Moments of vibrating string I have to calculate first 10 moments for vibrating string with damping, but I don't know how to do it. I read about moments and in definition they mention linear system $$x'=Ax+Bu, \qquad y=Cx$$ with transfer function $$W(s)=C(sI-A)^{-1}B.$$ Then kth moment of a system above is $$C(sI-A)^{-(k+1)}B.$$ But for a vibrating string I know only wave equation $$Mx''+Cx'+Kx=b(t).$$ How to transform it to linear system?
 A: You simply need to define a few state variables. Here is the standard, control theory recipe that should get you going:
Let $x_1(t) = x(t)$ and $x_2(t) = \dot{x_1}(t)$. Then your vibrating string equation becomes $M \mathrm{d}_t x_2(t) + C x_2(t) + K x_1(t) = b(t)$. That is:
$$\frac{\mathrm{d}}{\mathrm{d}\,t} \left(\begin{array}{c}x_1\\x_2\end{array}\right) = \left(\begin{array}{cc}0&1\\-\frac{K}{M}&-\frac{C}{M}\end{array}\right)\left(\begin{array}{c}x_1\\x_2\end{array}\right) +\left(\begin{array}{c}0\\\frac{1}{M}\end{array}\right) b(t)$$
and, if you want to learn about the behaviour of $x(t)$ in response to $b(t)$ (i.e. think of $x(t)$ as the output, $b(t)$ as the input), you define your output equation as:
$$y = (1,\,0)\left(\begin{array}{c}x_1\\x_2\end{array}\right)$$
So now, in the standard control theory notation of your first equation:
$$\dot{\vec{x}} = \mathbf{A} \vec{x} + \mathbf{B}\,\vec{u};\quad\vec{y} = \mathbf{C}\vec{x}$$
you can now make the following identifications:
$$\mathbf{A} =\left(\begin{array}{cc}0&1\\-\frac{K}{M}&-\frac{C}{M}\end{array}\right)$$
$$\mathbf{B} = \left(\begin{array}{c}0\\\frac{1}{M}\end{array}\right)$$
$$\mathbf{C} = (1,\,0)$$
$$y(t) = x_1(t);\quad u(t) = b(t)$$
and now you should be set to go. The Laplace transform transfer function is then, as you say, $\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}$, which, for your single input, single output (SISO) system, is a simple scalar transfer function. The frequency response, naturally, is given by putting $s=i\,\omega$, where $\omega$ is the natural frequency. The system's natural frequencies are the eigenvalues of $\mathbf{A}$.
