SATHEE: Wave Motion 3 Question 2

Wave Motion 3 Question 2

2. A travelling harmonic wave is represented by the equation $y (x, t) = 10^{- 3} \sin (50 t + 2 x)$ , where $x$ and $y$ are in metre and $t$ is in second. Which of the following is a correct statement about the wave?

(2019 Main, 12 Jan I)

(a) The wave is propagating along the negative $X$ -axis with speed $25 m s^{- 1}$ .

(b) The wave is propagating along the positive $X$ -axis with speed $25 m s^{- 1}$ .

(d) The wave is propagating along the negative $X$ -axis with speed $100 m s^{- 1}$ .

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Answer:

Correct Answer: 2. (a)

Solution:

Wave equation is given by,

$y = 10^{- 3} \sin (50 t + 2 x)$

Speed of wave is obtained by differentiating phase of wave.

Now, phase of wave from given equation is

$ϕ = 50 t + 2 x = constant$

Differentiating ’ $ϕ$ ’ w.r.t ’ $t$ ‘, we get

$\begin{array}{ll} \frac{d}{d x} (50 t + 2 x) = \frac{d}{d t} (constant) \\ \Rightarrow & 50 + 2 (\frac{d x}{d t}) = 0 \\ \Rightarrow & \frac{d x}{d t} = \frac{- 50}{2} = - 25 m s^{- 1} \end{array}$

So, wave is propagating in negative $x$ -direction with a speed of $25 m s^{- 1}$ .

Alternate Method

The general equation of a wave travelling in negative $x$ direction is given as

$y = a \sin (ω t + k x)$

Given equation of wave is

$y = 10^{- 3} \sin (50 + 2 x)$

Comparing Eqs. (i) and (ii), we get

$ω = 50 and k = 2$

Velocity of the wave, $v = \frac{ω}{k} = \frac{50}{2} = 25 m / s$

Wave Motion 3 Question 2

2. A travelling harmonic wave is represented by the equation $y (x, t) = 10^{- 3} \sin (50 t + 2 x)$ , where $x$ and $y$ are in metre and $t$ is in second. Which of the following is a correct statement about the wave?

Answer:

Alternate Method

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Wave Motion 3 Question 2

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

Answer:

Alternate Method

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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2. A travelling harmonic wave is represented by the equation $y (x, t) = 10^{- 3} \sin (50 t + 2 x)$ , where $x$ and $y$ are in metre and $t$ is in second. Which of the following is a correct statement about the wave?