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Find the number of times ${y_1} + {y_2} = 0$ at $x = 0$ in 1sec.
A.100
B.46
C.192
D.96

seo-qna
Last updated date: 13th Jun 2024
Total views: 393k
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Answer
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Hint: We know that the phase of a wave specifies the location of a point within a wave cycle of a repetitive waveform. Here, we use the expression of displacement y of a cosine wave having direction of propagation, phase. In this relation, we put the observed values from the given figure and get the required result.
Formula used:
$y = A\cos \left[ {kx - \omega t} \right]$

Complete answer:
Here, we have two wave equations, where ${y_1},{y_2}$ are representing the wave displacement, A is the amplitude of the wave, k is the wave vector, x is position, t is the time.
$y = A\cos \left[ {kx - \omega t} \right]$
Now we will use the above relation and put the given values, that are x=0 at t=1 sec.
$\eqalign{
  & {y_1} = A\cos \left[ {0.5\pi \times 0 + 100\pi } \right] \cr
  & \Rightarrow {y_1} = A\cos 100\pi \cr} $
For second wave we have:
$\eqalign{& {y_2} = A\cos \left[ {0.46\pi \times 0 - 92\pi } \right] \cr
  & \Rightarrow {y_1} = A\cos ( - 92\pi ) \cr
  & \Rightarrow {y_1} = A\cos 92\pi \cr} $
We have the relation:
${y_1} + {y_2} = 0$
By substituting the values in above equation, we get:
$ \Rightarrow A\left[ {cos100\pi + \cos 92\pi } \right] = 0$
$\therefore A2\sin 96\pi sin4\pi = 0$

Therefore, we get the required result and the correct option is A) i.e., the number of times in given condition is 100.

Additional information:
Waves involve the transfer of energy without the transfer of the matter. So, it can be said that waves can be described as a disturbance that travels through a medium, transporting energy from one location to another location without transfer of matter.
Here, we can see the basic diagram of propagation of a wave in the below diagram.
seo images

Further, the frequency is defined as the number of waves that pass a fixed point in unit time. It can also be defined as the number of cycles or vibrations undergone during one unit of time.
Two waves are said to be coherent if they are moving with the same frequency and have constant phase difference.
The summation or adding or subtraction of all the waves travelling in a particular medium, gives us the superposition of waves. If the direction or amplitude of the waves are opposite then the superposition of waves are calculated by subtracting the waves, whereas if the two waves are travelling in the same direction or have same amplitude the resultant is given by adding up the two or more waves.
The S.I unit of frequency is Hertz or Hz and the unit of wavelength is meter or m. Furthermore we also know the S.I unit of time which is given by second or s.

Note:
Superposition is calculated for two or more waves. The direction or amplitude of these waves will help in finding the resulting wave or the superposition of waves. A wave which has the same amplitude but opposite orientation will cancel out each other and thereby give zero output.