约瑟夫·布西内斯克Joseph Valentin Boussinesq

约瑟夫·布西内斯克出生于圣安德烈德桑戈尼斯，这是法国南部的一个小镇，位于贝济耶东北方向仅30多公里处，距蒙彼利埃以西大致相同距离。他的父亲Jacques Boussinesq是一位农民，在该地区拥有一个家庭农场，而他的母亲Anne-Marie Cavalier是一位实业家的女儿。父母都希望看到儿子接受教育，这在当时当然不是强制性的，他在当地接受了小学和中学教育。然而，他的大部分教育得益于他的一位叔叔，一位受过高等教育的牧师。从这位叔叔那里，他学会了拉丁语和希腊语，以及如何自学。完成中学教育后，布西内斯克想前往蒙彼利埃学习，这是最近的可接受高等教育的城镇，但他需要经济支持才能在那里学习。

问题在于，布西内斯克的母亲于1857年去世，他的父亲明确表示，他的职责是接管家庭农场的经营。他唯一的出路是违背父亲的意愿，在没有家庭支持的情况下学习他最喜欢的数学。他通过在蒙彼利埃中学担任无需教学职责的舍监来做到这一点，同时在蒙彼利埃理学院学习语言和数学。他于1861年获得数学学士学位。

获得中学教师资格后，布西内斯克开始了教学生涯，于1862年被任命到阿格德中学，1865年到勒维冈中学，1866年到加普中学。这些城镇都在法国南部：阿格德与布西内斯克出生的城镇一样，都在埃罗省——事实上它位于埃罗河畔，距地中海4公里；勒维冈在蒙彼利埃以北64公里；但加普要往东远得多——它是上阿尔卑斯省的首府，位于吕伊河右岸，海拔约800米。正是在阿格德的第一所学校任教期间，布西内斯克读了加布里尔·拉梅的两部著作，即Théorie mathématique de l'ElasticitéⓉ（《弹性数学理论》）和Leçons sur la théorie analytique de la chaleurⓉ（《热的解析理论讲义》），从而被吸引到数学研究中。受此启发，他于1865年由加布里尔·拉梅向Académie des Sciences提交了关于毛细现象的第一篇科学论文。他开始在埃米尔·韦尔代指导下攻读关于A mechanical theory of light的博士学位。然而，韦尔代于1866年去世，当时布西内斯克的学位论文尚未完成。随后他得到了另一位导师，为了适应新导师的兴趣，布西内斯克改变了论文题目，撰写了Études sur la propagation de la chaleur dans les milieux homogènesⓉ（《均匀介质中热的传播研究》）。加布里尔·拉梅在指导布西内斯克方面发挥了重要作用，这篇论文清楚地显示了布西内斯克处理其主题的方式受到他的影响。另一位影响这项工作的数学家是阿代马尔·巴雷·德·圣维南。然而，答辩委员会由约瑟夫·阿尔弗雷德·塞雷、约瑟·伯特兰和夏尔·布里奥组成，布西内斯克于1867年5月13日在巴黎向该委员会答辩了他的论文。

布西内斯克并没有只专注于自己的学位论文而排除其他研究，因为他同时研究了线性弹性理论。这个课题对阿代马尔·巴雷·德·圣维南特别有吸引力，他定期与布西内斯克通信，并给予他许多鼓励，使他相信自己有前途成为研究数学家，而不是数学教师[5]：-

布西内斯克不是一位清晰的作者，在给出逻辑解释时常常过于急躁，以至于有几次阿代马尔·巴雷·德·圣维南建议他在著作中给出清晰而详细的论证。

事实上从这时起，阿代马尔·巴雷·德·圣维南就是布西内斯克最坚定的支持者，并且在1868年当选接替让-维克托·彭赛列在Académie des Sciences力学部门的职位后，变得更有影响力。布西内斯克于1867年与Jeanne Giscard de la Roque结婚，若不是有阿代马尔·巴雷·德·圣维南的支持和建议，他很可能就安顿下来在中学教书了。阿代马尔·巴雷·德·圣维南很了解这个体制，能够建议布西内斯克，除非他同时具备数学和物理学的资格，否则要在大学获得一个力学职位会非常困难。他听从了这一建议，并于1872年获得物理学学士学位。Académie des Sciences于1872年授予他让-维克托·彭赛列奖，他因此为获得大学教职做好了充分准备。事实上他在第二年就成功了，被任命为里尔理学院的微分与积分微积分教授。

