Question

If rø = sin(pr) sin(pct) where p and c are constants, prove ​

Answer 1

Solution:

Given: r∅ = sin(pr) sin(pct)

Here, p and c are constants.

To prove: $\tt{ \dfrac{\partial^{2}\phi}{\partial\,r^{2}} + \dfrac{2}{r}\,\dfrac{\partial\,\phi}{\partial\,r} = \dfrac{1}{c^{2}}\,\dfrac{\partial^{2}\phi}{\partial\,t^{2}} }$

So, r∅ = sin(pr) sin(pct)       ---------[1]

On partial-differentiating both sides in equation [1] w.r.t. r (t will be constant), we get

$\tt{ r\dfrac{\partial\,\phi}{\partial\,r} + \phi = p\cos(pr)\,\sin(pct) }$        ---------[2]

On partial-differentiating both sides in equation [2] w.r.t. r (t will be constant), we get

$\tt{ r\dfrac{\partial^{2}\phi}{\partial\,r^{2}} + \dfrac{\partial\,\phi}{\partial\,r} + \dfrac{\partial\,\phi}{\partial\,r} = -p^{2}\sin(pr)\,\sin(pct) }$

$\tt{ \implies r\dfrac{\partial^{2}\phi}{\partial\,r^{2}} + 2\dfrac{\partial\,\phi}{\partial\,r} = -p^{2}\sin(pr)\,\sin(pct) }$

$\tt{ \implies r\dfrac{\partial^{2}\phi}{\partial\,r^{2}} + 2\dfrac{\partial\,\phi}{\partial\,r} = -p^{2}\,r\,\phi }$       ---------[3]

Now, on partial-differentiating both sides in equation [1] w.r.t. t (r will be constant), we get

$\tt{ r\dfrac{\partial\,\phi}{\partial\,t} = pc\sin(pr)\,\cos(pct) }$        ---------[4]

On partial-differentiating both sides in equation [4] w.r.t. t (r will be constant), we get

$\tt{ r\dfrac{\partial^{2}\phi}{\partial\,t^{2}} = -p^{2}\,c^{2}\sin(pr)\,\sin(pct) }$

$\tt{ \implies r\dfrac{\partial^{2}\phi}{\partial\,t^{2}} = -p^{2}\,c^{2}\,r\,\phi }$

$\tt{ \implies \dfrac{r}{c^{2}}\,\dfrac{\partial^{2}\phi}{\partial\,t^{2}} = -p^{2}\,r\,\phi }$        ---------[5]

Since, R.H.S. of equations [3] and [5] is equal.

$\tt{ \implies r\dfrac{\partial^{2}\phi}{\partial\,r^{2}} + 2\dfrac{\partial\,\phi}{\partial\,r} = \dfrac{r}{c^{2}}\,\dfrac{\partial^{2}\phi}{\partial\,t^{2}} }$

$\tt{ \implies \dfrac{\partial^{2}\phi}{\partial\,r^{2}} + \dfrac{2}{r}\,\dfrac{\partial\,\phi}{\partial\,r} = \dfrac{1}{c^{2}}\,\dfrac{\partial^{2}\phi}{\partial\,t^{2}} }$

Hence proved

Formulas used :-

1) $\tt{ \dfrac{\partial}{\partial\,x}\{f(x)\,\times\,g(x)\} = f(x)\dfrac{\partial}{\partial\,x}\{g(x)\} \ +\ g(x)\dfrac{\partial}{\partial\,x}\{f(x)\} }$

2) $\tt{ \dfrac{\partial}{\partial\,x}(\sin x) = \cos x }$

3) $\tt{ \dfrac{\partial}{\partial\,x}(\cos x) = - \sin x }$