基于Fourier分析理论的数理方程研究

Mathematical Physical Equation Research Based on Fourier Analysis Theory

摘要: Fourier分析不仅在数学中有重要的理论价值，在求解弦振动方程等经典的数理方程时也有着广泛的应用，本文基于Fourier思想，在理想的弦自由振动方程基础之上，从源自实际的弦振动问题中抽象出数学模型，构建二阶线性齐次双曲型偏微分方程，依据设定的初始条件与边界条件，采用与Fourier级数展开关联较高的分离变数方法对定解问题进行分析与求解，并对其中假设的合理性、模型的优缺点及使用前景作了评估，说明其具有较好的实用性。本文以琵琶为例，不仅给出了单次拨弦后弦上质点的振动方程，还对模型中的物理量的测量方法及对应误差来源做了分析，其研究意义是可以利用此模型将演奏弦乐器的物理原理和音乐方面的研究成果进行结合，在根据已知公式的基础上，充分考虑振动频率等因素，转而研究弦乐器的演奏技巧问题，从而使得本文讨论的主题在物理与音乐的交叉学科上具有较高的研究价值。

Abstract: Fourier analysis not only has important theoretical value in mathematics, but also has a wide range of applications in solving classical mathematical equations such as the string vibration equation. In this paper, based on the Fourier idea, the mathematical model is abstracted from the actual string vibration problem based on the ideal string free vibration equation. In this paper, we construct the second-order linear chi-square hyperbolic partial differential equation and adopt the separation variables method with high correlation with the Fourier series expansion according to the initial and boundary conditions. The separation variable method with high correlation with Fourier series expansion is used to analyze and solve the solution problem, and evaluate the rationality of the hypothesis, the advantages and disadvantages of the model and its prospects, indicating that it has good practicability. Taking pipa as an example, this paper not only gives the vibration equation of the particle on the string after a single plucked string, but also analyzes the measurement method of physical quantity in the model and the corresponding error source. The research significance is that this model can be used to combine the physical principle of playing string instruments with the research results in music, and take vibration frequency and other factors into full consideration on the basis of the known formula. It turns to study the playing skills of string instruments, which makes the topic discussed in this paper have high research value in the interdisciplinary subject of physics and music.