# Wave on string, bdry cond is mass on spring

1. Sep 23, 2012

### koopernikos

1. The problem statement, all variables and given/known data
A semi-infinite string (constant tension T and density ρ) lies along the positive
x-axis. The motion of the string is described by the linear wave equation

h_{xx}-c^2h_{tt}=0

with
c² ≡ T/ρ, where h(x,t) is the transverse displacement. Attached to the end of the string at
x = 0 is a mass m. Attached to the mass in the vertical direction is a spring with stiffness
constant k that has its other end attached to a rigid surface. The vertical displacement of
the mass m is designated by ξ (t) ≡ h(0,t). When the system is in equilibrium (i.e.,
when the spring is unextended and everything is at rest) ξ (t) ≡ 0. The radian frequency
of the free oscillator (when the string is not attached) is σ = k /m .
(a) By applying Newton's law to the motion of the mass at x = 0, find the boundary
condition on the string at x = 0. That is, find the equation of motion of the mass in
terms of ξ (t) . Ignore horizontal forces on the mass, and ignore gravity.
[HINT: the boundary condition should not have any terms showing h(0,t).]
(b) Prove that the total energy of the mechanical system (oscillator and string) is constant.
(c) Suppose that at t = 0 the initial conditions are

3. The attempt at a solution

a)
There are two forces (neglecting $\vec{g}$) acting on the mass: one due to the displacement (spring) and one stemming from the tension in the string.
\begin{align}
F_{net}
&=
\sum_i F_i\\
f_{net}
&=
\sum_i f_i\\
\ddot{\xi}
&=
-\sigma^2 \xi
+ c^2 h(0,t)_{xx}
\end{align}

b)
?
c)
?

....aaand that is about as much as i can come up with..

i played around with seperating variables a bit.. i don't even know what type of solution to expect.. something like

h(x,t)=e^{-At} (B \sin(kx-\omega t)+ C \cos(kx-\omega t))

?
obviously

\sigma \neq \omega