📚 Year 10 WJEC Geography: Formulae & Theorems Quick Reference Handbook | Year 10 WJEC 地理：公式定理速查手册
This quick reference handbook brings together the essential formulae and key theoretical relationships that underpin the WJEC Year 10 Geography course. From river discharge and population growth to urban models and climate water balances, mastering these computational and conceptual tools will help you analyse geographical data with confidence and answer exam questions accurately. Each section presents the core ideas in paired English and Chinese explanations, making this resource equally useful for native speakers and learners working in a bilingual environment.
这本速查手册汇集了 WJEC Year 10 地理课程的基础公式和核心理论关系。从河流流量、人口增长到城市模型和气候水量平衡，掌握这些计算与分析工具将帮助你自信地处理地理数据，并准确回答考试题目。每个小节都用英中配对解释呈现核心概念，使本手册对母语者和双语环境下的学习者同样实用。
1. River Velocity, Discharge and Hydraulic Radius | 河流流速、流量与水力半径
River fieldwork frequently requires calculations of velocity, discharge and hydraulic radius to understand channel efficiency. Velocity can be measured directly with a flowmeter or by timing a float, but it is also linked to channel shape through the hydraulic radius.
河流实地考察常需要计算流速、流量和水力半径，以理解河道效率。流速可用流速仪直接测量或通过计时浮标获得，但它也通过水力半径与河道形态相关联。
Velocity (v) = Distance (d) / Time (t)
流速 (v) = 距离 (d) / 时间 (t)
Once velocity is known, discharge – the volume of water passing a point per second – is given by the continuity equation. Cross-sectional area is calculated from channel width and mean depth.
已知流速后，流量——即每秒通过某点的水体体积——可由连续性方程得出。横截面积则根据河宽和平均水深算得。
Discharge (Q) = Cross-sectional Area (A) × Velocity (v)
流量 (Q) = 横截面积 (A) × 流速 (v)
The hydraulic radius indicates how efficiently a channel can convey flow, with a higher value meaning less friction relative to the volume of water. It is the ratio of cross-sectional area to wetted perimeter.
水力半径反映河道输水效率，数值越高表示相对于水体体积的摩擦力越小。它是横截面积与湿周的比值。
Hydraulic Radius (R) = Area (A) / Wetted Perimeter (P)
水力半径 (R) = 面积 (A) / 湿周 (P)
2. Horton's Laws of Stream Ordering | 霍顿河流分级定律
Horton's laws, often applied using the Strahler ordering system, describe the geometric relationships within a drainage basin. These empirical regularities help predict how stream number, length and basin area change with increasing stream order.
霍顿定律通常结合 Strahler 分级法使用，它描述了流域内部的几何关系。这些经验规律有助于预测随着河流级别升高，支流数量、长度和流域面积如何变化。
In Strahler ordering, first-order streams have no tributaries; when two first-order streams join, they form a second-order segment, and so on. The law of stream numbers states that the number of streams of a given order decreases in a geometric progression.
在 Strahler 分级中，一级支流没有支流汇入；两条一级支流汇合形成二级河段，以此类推。河流数量定律指出，某一级别的支流数量按几何级数递减。
Bifurcation Ratio (Rb) = Number of streams of order u / Number of streams of order (u+1)
分叉比 (Rb) = u 级支流数量 / (u+1) 级支流数量
Typical Rb values range between 3 and 5. The law of stream length relates average length of streams to order, while the law of basin area links total contributing area to order. Together they underpin the concave long profile of a graded river.
典型的 Rb 值在 3 至 5 之间。河流长度定律将支流平均长度与级别关联，流域面积定律则将总汇水面积与级别关联。三者共同构成均衡河流的凹形纵剖面基础。
3. The Hjulström Curve: Erosion, Transport and Deposition | Hjulström 曲线：侵蚀、搬运与沉积
The Hjulström diagram is not a single formula but a fundamental theorem in fluvial geomorphology. It illustrates the critical erosion velocity and settling velocity for particles of different sizes, defining the boundary between erosion, transportation and deposition.
Hjulström 曲线图并非单一公式，而是河流地貌学的基本定理。它展示了不同粒径颗粒的临界侵蚀流速和沉降流速，划分出侵蚀、搬运和沉积的边界。
The curve shows that fine cohesive clays require higher velocities to be eroded than sands, because of inter-particle electrostatic forces. Once entrained, clay can stay in suspension at very low velocities. Sand and gravel have lower erosion thresholds but settle quickly as flow wanes.
该曲线表明，细小粘性黏土因颗粒间静电引力，其启动侵蚀所需流速反而高于砂粒。一旦被携带，黏土可在极低流速下保持悬浮。砂和砾石的侵蚀临界值较低，但当水流减弱时沉降很快。
This theorem helps explain features like river meanders, floodplain deposition and the formation of levees. It is particularly useful when interpreting the energy conditions needed to erode resistant channel beds.
