파동 방정식(wave equation) #4

Pixabay

[모바일에서는 수식이 모두 LaTeX 그대로 나옵니다. 따라서 PC로 보실 것 적극 권장 드립니다.]

 

$$\frac{d^{2}R(r)}{d{r}^2}+\frac{2}{r}\frac{dR(r)}{dr}+\frac{{\omega }^2}{c^2}R(r)-\frac{n(n+1)}{r^2}R(r)=0$$

 

Spherical Bessel function

$$x^2\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}+(x^2-n(n+1))y=0$$

 

And this is the eqaution we derived from spherical wave equation:

 

$$r^2\frac{d^{2}R(r)}{d{r}^2}+2r\frac{dR(r)}{dr}+r^2\frac{{\omega }^2}{c^2}R(r)-n(n+1)R(r)=0$$

 

First, bring physical concept 'wavenumber($\kappa$)'

$$\frac{{\omega }^2}{c^2}={\kappa }^2$$

 

Then,

$$r^2\frac{d^{2}R(r)}{d{r}^2}+2r\frac{dR(r)}{dr}+[{\kappa }^2{r}^2-n(n+1)]R(r)=0\cdots (a)$$

 

Now, let's modify 'spherical Bessel function' to our equation $(a)$

 

Assume $x=\kappa r$

$$\frac{dx}{dr}=\frac{d}{dr}(\kappa r)=\kappa$$

$$dx=\kappa dr$$

 

$${\kappa }^2{r}^2\frac{d^{2}y}{d{x}^2}+2\kappa r\frac{dy}{dx}+(({\kappa }^2{r}^2)-n(n+1))y=0$$

$${\kappa }^2{r}^2\frac{d}{dx}\frac{dy}{dx}+2\kappa r\frac{dy}{dx}+({\kappa }^2{r}^2-n(n+1))y=0$$

 

Before expanding our equation, arrange the first term.

$$\kappa r^2\frac{d}{dr}\frac{dy}{dx}=\kappa r^2\frac{d}{dr}\frac{dy}{\kappa dr}=r^2\frac{d^{2}y}{d{r}^2}$$

 

Then,

$$r^2\frac{d^{2}y}{d{r}^2}+2r\frac{dy}{dr}+({\kappa }^2{r}^2-n(n+1))y=0$$

 

Which is same with equation $(a)$ when we substitue $y$ to $R(r)$

$$r^2\frac{d^{2}R(r)}{d{r}^2}+2r\frac{dR(r)}{dr}+[{\kappa }^2{r}^2-n(n+1)]R(r)=0\cdots (a)$$

 

Conclusion

Solution of equation $(a)$: 

$$j_n(x)=j_n(\kappa r)$$

 

Reference

Wikipedia, Bessel function