Author Topic: Bonus Topic 1 (Vibrating string)  (Read 203 times)

Victor Ivrii

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Bonus Topic 1 (Vibrating string)
« on: September 22, 2016, 07:46:45 AM »
Consider a string with linear density $\mu$ and tension $T$ like in https://en.wikipedia.org/wiki/String_vibration and explain equation
\begin{equation}
\mu u_{tt} - (Tu_x)_x=0.
\end{equation}
This equation is valid even if $\mu=\mu(x)$ and $T=T(x)$ are variable. Here $u(x,t)$ is a small displacement of the string.
« Last Edit: September 22, 2016, 08:15:54 AM by Victor Ivrii »

XinYu Zheng

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Re: Bonus Topic 1 (Vibrating string)
« Reply #1 on: September 22, 2016, 09:20:34 AM »
Please refer to attached diagram. Consider a infinitesimal segment of the string located with the left end point at location $x$ and let the tension in the string be $T(x)$. We will assume the string is not accelerating horizontally, so we have $T(x)\cos\phi=T(x+dx)\sin\theta$. We will also assume that the string is not displaced far from equilibrium, so that both $\theta$ and $\phi$ are small such that the small angle approximation holds (ie. $\theta\approx\sin\theta\approx\tan\theta$ and same thing for $\phi$).
The net vertical force on this segment of string is $T(x+dx)\sin\theta-T(x)\sin\phi$. By the small angle approximation, this is approximately $T(x+dx)\tan\theta-T(x)\tan\phi$. But by the definition of slope, note that $\left.\frac{\partial u}{\partial x}\right|_x=\tan\phi$ and $\left.\frac{\partial u}{\partial x}\right|_{x+dx}=\tan\theta$, thus we have
\begin{equation}
T(x+dx)\left.\frac{\partial u}{\partial x}\right|_{x+dx}-T(x)\left.\frac{\partial u}{\partial x}\right|_{x}=F
\end{equation}
By newton's second law, we have $F=m\frac{\partial^2 u}{\partial t^2}$. Note that $m=\mu(x) dx$, and finally we have
\begin{equation}
T(x+dx)\left.\frac{\partial u}{\partial x}\right|_{x+dx}-T(x)\left.\frac{\partial u}{\partial x}\right|_{x}=\mu(x) \frac{\partial^2 u}{\partial t^2} dx
\end{equation}
Moving the $dx$ to the LHS we recognize the LHS as the derivative of $T(x)\frac{\partial u}{\partial x}$ at $x$ and we obtain
\begin{equation}
\frac{\partial}{\partial x}\left(T(x)\frac{\partial u}{\partial x}\right)=\mu(x)\frac{\partial^2 u}{\partial t^2}
\end{equation}

« Last Edit: September 22, 2016, 12:45:03 PM by XinYu Zheng »