Wave Motion 5 Question 7
7. The displacement $y$ of a particle executing periodic motion is given by
$$ y=4 \cos ^{2}\left(\frac{1}{2} t\right) \sin (1000 t) $$
This expression may be considered to be a result of the superposition of independent harmonic motions.
$(1992,2 M)$
(a) two
(b) three
(c) four
(d) five
Passage Based Questions
Passage 1
Two trains $A$ and $B$ are moving with speeds $20 m / s$ and $30 m / s$ respectively in the same direction on the same straight track, with $B$ ahead of $A$. The engines are at the front ends. The engine of train $A$ blows a long whistle.
Assume that the sound of the whistle is composed of components varying in frequency from $f _1=800 Hz$ to $f _2=1120 Hz$, as shown in the figure. The spread in the frequency (highest frequency-lowest frequency) is thus $320 Hz$. The speed of sound in air is $340 m / s$.
Show Answer
Solution:
- The given equation can be written as
$$ \begin{aligned} y & =2\left(2 \cos ^{2} \frac{t}{2}\right) \sin (1000 t) \\ y & =2(\cos t+1) \sin (1000 t) \\ & =2 \cos t \sin 1000 t+2 \sin (1000 t) \\ & =\sin (1001 t)+\sin (999 t)+2 \sin (1000 t) \end{aligned} $$
i.e. the given expression is a result of superposition of three independent harmonic motions of angular frequencies 999, 1000 and $1001 rad / s$.
