数学物理方法 2025-1

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1. 考虑两端固定的弦在初始激励下的自由振动，即是给定初始位移和初始速度，求以后各时刻的位移 $u(x,t)$ 。定解问题为

\frac{\partial^2 u}{\partial t^2} - a^2 \frac{\partial^2 u}{\partial x^2} = 0, \quad (0 < x < l,\ t > 0) \tag{1a}

u\big|_{x=0} = 0,\quad u\big|_{x=l} = 0, \quad (t \ge 0) \tag{1b}

u\big|_{t=0} = \varphi(x),\quad \left.\frac{\partial u}{\partial t}\right|_{t=0} = \psi(x), \quad (0 \le x \le l) \tag{1c}

其中 $\varphi(x)$ 和 $\psi(x)$ 是已知函数，分别给出弦上各点的初始位移和初始速度。

Fixed–ends string (Fourier animation)

Space: x ∈ [0,l]. Ends fixed at x=0,l.

Wave speed a

Length l

Modes N

Vertical scale (px)

Time speed ×

Resolution (x points)

ϕ(x) preset

ψ(x) preset

x₀ (for pluck / Gaussian)

σ (Gaussian width, fraction of l)

ϕ amplitude

ψ amplitude

Custom ϕ(x) — use x, l, Math.*

Custom ψ(x) — use x, l, Math.*

Ready.

2. 考虑弦振动问题， $x = 0$ 端固定， $x = l$ 端作已知的简谐振动 $A \sin \omega t$ （其中振幅 $A$ 很小），弦的初始位移和初始速度均为零，且不受外力作用，求解弦的振动。定解问题为

\frac{\partial^2 u}{\partial t^2} - a^2 \frac{\partial^2 u}{\partial x^2} = 0, \quad (0 < x < l,\ t > 0) \tag{51a}

u\big|_{x=0} = 0,\quad u\big|_{x=l} = A \sin \omega t, \quad (t \ge 0) \tag{51b}

u\big|_{t=0} = 0,\quad \left.\frac{\partial u}{\partial t}\right|_{t=0} = 0, \quad (0 \le x \le l) \tag{51c}

Driven string (boundary forcing)

Space: x ∈ [0,l]. Left end fixed at x = 0, right end driven: u(l,t) = A sin(ω t).