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 $25 ms^{-1}$.

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

(c) The wave is propagating along the positive $X$-axis with speed $100 ms^{-1}$.

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

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Answer:

Correct Answer: 2. (a)

Solution:

  1. 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

$$ \phi=50 t+2 x=\text { constant } $$

Differentiating ’ $\phi$ ’ w.r.t ’ $t$ ‘, we get

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

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

Alternate Method

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

$$ y=a \sin (\omega t+k x) $$

Given equation of wave is

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

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

$$ \omega=50 \text { and } k=2 $$

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



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