这一理论有助于解释河曲、河漫滩沉积和天然堤的形成。在分析侵蚀坚硬河床所需的能量条件时，该定理尤为实用。
4. Bradshaw Model: Downstream Changes | 布拉德肖模型：下游变化
The Bradshaw model is a conceptual theorem that summarises the expected changes in river characteristics as one moves downstream from source to mouth. It identifies how factors such as discharge, channel width, depth, velocity and sediment particle size typically vary along the long profile.
布拉德肖模型是一个概念性定理，它总结了从河源到河口沿下游方向河流特征的变化预期。该模型指出流量、河宽、水深、流速和沉积物粒径等因素沿纵剖面如何典型变化。
According to the model, discharge increases downstream as tributaries add water; width and depth increase while gradient and bed roughness decrease. Sediment particle size becomes finer and more rounded. However, velocity does not always increase monotonically; local channel variations and riffle-pool sequences can cause fluctuations.
依据该模型，随着支流汇入，流量向下游递增；宽度和深度增加，而坡降和河床粗糙度减小。沉积物粒径变小、圆度变好。但流速并非总呈单调递增；局部河道变化和浅滩-深潭交替可能引起波动。
This model provides a valuable framework for designing fieldwork investigations and for understanding how erosion and deposition create the distinctive landforms of the upper, middle and lower courses.
该模型为设计实地调查方案、理解侵蚀与沉积如何塑造上、中、下游独特地貌提供了宝贵的框架。
5. Wave Refraction and Longshore Drift | 波浪折射与沿岸漂移
Wave refraction is a key coastal theorem that explains how wave energy is concentrated on headlands and dispersed in bays, leading to differential erosion. As waves approach an irregular coastline, the part of the wave in shallower water slows down, causing the wave crest to bend and align roughly parallel to the shoreline.
波浪折射是解释波能如何在岬角集中、在港湾分散从而导致差异侵蚀的重要海岸定理。当波浪接近不规则海岸线时，浅水区波段的传播速度减慢，导致波峰弯曲并大致平行于岸线排列。
The orthogonals – lines drawn perpendicular to wave crests – converge on headlands, increasing energy and erosion, and diverge in bays, encouraging deposition. This theorem explains the formation of cliffs, wave-cut platforms, caves, arches and stacks on resistant headlands, and beaches in sheltered bays.
垂直于波峰的波向线在岬角处汇聚，增强能量和侵蚀；在海湾处发散，促进沉积。该定理解释了抗蚀岬角上悬崖、波切台、洞穴、拱门和海蚀柱的形成，以及遮蔽海湾里海滩的发育。
Longshore drift, the movement of sediment along the coast, is driven by waves approaching at an oblique angle. The swash carries material up the beach in the direction of the wave, while the backwash returns perpendicular to the slope, creating a zigzag net transport.
沿岸漂移（沉积物沿海岸移动）由斜向入射波浪驱动。冲流沿波浪方向将物质带上海滩，回流则垂直于坡度返回，形成锯齿状净搬运。
The rate of longshore drift can be estimated using the simple relationship:
沿岸漂移速率可用以下简单关系估算：
Longshore Drift Rate ≈ Wave Energy Flux × sin(2α)
沿岸漂移速率 ≈ 波能通量 × sin(2α)
where α is the angle between wave crest and shoreline. This formula highlights that drift is maximised when waves break at around 45° to the coast.
式中 α 为波峰与海岸线的夹角。该公式表明，当波浪以约 45° 角度破碎时漂移量最大。
6. Coastal Erosion and Mass Movement Formulas | 海岸侵蚀与块体运动公式
Coastal retreat rates are essential for managing erosion-prone areas. The average annual cliff recession can be calculated by dividing the total distance lost over a measured period. Although local geology and storm events cause variability, this basic formula underpins Shoreline Management Plans.
海岸后退速率对管理侵蚀敏感区域至关重要。年均悬崖后退量可用某一时期内总损失距离除以年限得出。尽管局地地质和风暴事件导致年际变化，但这一基本公式构成海岸线管理计划的基础。
Cliff Retreat Rate (R) = Total Recession Distance / Time Period
悬崖后退速率 (R) = 总后退距离 / 时段长度
Mass movement on coastal slopes is often analysed through the factor of safety concept. A slope becomes unstable when the downslope shear stress exceeds the shear strength of the material. In simple terms, the factor of safety (F) compares resisting forces to driving forces.
海岸边坡上的块体运动常通过安全系数概念分析。当下滑剪应力超过物质抗剪强度时，边坡失稳。简言之，安全系数 (F) 比较抗滑力与下滑力。
Factor of Safety (F) = Total Resisting Force / Total Driving Force
安全系数 (F) = 总抗滑力 / 总致滑力
If F > 1, the slope is stable; if F < 1, failure is likely. Triggers include heavy rainfall increasing pore-water pressure, undercutting by waves, or human loading. This theorem is central to understanding rotational slumps and landslides.
