Question

A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest a total distance of 0.62m. The fisherman sees that the wave crests are spaced 6.0 m apart. What would be the equation of such a wave?

• A. $y=0.31sin(\frac{2\pi }{5}t-\frac{\pi }{3}x)$
• B. $y=0.62sin(\frac{2\pi }{7}t-\frac{2\pi }{5}x)$
• C. $y=6sin(2.5t-0.62x)$
• D. $y=2.5sin(0.62t+2.5x)$

Answer 1

The correct option is A

$y=0.31sin(\frac{2\pi }{5}t-\frac{\pi }{3}x)$

(Note:- This is a snapshot of wave the boat will oscillate periodically (up/down) at one fixed x coordinate and the wave will move below it)
Distance between a crest and consecutive through is$\frac{\lambda }{2}$ time taken for wave to travel $\frac{\lambda }{2}$ is half of a time period = $\frac{T}{2}$

So given $\frac{T}{2}$= 2.5 sec

So given $T=5sec=\frac{2\pi }{\omega }$

$\Rightarrow \omega =\frac{2\pi }{5}$

$2A=0.62$

$A=0.31$ m

$\lambda =6m$(distance between 2 consecutive crests $k=\frac{2\pi }{\lambda }=\frac{2\pi }{6}=\frac{\pi }{3}$ general equation of wave

$y=Asin(\omega t-kx)$

$y=0.31sin(\frac{2\pi }{5}t-\frac{\pi }{3}x)$