The Klein-Gordon equation is ${\mathbf{\nabla }}^{\mathbf{2}}\mathit{u}\mathbf{=}\left(1/V^2\right){\mathbf{\partial }}^{\mathbf{2}}\mathit{u}\mathbf{/}\mathbf{\partial }{\mathit{t}}^{\mathbf{2}}\mathbf{+}{\mathit{\lambda }}^{\mathbf{2}}\mathit{u}$. This equation is of interest in quantum mechanics, but it also has a simpler application. It describes, for example, the vibration of a stretched string which is embedded in an elastic medium. Separate the one-dimensional Klein-Gordon equation and find the characteristic frequencies of such a string.

Klein-Gordon equation is ${\nabla }^{2}u=\left(\frac{1}{v^2}\right)\frac{{\partial }^{2}u}{\partial t^2}+{\lambda }^{2}u$.

Any disturbance or energy transfer from one location to another is referred to as a wave. Each wave is guided by a mathematical formula. A wave can be either standing or stationary.

Consider the Klein Gordon equation.

${\nabla }^{2}u=\left(\frac{1}{c^2}\right)\frac{{\partial }^{2}u}{\partial t^2}+\left(\frac{m^2{c}^2}{{\hslash }^2}\right)u$

The above equation of the form $w\left(r,t\right)=e^{i\left(kr-\omega t\right)}$ where$\omega \in R$ and $k\in R^3$.

Use the above given terms to write the dispersion relation.

….. (1)

$-{\left|k\right|}^2+\frac{{\omega }^2}{c^2}=\frac{m^2{c}^2}{{\hslash }^2}$

Write the givenKlein-Gordon equation as below.

${\nabla }^{2}u=\left(\frac{1}{v^2}\right)\frac{{\partial }^{2}u}{\partial t^2}+{\lambda }^{2}u$

Compare the above equation with equation (1).

$$\begin{gathered} {\lambda }^2=\frac{m^2{c}^2}{{\hslash }^2} \\ c=v \end{gathered}$$

Put the value of ${\lambda }^2$ and c in equation (1).

$-{\left|k\right|}^2+\frac{{\omega }^2}{v^2}={\lambda }^2$

$v^2\left({\lambda }^2+k^2\right)={\omega }^2---\left(2\right)$

Use the following values and put them in equation (2).

$$\begin{gathered} \omega =2\upsilon \pi \\ k=\frac{n\pi }{l} \end{gathered}$$

Put the values.

$v{\left(2\upsilon \pi \right)}^2=v^2\left({\lambda }^2+k^2\right)$

$$\begin{gathered} {\upsilon }^2=\frac{v^2}{4\pi }\left({\lambda }^2+{\left(\frac{n\pi }{l}\right)}^2\right) \\ =\frac{v^2{\pi }^2}{4{\pi }^2}\left({\left(\frac{\lambda }{\pi }\right)}^2+{\left(\frac{n}{l}\right)}^2\right) \\ \upsilon =\frac{v}{2}\sqrt{\left({\left(\frac{\lambda }{\pi }\right)}^2+{\left(\frac{n}{l}\right)}^2\right)} \end{gathered}$$

Hence the characteristic frequencies of a string is$\upsilon =\frac{v}{2}\sqrt{\left({\left(\frac{\lambda }{\pi }\right)}^2+{\left(\frac{n}{l}\right)}^2\right)}$