91影视

Klein-Gordon equation is given as below.

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

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 $w(r,t)=e^{i(kr-蝇t)}$ where $w鈭�R$and $k鈭�R^3$.

Use the above given terms to write the dispersion relation.

$-{\left|k\right|}^2+\frac{{蝇}^2}{c^2}=\frac{m^2{c}^2}{{\mathrm{鈩�}}^2}$ 鈥�.. (1)

Write the givenKlein-Gordon equation is as given 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).

${位}^2=\frac{m^2{c}^2}{{\mathrm{鈩�}}^2}$

Here, the velocity c=v,

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

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

$v^2\left({位}^2+k^2\right)={蝇}^2$

鈥�.. (2)

The lengths of the rectangular membrane is given by,

.$\left(k_1,k_2\right)=\left(\frac{n蟺}{a},\frac{m蟺}{b}\right)$

Use equation (2) to find the frequency.

$$\begin{gathered} {蝇}^2=v^2\left({位}^2+k_1^2+k_2^2\right) \\ (2谓蟺{)}^2=v^2\left({位}^2+k_1^2+k_2^2\right) \\ {谓}^2=\frac{v^2}{4蟺}\left({位}^2+{\left(\frac{n蟺}{a}\right)}^2+{\left(\frac{m蟺}{b}\right)}^2\right) \\ 谓=\frac{v}{2}\sqrt{{\left(\frac{n}{a}\right)}^2+{\left(\frac{m}{b}\right)}^2+{\left(\frac{位}{蟺}\right)}^2} \end{gathered}$$

Hence the frequency for a rectangular membrane is $谓=\frac{v}{2}\sqrt{{\left(\frac{n}{a}\right)}^2+{\left(\frac{m}{b}\right)}^2+{\left(\frac{位}{蟺}\right)}^2}$.