布西内斯克到此为止的研究贡献非常可观。除了为学位论文所做的研究外，他已经发表了诸如Mémoire sur l'influence des frottements dans les mouvements réguliers des fluidesⓉ（关于摩擦在流体规则运动中的影响的回忆录）（1868年）；Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaireⓉ（论称为孤立波或平移波的液体扰动在矩形渠道中的传播）（1871年）；Étude nouvelle sur l'équilibre et le mouvement des corps solides élastiques dont certaines dimensions sont très petites par rapport à d'autres. Premier mémoire : des tiges ; deuxième mémoire : des plaques planesⓉ（爱德华·斯图迪论弹性固体的新平衡与运动，某些尺寸远小于其他尺寸。第一回忆录：杆；第二回忆录：平板）（1871年）；Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontalⓉ（论沿水平矩形渠道传播的波与涡旋的理论）（1872年）；Théorie des ondes liquides périodiquesⓉ（周期液体波理论）（1872年）；以及Sur les lois qui régissent, à une première approximation, les ondes lumineuses propagées dans un milieu homogène et transparent d'une contexture quelconqueⓉ（论在一级近似下支配在任意结构的均匀透明介质中传播的光波的定律）（1872年）。事实上，布西内斯克在1871年的第一篇论文中首次对孤立波的稳定性问题作了数学处理。Darrigol写道[3]：-

……布西内斯克研究明渠理论已有一段时间。他追随其导师阿代马尔·巴雷·德·圣维南的脚步，试图将河流与运河中水运动的每一个方面都纳入数学分析。他了解约翰·史考特·罗素的观察，也了解法国水文学家Henry Bazin对孤立波所作的更精确测量。他已经写了一篇关于恒定深度水面上小高度水波的长篇回忆录。除了在乔治·格林、菲利普·凯兰和乔治·比德尔·艾里早期回忆录中能找到的结果（他并不知道这些）之外，他还对有限高度的波提出了一些初步考虑，这些考虑可能引导他思考约翰·史考特·罗素的波。在他于1871年发表在《Comptes rendus》上的孤立波首次推导中，布西内斯克寻求莱昂哈德·欧拉方程的一个近似解，该解在矩形渠道中以恒定速度c传播而不变形。他在这项困难任务上的成功，取决于他在估计其展开式中各项相对重要性方面的特殊才能。他的基本策略是将速度分量按距渠道底部的垂直距离的幂展开，并通过边界条件确定该展开式的系数。约瑟夫·拉格朗日已经尝试过这条路径并写出了由此得到的微分方程组，但发现除非舍弃非线性项，否则其积分超出了当时分析的可能性。一个世纪后，布西内斯克设法将这些项包括进来。

布西内斯克在里尔任教约十五年，在此期间他继续在应用数学的众多领域做出重大贡献。在[6]中，Bois将这些贡献归入以下标题：土壤静应力问题；湍流（第一阶段）；表面波与布西内斯克方程；BBO方程与Basset-布西内斯克的“历史项”；以及势方法、“布西内斯克问题”与杆的振动。此外，他还开展了关于光传播和科学哲学方面的研究，这些课题贯穿了他的整个职业生涯。这些杰出贡献使他在1886年1月18日被任命为科学院Eugene Rolland讲席。这项任命的一个条件是布西内斯克必须迁往巴黎，因此他请求从里尔理学院转到巴黎理学院。这一请求获得批准，他被任命为巴黎理学院物理与实验力学教授。Pierre-路易·安托万 Bois写道[6]：-

布西内斯克在1886年初抵达巴黎时，已不再是风华正茂的年轻人。他外表朴素。他的前导师阿代马尔·巴雷·德·圣维南未能前来迎接，因为命运不济，这位从布西内斯克在阿格德开始职业生涯起，一直到科学院，都如此坚持不懈地支持他的人，已于1月6日去世，享年89岁，比他的学生当选早十二天。在加入新同事后，布西内斯克着手撰写前导师的讣告，与阿代马尔·巴雷·德·圣维南的第二得意门生Alfred Flamant合作。这些事件促使他更加内向；他唯一的消遣是研究，他开始更加不懈地工作。布西内斯克在索邦大学担任物理与实验力学教授十年，直到他继承了更有声望的数学物理与概率论讲席，这一职位他一直担任到1918年76岁退休。

在巴黎，他利用Bazin的实验结果和奥斯鲍恩·雷诺的思想，对湍流进行了另一项研究。这使他提出了“布西内斯克假设”，该假设后来被Bazin实验证实，为他赢得了相当大的声誉。他还发展了热的分析理论，并提出了所谓的“布西内斯克近似”。在Théorie analytique de la chaleur mise en harmonie avec la thermodynamique et avec la théorie mécanique de la lumière, Tome II : Refroidissement et échauffement par rayonnement. Conductibilité des tiges, lames et masses cristallines. Courants de convection. Théorie mécanique de la lumière Ⓣ（与热力学和光的力学理论相协调的热的分析理论，第二卷：辐射冷却与加热。电导率茎、叶片和晶体块。对流。光的力学理论）（1903年）中，他解释了他的近似成立的条件：-

