파동 방정식(wave equation) #4

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

 

Spherical Bessel function

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

 

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

 

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

 

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

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

 

Then,

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

 

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

 

Assume \(x=\kappa r\)

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

$$dx=\kappa dr$$

 

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

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

 

Before expanding our equation, arrange the first term.

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

 

Then,

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

 

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

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

 

Conclusion

Solution of equation \((a)\): 

$$j_{n}(x)=j_{n}(\kappa r)$$

 

Reference

Wikipedia, Bessel function