Question

The equation of a cylindrical progressive wave is :

• A. $y=asin\omega t$
• B. $y=asin(\omega t-kr)$
• C. $y=\frac{a}{\sqrt{r}}sin(\omega t-kr)$
• D. $y=\frac{a}{r}sin(\omega t-kr)$

Answer 1

Answer: (C)

$y=\frac{a}{\sqrt{r}}sin(\omega t-kr)$

The surface area of cylindrical surface is $2\pi rl$, where r is radius of base and l is length of cylinder.
We know that with increasing distance from source , the total energy or
power transmitted remains same, but intensity decreases. For any source of power P, intensity I at distance r from it will be
I=P/S , where S is surface area.
(i) For spherical wave front $s=4\pi r^2$ (surface)
So $I=\frac{P}{4\pi r^2}\Rightarrow I\propto \frac{1}{r^2}\Rightarrow A\propto \frac{1}{r}$
(ii) For cylindrical wave front $S=2\pi rl$
so

$I=\frac{P}{2\pi rl}\Rightarrow I\propto \frac{1}{r}\Rightarrow A\propto \frac{1}{\sqrt{r}}$
Ans: C