16. (JEE Main 2019 (Online) 12th January Morning Slot)

A travelling harmonic wave is represented by the equation y(x,t) = 10^–3sin (50t + 2x), where, x and y are in mater and t is in seconds. Which of the following is a correct statement about the wave ?

A. The wave is propagating along the positive x-axis with speed 100 ms^–1

B. The wave is propagating along the positive x-axis with speed 25 ms^–1

C. The wave is propagating along the negative x-axis with speed 25 ms^–1

D. The wave is propagating along the negative x-axis with speed 100 ms^–1

Correct Answer is Option (C)

y = a sin( $ω$ t + kx)

$\Rightarrow$ wave is moving along $-$ ve x-axis with speed

v = $\frac{ω}{K}$ $\Rightarrow$ v = $\frac{50}{2}$ = 25m/sec

17. (AIEEE 2011)

The transverse displacement $y (x, t)$ of a wave on a string is given by $y (x, t) = e^{- (a x^{2} + b t^{2} + 2 \sqrt{a b} x t)} .$ This represents $a :$

A. wave moving in $- x$ direction with speed $\sqrt{\frac{b}{a}}$

B. standing wave of frequency $\sqrt{b}$

C. standing wave of frequency $\frac{1}{\sqrt{b}}$

D. wave moving in $+ x$ direction speed $\sqrt{\frac{a}{b}}$

Correct Answer is Option (A)

Given wave equation is

$y (x, t) =_{e} (- a x^{2} + b t^{2} + 2 \sqrt{a b} x t)$

$= e^{- [{(\sqrt{a x})}^{2} + {(\sqrt{b t})}^{2} + 2 \sqrt{a} x . \sqrt{b} t]}$

$= e^{- {(\sqrt{a} x + \sqrt{b} t)}^{2}}$

$= e^{- {(x + \sqrt{\frac{b}{a}} t)}^{2}}$

It is a function of type $y = f (x + v t)$

$\Rightarrow$ Speed of wave $= \sqrt{\frac{b}{a}}$

18. (AIEEE 2008)

A wave travelling along the $x$ -axis is described by the equation $y (x, t) = 0.005$ $\cos (α x - β t) .$ If the wavelength and the time period of the wave are $0.08$ $m$ and $2.0 s$ , respectively, then $α$ and $β$ in appropriate units are

A. $α = 25.00 π, β = π$

B. $α = \frac{0.08}{π}, β = \frac{2.0}{π}$

C. $α = \frac{0.04}{π}, β = \frac{1.0}{π}$

D. $α = 12.50 π, β = \frac{π}{2.0}$

Correct Answer is Option (A)

$y (x, t) = 0.005 \cos (α x - β t)$ (Given)

Comparing it with the standard equation of wave

$y (x, t) = a \cos (k x - ω t)$ we get

$k = α$ and $ω = β$

$∴$ $\frac{2 π}{γ} = α$ and $\frac{2 π}{T} = β$

$∴$ $α = \frac{2 π}{0.08} = 25 π$ and $β = \frac{2 π}{2} = π$

19. ( AIEEE 2004)

The displacement $y$ of a particle in a medium can be expressed as, $y = 10^{- 6} \sin$ $(100 t + 20 x + \frac{π}{4})$ $m$ where $t$ is in second and $x$ in meter. The speed of the wave is

A. 20 m/s

B. 5 m/s

C. 2000 m/s

D. 50 m/s

Correct Answer is Option (B)

From equation given,

$ω = 100$ and $k = 20,$ $v = \frac{ω}{k} = \frac{100}{20} = 5 m / s$

20. (AIEEE 2003)

The displacement $y$ of a wave travelling in the $x$ -direction is given by $x$ is expressed in metres and $t$ in seconds. The speed of the wave - motion, in $m s^{- 1}$ , is :

A. 300

B. 600

C. 200

D. 1200

Correct Answer is Option (A)

$y = 10^{- 4} \sin (600 t - 2 x + \frac{π}{3})$

But $y = A \sin (ω t - k x + ϕ)$

On comparing we get $ω = 600; k = 2$

$v = \frac{ω}{k} = \frac{600}{2} = 300 m s^{- 1}$

Waves