String Standing Waves

弦的驻波

For an ideal string of length L which is fixed at both ends, the solutions to the wave equation can take the form of standing waves:

对于一根理想弦，其长度为L，两端固定，波动方程的解可以表现为驻波形式：

This kind of solution can be verified by direct substitution into the wave equation:

这种解可以通过直接代入波动方程来验证：

Substituting:

代入：

These two expressions are equal for all values of x and t provided

这两个表达式在所有x和t的值下都相等

基频

弦谐振

索引波的概念

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Fourier Series

傅里叶级数

If a continuous function f(t) is periodic with period T, then it may be approximated arbitrarily well by a Fourier series

如果一个连续函数 f(t) 具有周期 T，那么它可以被傅里叶级数任意逼近

where the coefficients are given by:

其中系数由下式给出：

索引波的概念

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