We present an analytical solution describing steady, two-dimensional groundwater flow in the vertical plane in an unconfined, coastal aquifer of finite depth. An interface separates flowing freshwater from stagnant seawater. The boundary conditions along both the phreatic surface and the interface are treated exactly by the use of the hodograph method and conformal mapping. Explicit expressions for the distance of the toe from the coast, and the distance of the tip from the coast, are presented as asymptotic expansions about small gradients valid for isotropic and vertically anisotropic aquifers. A new, and simpler form of the solution to the classical problem of confined interface flow is obtained as a limiting case of the unconfined solution. Both new solutions avoid the problem of crowding, which occurs in the existing solutions for confined interface flow, and is commonly encountered in conformal mappings of elongated domains. A comparison of the tip and toe locations is made for the unconfined and confined cases; the results compare favorably for typical values of seawater density. The exact equation for the toe length as a function of the gradient, truncated after two terms, provides a simple, accurate, and invertible expression for estimating the freshwater discharge to the coast based on field observations of the toe length.

中文翻译：

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用Hodograph方法求解有限制和无限制海岸界面流的解析解

我们提出了一个分析解决方案，描述了有限深度的非承压沿海含水层中垂直平面上稳定的二维地下水流。界面将流动的淡水与停滞的海水分开。沿着潜水面和界面的边界条件通过使用hodograph方法和保角映射来精确处理。尖端到海岸的距离和尖端到海岸的距离的显式表达式被表示为关于小梯度的渐近扩展，对各向同性和垂直各向异性含水层有效。受限界面流经典问题的一种新的、更简单的解法作为非受限解法的极限情况获得。两种新的解决方案都避免了拥挤的问题，这发生在现有的受限界面流解决方案中，并且通常在细长域的共形映射中遇到。对非受限和受限情况下的尖端和脚趾位置进行了比较；与典型的海水密度值相比，该结果具有优势。作为梯度函数的脚趾长度的精确方程，在两个项之后截断，提供了一个简单、准确和可逆的表达式，用于根据脚趾长度的实地观察估计向海岸的淡水排放。