s [13]. At the beginning, transient flow wave communication technology was applied to layered oil production and water detecting/plugging on a small scale. In the late 2010s, it was applied to separated zone water injection, forming a transient flow wave-controlled separated zone water injection technology [13]. The theory of transient flow wave (water hammer) has been well developed, especially by the valuable research contributed by [PERSON] and [PERSON]. [PERSON] [14] produced the best-known equation in transient flow theory called the \"fundamental equation of water hammer\". [PERSON] [15] developed a general theory of water hammer from first principles and showed that the convective term in the momentum equation was negligible. He introduced two important dimensionless parameters widely used to characterize pipelines and valve behavior. Further refinements to the governing equations of water hammer appeared in [PERSON] [16] and [PERSON] [17]. Their combined efforts have resulted in the following classical mass and momentum equations for one-dimensional (1D) water hammer. For continuous transient flow waves in pipelines, [PERSON] proposed the use of transfer functions to describe the transmission process of fluctuations. However, [PERSON]'s model is based on a simplified vibration model, and the error is relatively large when calculating long pipes [18]. Due to the low flow velocity of the injection tube, the generation and transmission of transient flow wave signals in the injection tube are different from that in measurement with drilling. Therefore, it is significant to study the model of the transient flow wave in a water injection tubular string. During the water injection process, the tubular string can be considered as a vertical hollow tube. A mechanical device is used to generate the transient flow wave at the bottom of the well, and the wave signal is encoded. Then, the wave signal is transmitted by the tubular string, measured and decoded on the wellhead. Therefore, usage of the transient flow waves can pass the downhole measurement information to the wellhead [19]. However, the propagation mechanism of the transient flow wave is complicated [20]. Many scholars have studied the propagation mechanism and coding method of the transient flow wave signal [21; 22].