人们仍需观察到，在我们重流体的大多数热诱导运动中，体积或密度大致守恒，尽管单位体积重量的相应变化实际上是我们所研究现象的成因。由此产生一种可能性：在密度变化未乘以重力g的地方忽略这些变化，而在计算中保留其与重力的乘积。

他已经在里尔时发表了其他一些重要著作，如Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiquesⓉ（势在弹性固体平衡和运动研究中的应用）（1885年），搬到巴黎后：Cours d'analyse infinitésimale en vue de ses applications mécaniques et physiquesⓉ（力学和物理应用的微积分课程）（2卷）（1887年和1890年）；Leçons synthétiques de Mécanique générale servant d'introduction au cours de Mécanique physiqueⓉ（作为物理力学导论的一般力学课程）（1889年）；以及Théorie analytique de la chaleur mise en harmonie avec la thermodynamique et avec la théorie mécanique de la lumière, Tome I : Problèmes générauxⓉ（热分析理论与热力学和光的力学理论相协调，第一卷：一般问题）（1901年）。

布西内斯克抵达巴黎八年后，他的妻子Jeanne去世；他们没有孩子。次年，1895年，他与Claire Onfroy de Véretz结婚。这段婚姻持续了10年，直到Claire于1905年去世。布西内斯克在妻子去世后的第二年再次结婚，这第三次是与Jeanne Le Bouteiller。这段婚姻只持续了三年；他们于1909年分居。1918年，76岁的他从大学职位退休[6]：-

他天性内向，变得越来越孤独。在1933年的讣告中，他的同事埃米尔·皮卡只谈到了布西内斯克作为学者的最后几年。然而，即使在学术上，Picard也强调了他的同事是多么内向。……然而，埃米尔·皮卡指出，布西内斯克的孤独中并无苦涩。尽管害羞和内向，他从不严厉评判他人。几乎直到最后，他每天都会来到科学院的图书馆，坐在同一张桌子旁。那是他与外界唯一的联系。

至于布西内斯克在数学之外的兴趣，这些似乎主要是对哲学和宗教问题的兴趣[1]：-

……特别是关于决定论与自由意志的调和。他谦卑地承认“我们清晰知识的集合之渺小，消逝在黑暗的海洋中。”

布西内斯克在数学物理发展中的地位，由Félix总结如下[1]：-

他的结论之一是，简单性在科学组织中不可或缺，而直觉是有价值的指南。布西内斯克厌恶引入诸如无导数的连续函数和非欧空间这样的怪物。他敌视相对主义的创新，忠于经典力学和以太的可靠实在。……凭借其研究的精神，他可被视为十九世纪古典科学的最后人物之一。

Joseph Boussinesq was born in Saint-André de Sangonis which is a small town in the south of France, just over 30 km north east of Beziers and about the same distance west of Montpellier. His father, Jacques Boussinesq, was a farmer who owned a family farm in the district, while his mother, Anne-Marie Cavalier, was the daughter of an industrialist. Both parents wanted to see their son educated, which was certainly not compulsorily at this time, and he attended both primary and secondary education locally. However, much of his education was due to one of his uncles, a highly educated priest. From this uncle he learnt Latin and Greek as well as how to study on his own. After completing his secondary education, Boussinesq wanted to study at Montpellier, the closest town where he could undertake higher education, but he required financial support to enable him to study there.

The problem was that Boussinesq's mother died in 1857 and his father made it very clear that his duty was to take over the running of the family farm. His only route was to defy his father's wishes, and to study his favourite subject of mathematics without family support. He managed this by taking employment as a housemaster without teaching duties at the Lycée de Montpellier, while he studied languages and mathematics at the Faculty of Sciences at Montpellier. He was awarded a Bachelor's degree in mathematics in 1861.

