Two travelling waves produces a standing wave represented by equation, $y=1.0 \mathrm{~mm} \cos \left(1.57 \mathrm{~cm}^{-1}\right) x \sin \left(78.5 \mathrm{~s}^{-1}\right) t$. The node closest to the origin in the region $x>0$ will be at $x=$\_\_\_\_ cm.

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Waves & Sound

Two travelling waves produces a standing wave represented by equation,

y = 1.0 mm cos (1.57 cm^{- 1}) x sin (78.5 s^{- 1}) t

. The node closest to the origin in the region

x > 0

will be at

x =

____ cm.

Answer: 1

The standing wave equation is given as

y = 1.0 mm cos (1.57 cm^{- 1}) x sin (78.5 s^{- 1}) t

.

For a standing wave of the form

y = A cos (k x) sin (ω t)

, nodes occur where

cos (k x) = 0

.

This happens when

k x = \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, ...

For the first node (closest to origin in

x > 0

k x = \frac{π}{2}

Given

k = 1.57 cm^{- 1} = \frac{π}{2} cm^{- 1}

(since

1.57 \approx \frac{π}{2}

)

x = \frac{π}{2 k} = \frac{π}{2 \times \frac{π}{2}} = 1.0 cm

Therefore, the node closest to the origin is at

x = 1.0

cm.

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Integer answer