Question

Find the intensity ratio of waves if in the interference phenomenon of two waves it is observed that maximum amplitude to the minimum amplitude ratio is $9:7$.

Answer 1

Hint Intensity of the wave at a point can be defined as the amount of sound passing per second normally through unit area at that point. As the ratio of amplitude is given so, intensity is directly proportional to the square of amplitude.

Complete step by step answer:
The phenomenon in which two waves superpose and form a great resultant wave of greater, lower or the same amplitude is known as interference. Constructive and destructive interference occurs when the two waves interact which are coherent with each other because they came back from the same source or they have nearly the same frequency.
As the ratio of amplitude is given in the question, so, let’s see something about amplitude –
The amplitude of the wave can be defined as the displacement of the particle on the medium from its rest position. In other words, it can also be defined as the distance from rest to crest. Similarly, the amplitude can be measured from the rest position to the trough position.
As we have to find the intensity so, let’s look at the intensity –
Intensity of the wave at a point can be defined as the amount of sound passing per second normally through the unit area at that point. It can also be the combination of rate of wave and density of transfer of energy.
Now, we know that intensity is directly proportional to the square of amplitude. Mathematically, it can be represented as –
$\Rightarrow I\alpha A^2$
This can be rewritten as –
$I\alpha {\displaystyle \frac{{\left(A_{max}+A_{min}\right)}^2}{{\left(A_{max}-A_{min}\right)}^2}}$
Neglecting constant value –
$I={\displaystyle \frac{{\left(A_{max}+A_{min}\right)}^2}{{\left(A_{max}-A_{min}\right)}^2}}$
Putting the value of maximum amplitude and minimum amplitude, we get –
$$\begin{gathered} \Rightarrow I={\displaystyle \frac{{\left(9+7\right)}^2}{{\left(9-7\right)}^2}} \\ \Rightarrow I={\displaystyle \frac{{16}^2}{2^2}}={\displaystyle \frac{256}{4}} \\ \therefore I=64 \end{gathered}$$
Hence, we got the value of intensity as $64$.

Note The relation between intensity and amplitude is –
$$\begin{gathered} I={\displaystyle \frac{1}{2}}\tau k\omega A^2 \\ \Rightarrow I\alpha A^2 \end{gathered}$$
The factor ${\displaystyle \frac{1}{2}}\tau k\omega$ is replaced by the proportionality constant $\alpha$. This is done because for waves which are not waves on a string, we end up with factors other than $\tau$ to describe the medium of propagation.