# Infinite string attached to a spring [closed]

I'm trying to solve a problem related to waves on a string.

Say I have an infinite string, with tension $$T$$ and mass density $$\mu$$.

To the string, at $$x=0$$ (seeing as it's infinite, the specific point doesn't actually matter, so long as it's constant), I attach a spring $$k$$ (the string is horizontal whilst the spring is vertical).

I am attempting to calculate the reflected and transmitted waves, resulting from an incident wave: $$y_{inc}(x,t)=A_{inc}\,e^{i(x-\omega\,t)}$$

My attempt at a solution was to write

$$\mu\,\frac{\partial^2y}{\partial t^2}|_{x=0}=T\,\frac{\partial^2y}{\partial x^2}|_{x=0}-k\,y(0,t)$$

which (I think) is, generally speaking, true. I am not sure how (if at all) I can use this to find an expression for the reflected wave.

So, my question is how does the spring affect the reflection?

## closed as off-topic by sammy gerbil, Gert, stafusa, Jon Custer, Kyle KanosNov 22 '17 at 11:04

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – sammy gerbil, Gert, stafusa, Jon Custer, Kyle Kanos
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• If I'd try to solve it, my first attempt would be to impose boundary conditions. In case the other end of this spring is not fixed, I expect the problem to be much more complicated than otherwise. – stafusa Nov 22 '17 at 0:26
• Please note that this is not a homework help site. Please see this Meta post on asking homework questions and this Meta post for "check my work" problems. – Kyle Kanos Nov 22 '17 at 11:04
• @KyleKanos this isn't homework. I'm trying to build this system and measure some of these things, but I had difficulty solving the analytical part. – ItamarG3 Nov 22 '17 at 11:06
• @ItamarG3: please check out the links I provided for a discussion on how we define homework/homework-like problems and how to post such questions. – Kyle Kanos Nov 22 '17 at 11:07
• @KyleKanos no problem :) – ItamarG3 Nov 22 '17 at 11:10

There must be a body attached to the spring. Otherwise with $m = 0$ ideal spring, the motion of the string could never strech the spring (due to no 'inertia') and the problem would be equivalent to the one without the spring.
Say I attach a body of mass $m$ to the spring and let $s_0(t)$ describe its position relative to $y = 0$ string equilibrium position. $$m \ddot{s_0}(t) + k(-y(0,t) + s_0(t) - l_0) = 0$$ where $l_0$ is the equilibrium length of the spring. It's easier to use relative displacements: $s(t) = s_0(t) - l_0$ so $$m \ddot{s}(t) + k s(t) = ky(0,t). \tag1$$ Resultant force on the point $x = 0$ on the spring is 0: $$k(s(t) - y(0,t)) + T (\tan{\alpha_+} - \tan{\alpha_-}) = 0 \tag2$$ where $\tan{\alpha_\pm} = \partial_x y(0^\pm,t)$ Seek the solution of the wave-equation in the form \begin{align} y_{\leftarrow}(x,t) &= e^{i (\omega t - \bar k x)} + R e^{i (\omega t + \bar k x)}\\ \tag{As} y_{\rightarrow}(x,t) &= \Theta\, e^{i (\omega t - \bar k x)}. \end{align} where $R, \Theta \in \mathbb{C}$. Writing $s(t) = S e^{i \omega t}$ and using the $(\text{As})$ ansatz for (1) and (2) and imposing $$1 + R = \Theta \tag3$$ continuity gives three equations for three unknowns $(S,R,\Theta)$ in terms of $(m,\, \bar k,\, \omega = \bar{k}\sqrt{\frac{T}{\mu}}, \,T, \,\mu)$. An interesting fact is that for a configuration there can exist a wavenumber $\bar k_*$ for which the system isn't invertible. What is the physical explaination for it?