若 F > 1，边坡稳定；若 F < 1，很可能发生失稳。触发因素包括强降雨增加孔隙水压力、波浪掏蚀底部或人类加载。这一定理是理解旋转滑动和滑坡的核心。
7. Population Density, Natural Increase and Net Migration | 人口密度、自然增长率与净迁移
Demographic calculations start with population density, a simple ratio that allows comparison between countries or regions. It is usually expressed as people per square kilometre.
人口统计计算始于人口密度，这一简单比率可用于不同国家或区域之间的比较，通常以每平方公里人数表示。
Population Density = Total Population / Land Area (km²)
人口密度 = 总人口 / 土地面积 (km²)
Natural increase measures the difference between live births and deaths, ignoring migration. When combined with net migration, it gives the total population growth rate.
自然增长率衡量出生人口与死亡人口之差，不计入迁移。将其与净迁移相加，即可得到总人口增长率。
Natural Increase Rate = Birth Rate - Death Rate
自然增长率 = 出生率 - 死亡率
Net Migration = Immigration - Emigration
净迁移 = 迁入人数 - 迁出人数
Total Population Growth Rate = (Births - Deaths) + (Immigration - Emigration)
人口总增长率 = (出生人数 - 死亡人数) + (迁入人数 - 迁出人数)
These rates are often expressed per thousand people per year. Understanding these components is essential for interpreting the Demographic Transition Model and for evaluating policies related to population management.
这些比率通常以每年每千人的数值表示。理解这些构成要素对于解释人口转变模型和评估人口管理政策至关重要。
8. Dependency Ratio and Age Structure | 依赖比率与年龄结构
The dependency ratio is a key indicator of the economic pressure on the productive population. It compares the number of people typically not in the labour force – the young (0–14) and the elderly (65+) – with those of working age (15–64).
依赖比率是衡量生产性人口所承受经济压力的关键指标。它比较通常不参与劳动的人口——少年儿童（0–14 岁）和老年人口（65 岁以上）——与劳动年龄人口（15–64 岁）的数量关系。
dency Ratio = [(Population aged 0-14 + Population aged 65+) / Population aged 15-64] × 100
依赖比率 = [(0-14 岁人口 + 65 岁以上人口) / 15-64 岁人口] × 100
A high dependency ratio, often seen in LEDCs with a youthful population or in ageing MEDCs, implies greater spending on schools, healthcare and pensions. The ratio can be split into young-age and old-age components to analyse the specific challenges facing a society.
高依赖比率常见于人口年轻化的欠发达国家或老龄化的发达国家，意味着更多的教育、医疗和养老金支出。该比率可细分为少儿依赖比和老年依赖比，以分析社会面临的具体挑战。
Additionally, the ageing index compares the older population to the young, giving another perspective on population structure:
此外，老龄化指数比较老年人口与少儿人口，从另一角度揭示人口结构：
Ageing Index = (Population 65+ / Population 0–14) × 100
老龄化指数 = (65 岁以上人口 / 0-14 岁人口) × 100
9. Urban Models: Burgess, Hoyt and Harris-Ullman | 城市模型：伯吉斯、霍伊特和哈里斯-乌尔曼
Urban land-use models are theoretical frameworks that help geographers understand and predict the spatial patterns of cities. These models are often treated as geographical theorems because they summarise observed regularities into generalised diagrams.
城市土地利用模型是帮助地理学家理解和预测城市空间格局的理论框架。这些模型常被视为地理定理，因为它们将观察到的规律归纳为一般化图示。
The Burgess concentric zone model (1925) divides a city into five concentric rings: the CBD, transition zone, low-class residential, middle-class residential, and commuter zone. It assumes an outward expansion process similar to ecological succession.
伯吉斯同心圆模型（1925）将城市划分为五个同心环带：中央商务区、过渡带、低收入住宅区、中等收入住宅区和通勤区。它假定类似于生态演替的向外扩张过程。
Hoyt's sector model (1939) modifies Burgess by suggesting that land uses develop in wedge-shaped sectors radiating from the CBD along transport routes. High-rent residential sectors, for example, extend along favourable axes, while industry follows rail lines or rivers.
霍伊特扇形模型（1939）修正了伯吉斯模型，提出土地利用沿交通线从中央商务区呈楔形扇形发展。例如，高租金住宅区沿优越轴线延伸，而工业区沿铁路或河流分布。
The Harris–Ullman multiple nuclei model (1945) argues that cities grow around several distinct nodes rather than a single centre. Specialised activities cluster, forming nuclei such as an airport district, industrial park or university campus, due to agglomeration economies and repulsive forces between incompatible land uses.
哈里斯-乌尔曼多核心模型（1945）主张城市围绕多个不同节点而非单一中心发展。专门化活动因集聚经济及互不相容土地利用间的排斥力而聚集，形成如机场区、工业园或大学城等核心。
10. The Demographic Transition Model (DTM) | 人口转变模型
The Demographic Transition Model is a fundamental theorem in population geography. It describes how birth and death rates change as a country develops from a pre-industrial to a post-industrial society,
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