Having been awarded a degree which qualified him to teach in secondary schools, Boussinesq followed a teaching career being appointed to the Lycée in Agde in 1862, that in Le Vigan in 1865, then in Gap in 1866. These towns are all in the south of France: Agde is, like the town in which Boussinesq was born, in Hérault - in fact it is on the river Hérault, 4 km from the Mediterranean Sea; Le Vigan is 64 km north of Montpellier; but Gap is much further east - it is the capital of the Hautes-Alpes, about 800 metres above sea level on right bank of the Luye River. It was while teaching at the first of these schools in Agde that Boussinesq became drawn towards research in mathematics when he read two texts by Gabriel Lamé, namely Théorie mathématique de l'Elasticité Ⓣ and Leçons sur la théorie analytique de la chaleur Ⓣ. Inspired to undertake research, his first scientific paper on capillarity was presented to the Académie des Sciences by Lamé in 1865. He began working for a doctorate on A mechanical theory of light supervised by Émile Verdet. However, Verdet died in 1866 before Boussinesq's thesis was completed. He was then given another supervisor and, to suit his new supervisor's interests, Boussinesq changed the topic of his thesis and wrote Études sur la propagation de la chaleur dans les milieux homogènes Ⓣ. Lamé had played a major role in advising Boussinesq and the thesis clearly shows his influence in the way Boussinesq approached his subject. Another mathematician who influenced the work was Saint-Venant. The examining committee, however, comprised of Joseph Serret, Joseph Bertrand and Charles Auguste Briot and Boussinesq defended his thesis in Paris before this committee on 13 May 1867.

Boussinesq had not concentrated on his thesis to the exclusion of other research, for he had worked on the theory of linear elasticity at the same time. This topic was particularly fascinating to Saint-Venant who corresponded regularly with Boussinesq and gave him much encouragement to believe that he had a future as a research mathematician rather than as a school teacher of mathematics [5]:-

Boussinesq was not a lucid writer and was often too impatient in giving logical explanations so that, on several occasions, Saint-Venant advised him to give clear and detailed arguments in his work.

In fact from this time on Saint-Venant was Boussinesq's staunchest supporter and became even more influential in 1868 when he was elected to succeed Poncelet in the mechanics section of the Académie des Sciences. Boussinesq had married Jeanne Giscard de la Roque in 1867 and, but for Saint-Venant's support and advice, might well have settled down to teaching in secondary schools. Saint-Venant understood the system well and was able to advise Boussinesq that to obtain a university position in mechanics would be very difficult unless he had both a mathematics and a physics qualification. He followed this advice and in 1872 was awarded a Bachelor's degree in physics. The Académie des Sciences awarded him their Poncelet Prize in 1872 and he was well set up for a university appointment. Indeed he succeeded in the following year when he was appointed Professor of Differential and Integral Calculus at the Faculty of Science in Lille.

Boussinesq's research contributions up to this point had been very substantial. In addition to the research undertaken for his thesis, he had already published works such as Mémoire sur l'influence des frottements dans les mouvements réguliers des fluides Ⓣ (1868); Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire Ⓣ (1871); Étude nouvelle sur l'équilibre et le mouvement des corps solides élastiques dont certaines dimensions sont très petites par rapport à d'autres. Premier mémoire : des tiges ; deuxième mémoire : des plaques planes Ⓣ (1871); Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal Ⓣ (1872); Théorie des ondes liquides périodiques Ⓣ (1872); and Sur les lois qui régissent, à une première approximation, les ondes lumineuses propagées dans un milieu homogène et transparent d'une contexture quelconque Ⓣ (1872). In fact Boussinesq gave the first mathematical treatment of the problem of the stability of solitary waves in the first of these 1871 papers. Darrigol writes [3]:-

... Boussinesq had been working on open-channel theory for some time. In the steps of his mentor Saint-Venant, he tried to subject every aspect of the motion of water in rivers and canals to mathematical analysis. He was aware of Russell's observations, and also of the more precise measurements of solitary waves performed by the French hydraulician Henry Bazin. He had already written a long memoir on water waves of small height on water of constant depth. In addition to results that could be found in earlier memoirs by Green, Kelland, and Airy (of which he was unaware), he offered a few preliminary considerations on waves of finite height that may have led him to reflect on Russell's wave. In his first derivation of the solitary wave, published in 1871 in the 'Comptes rendus', Boussinesq sought an approximate solution of Euler's equations that propagated at the constant speed c without deformation in a rectangular channel. His success in this difficult task depended on his special flair in estimating the relative importance of the various terms of his developments. His basic strategy was to develop the velocity components in powers of the vertical distance from the bottom of the channel, and to determine the coefficients of this development through the boundary conditions. Lagrange had already tried this route and written the resulting series of differential equations, but had found their integration to exceed the possibilities of contemporary analysis unless nonlinear terms were dropped. A century later Boussinesq managed to include these terms.

