Mechanical waves

Is \(u_n(t)\) the voltage across capacitor \(C\) of the cell number \(n\) and \(i_n(t)\) the current in the coil follows (see figure), we can write :

• Kirchhoff's Voltage Law :

  \(u_{n-1}(t)=u_{n}(t)+L\frac{di_{n-1}(t)}{dt}\;\;\;\;and\;\;\;\;u_n(t)=u_{n+1}(t)+L\frac{di_n(t)}{dt}\)

• Kirchhoff's Current Law :

  \(i_{n-1}(t)=i_{n}(t)+C\frac{du_{n}(t)}{dt}\)

The last equation is derived with respect to time :

\(\frac {di_{n-1}(t)}{dt}=\frac{di_{n}(t)}{dt}+C\frac{d^2u_{n}(t)}{dt^2}\)

And used two loop laws :

\(\frac{u_{n-1}(t)-u_n(t)}{L}=\frac{u_{n}(t)-u_{n+1}(t)}{L}+C\frac{d^2u_{n}(t)}{dt^2}\)

Whence :

\(LC\frac{d^2u_{n}(t)}{dt^2}=u_{n-1}(t)+u_{n+1}(t)-2u_{n}(t)\)

We are interested in the propagation of plane waves that we write in the form :

\(u_n(t)=A\;exp(i(\omega t - kna))\)

where \(a\) is the "width" of a cell.

The previous differential equation then becomes :

\(-LC\omega^2=e^{-ika}+e^{ika}-2=2cos(ka)-2\)

Or, finally :

\(\omega=\frac{2}{\sqrt{LC}}sin(\frac{ka}{2})\)

It is requested dispersion relation.

The cut-off frequency is obtained when  \(sin(ka/2)\) is equal to \(1\) :

\(\omega_c=\frac{2}{\sqrt{LC}}\)

It is assumed that the distance between the cells \(a\) is small compared to the wavelength (\(a<<\lambda)\).

We can then define a function \(u(x,t)\) such as (Taylor-Young Development of the second order) :

\(\left\{ \begin{array}{l}u_{n + 1} (t) = u((n+1)a ,t)  = u(na ,t) + \frac{{\partial u}}{{\partial x}}a + \frac{1}{2}\frac{{\partial ^2 u}}{{\partial x^2 }}a^2 \\u_{n - 1} (t) = u((n-1)a,t)=u(na ,t) - \frac{{\partial u}}{{\partial x}}a + \frac{1}{2}\frac{{\partial ^2 u}}{{\partial x^2 }}a^2 \\\end{array} \right.\)

And the propagation equation becomes :

\(LC\frac{{\partial ^2 u(x,t)}}{{\partial t^2 }} = \frac{{\partial ^2 u(x,t)}}{{\partial x^2 }}\;a^2 \)

Is :

\(\frac{{\partial ^2 u(x,t)}}{{\partial x^2 }} - \frac{1}{{c^2 }}\frac{{\partial ^2 u(x,t)}}{{\partial t^2 }} = 0\;\;\;\;\;\;\;\;\;\;with\;\;\;\;\;\;\;\;\;\;c = \frac{{a }}{\sqrt{LC}}\)

We find a d'Alembert equation, characteristic of a propagation without dissipation at the speed \(c\).