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:
- 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, $v=\frac{\omega}{k}=\frac{50}{2}=25 m / s$