Boussinesq taught at Lille for about fifteen years during which he continued to make major contributions to a wide variety of areas of applied mathematics. In [6] Bois collects these under the following headings: The problem of static stresses in soils; Turbulent flows (first phase); Surface waves and Boussinesq's equation; The BBO equation and the 'historical term' of Basset-Boussinesq; and The method of potential, the 'Boussinesq problem' and the vibrations of bars. In addition he undertook research on light propagation and on philosophical aspects of science, topics on which he continued to work throughout his career. These outstanding contributions led to his appointment to Eugene Rolland's chair in the Académie des Sciences on 18 January 1886. A condition of this appointment was that Boussinesq move to Paris so he requested that he be given a transfer from the Faculty of Science in Lille to the Faculty of Science in Paris. This was agreed and he was appointed Professor of Physical and Experimental Mechanics at the Faculty of Science in Paris. Pierre-Antoine Bois writes [6]:-

Boussinesq was no dashing young man when he arrived in Paris at the beginning of 1886. He was a modest looking man. Saint-Venant, his former mentor, was not there to greet him because, by an unlucky stroke of fate, he who supported Boussinesq with so much perseverance from the start of his career in Agde up to the Academy of Sciences, had just died, on 6 January, at the age of 89, twelve days before his pupil's election. Upon arrival among his new colleagues, Boussinesq proceeded to write his former mentor's obituary, in collaboration with Alfred Flamant, Saint-Venant's second favorite disciple. Such events contributed to his further withdrawing into himself; his only distraction was his research and he started working even more relentlessly. Boussinesq remained Professor of Physical and Experimental Mechanics at the Sorbonne for ten years, until he inherited the more prestigious chair of Mathematical Physics and Theory of Probabilities, a position which he kept until his retirement in 1918 at the age of 76.

In Paris he made another study of turbulent flows, using experimental results due to Bazin and ideas due to Reynolds. This led him to make 'Boussinesq's hypothesis' which gained him considerable credit when it was later confirmed experimentally by Bazin. He also developed an analytical theory of heat and made what has been called the 'Boussinesq Approximation'. In the text Théorie analytique de la chaleur mise en harmonie avec la thermodynamique et avec la théorie mécanique de la lumière, Tome II : Refroidissement et échauffement par rayonnement. Conductibilité des tiges, lames et masses cristallines. Courants de convection. Théorie mécanique de la lumière Ⓣ (1903) he explained when his approximation held:-

One still had to observe that in most heat-induced motions of our heavy fluids, the volumes or densities are approximately conserved, although the corresponding variation of the weight of the unit of volume is actually the cause of the phenomena we are studying. One possibility stems from there: neglecting the variations of the density where they are not multiplied by the gravity g, while conserving its product by the gravity in the calculation.

He had already published a number of other important texts in such as Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques Ⓣ (1885) while still in Lille, and, after moving to Paris: Cours d'analyse infinitésimale en vue de ses applications mécaniques et physiques Ⓣ (2 volumes) (1887 and 1890); Leçons synthétiques de Mécanique générale servant d'introduction au cours de Mécanique physique Ⓣ (1889); and Théorie analytique de la chaleur mise en harmonie avec la thermodynamique et avec la théorie mécanique de la lumière, Tome I : Problèmes généraux Ⓣ (1901).

Eight years after Boussinesq arrived in Paris, his wife Jeanne died; they had no children. In the following year, 1895, he married Claire Onfroy de Véretz. This marriage lasted for 10 years until Claire died in 1905. Boussinesq again married in the year following his wife's death, on this third occasion to Jeanne Le Bouteiller. This marriage only lasted for three years; they separated in 1909. In 1918, at the age of 76, he retired from his university positions [6]:-

Withdrawn by nature, he became more and more solitary. In his obituary in 1933, his colleague Émile Picard only talked about Boussinesq's last years as an academic. However, even academically, Picard stressed how introverted his colleague had been. ... Picard pointed out, however, that there was no bitterness in Boussinesq's solitude. Although shy and withdrawn, he never judged people severely. Every day, almost until the end, he would come to the library at the Academy and sit at the same table. That was his only link with the outside world.

As to Boussinesq's interests outside mathematics, these seem to have been mainly an interest in philosophical and religious problems [1]:-

... particularly on the conciliation of determinism and free will. He humbly admitted "the smallness of the ensemble of our unclouded knowledge is lost in an ocean of darkness."

Boussinesq's place in the development of mathematical physics is summed up by Félix as follows [1]:-

One of his conclusions was that simplicity is indispensable in scientific organisation and that intuition is a valuable guide. Boussinesq loathed the introduction of such monsters as continuous functions without derivatives and of non-Euclidean spaces. Hostile to relativistist innovations, he remained loyal to classical mechanics and the sure reality of the aether. ... By virtue of the spirit of his research he can be considered one of the last figure of classical science in the nineteenth century.