Waves - Result Question 18
18. The equation of a travelling wave is
$y=60 \cos (180 t-6 x)$
where $y$ is in microns, $t$ in second and $x$ in metres. The ratio of maximum particle velocity
to velocity of wave propagation is [1997]
(a) 3.6
(b) $3.6 \times 10^{-4}$
(c) $3.6 \times 10^{-6}$
(d) $3.6 \times 10^{-11}$
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Answer:
Correct Answer: 18. (b)
Solution:
- (b) $y=60 \cos (180 t-6 x)$ $\omega=180, k=6 \Rightarrow \frac{2 \pi}{\lambda}=6$
Wave velocity,
$v=\frac{\omega}{k}=\frac{2 \pi}{T} \times \frac{\lambda}{2 \pi}=\frac{180}{6}=30 m / s$
Differentiating (1) w.r.t. $t$,
Particle velocity,
$v_p=\frac{d y}{d t}=-60 \times 180 \sin (180 t-6 x)$
$v _{p \text{ max }}=60 \times 180 \mu m / s$
$=10800 \mu m / s=0.0108 m / s$
$\frac{v _{p \text{ max }}}{v}=\frac{0.0108}{30}=3.6 \times 10^{-4}$
Wave velocity $(v)$ is the distance travelled by the disturbance in one second. It only depends on the properties of the medium and is independent of time and position.
$ V=n \lambda=\frac{\lambda}{T}=\frac{\omega \lambda}{2 \pi}=\frac{\omega}{k} $
