Question

Beat phenomenon is physically meaningful, only if
A)$\left| {{\omega _1} - {\omega _2}} \right| \gg \left| {{\omega _1} + {\omega _2}} \right|$
B) $\left| {{\omega _1} - {\omega _2}} \right| \ll \left| {{\omega _1} + {\omega _2}} \right|$
C)$\dfrac{{{\omega _1}}}{{{\omega _2}}} \leqslant 17$
D)$\left| {{\omega _1} + {\omega _2}} \right| \gg \dfrac{{{\omega _1}}}{{{\omega _2}}}$

Answer 1

Hint: Superimposition of two waves having comparable frequency lead to periodic changes in the amplitude of the final wave this phenomenon is called Beats. We take two progressive waves and apply trigonometric formulas to solve the above question.

Complete answer:
The beat phenomenon is physically meaningful, only if superimposition of 2 progressive waves having a comparable frequency.
Let's consider 2 waves
${y_1} = A\sin {\omega _1}t$
${y_2} = A\sin {\omega _2}t$
By adding these two waves we get
Y=y1+y2
${y_1} + {y_2} = A\left( {\sin {\omega _1}t + \sin {\omega _2}t} \right)$
$\sin x + \sin y = 2\sin \dfrac{{x + y}}{2}\cos \dfrac{{x - y}}{2}$ applying this trigonometry formula to the above equation we get.
$y = 2A\sin \left( {\dfrac{{{\omega _1} + {\omega _2}}}{2}} \right)t\cos \left( {\dfrac{{{\omega _1} - {\omega _2}}}{2}} \right)t$
We take,
$\omega ' = \dfrac{{{\omega _1} + {\omega _2}}}{2}$
$A' = 2A\cos \left( {\dfrac{{{\omega _1} - {\omega _2}}}{2}} \right)t$
So we can write it as
$y = A'\sin \omega 't$
beat to be meaningful when we have $\left| {{\omega _1} - {\omega _2}} \right| \gg \left| {{\omega _1} + {\omega _2}} \right|$

Hence, option A is the correct answer.

Additional information:
A wave is a physical phenomenon that is characterized by its frequency, wavelength, and amplitude.
Wave can transport energy without transporting matter, which means the wave can move energy from one place to another without transferring matter.
If constructive interference occurs between two crests of a wave, then a loud sound is heard.
The beat is nothing but a sound effect, there is an interference between two sounds but a slight variation in frequency.
A normal and healthy human body should have a resonant power with a natural frequency is 65M HZ to 75M HZ.
Students have to remember all the trigonometric formulas, it would be better for getting more marks in the examination.
In this phenomenon superimposition, the meaning is the placement of one wave over another.

Note:
In the resultant wave there are changes of amplitude when two progressive waves superimposed. With the help of considering 2 waves, we can analyze the physical significance of the beat phenomena.