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Klein-Gordon equation is ${鈭�}^{2}u=\left(\frac{1}{v^2}\right)\frac{{鈭�}^{2}u}{鈭�t^2}+{位}^{2}u$.

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.

${鈭�}^{2}u=\left(\frac{1}{c^2}\right)\frac{{鈭�}^{2}u}{鈭�t^2}+\left(\frac{m^2{c}^2}{{\mathrm{鈩�}}^2}\right)u$

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

Use the above given terms to write the dispersion relation.

鈥�.. (1)

$-{\left|k\right|}^2+\frac{{蝇}^2}{c^2}=\frac{m^2{c}^2}{{\mathrm{鈩�}}^2}$

Write the givenKlein-Gordon equation as below.

${鈭�}^{2}u=\left(\frac{1}{v^2}\right)\frac{{鈭�}^{2}u}{鈭�t^2}+{位}^{2}u$

Compare the above equation with equation (1).

$$\begin{gathered} {位}^2=\frac{m^2{c}^2}{{\mathrm{鈩�}}^2} \\ c=v \end{gathered}$$

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

$-{\left|k\right|}^2+\frac{{蝇}^2}{v^2}={位}^2$

$v^2\left({位}^2+k^2\right)={蝇}^2---\left(2\right)$

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

$$\begin{gathered} 蝇=2蠀蟺 \\ k=\frac{n蟺}{l} \end{gathered}$$

Put the values.

$v{\left(2蠀蟺\right)}^2=v^2\left({位}^2+k^2\right)$

$$\begin{gathered} {蠀}^2=\frac{v^2}{4蟺}\left({位}^2+{\left(\frac{n蟺}{l}\right)}^2\right) \\ =\frac{v^2{蟺}^2}{4{蟺}^2}\left({\left(\frac{位}{蟺}\right)}^2+{\left(\frac{n}{l}\right)}^2\right) \\ 蠀=\frac{v}{2}\sqrt{\left({\left(\frac{位}{蟺}\right)}^2+{\left(\frac{n}{l}\right)}^2\right)} \end{gathered}$$

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