Question

Two waves represented by ${y_1} = a\sin \left( {\omega t} \right)$ and ${y_2} = a\cos \left( {\omega t} \right)$ have a phase difference of:

Answer 1

Hint: In this question, we will use the equation of displacement of waves to get the required path difference between the two waves. Further, by substituting the values of each wave, will give us the required answer. Also, we will study the basics of path difference, wave and its propagation for our better understanding.

Formula used:
$\Delta \phi = {\phi _2} - {\phi _1}$

Complete step-by-step answer:
As we know that path difference can be defined as the difference of the distance between two waves.
Here, in the given question the two waves are represented by:
${y_1} = a\sin \left( {\omega t} \right)$
${y_2} = a\cos \left( {\omega t} \right)$
$ \Rightarrow {y_2} = a\sin \left( {\omega t + \dfrac{\pi }{2}} \right)$
Now, using this we have the two wave equations as:
$\eqalign{& {\phi _2} = \left( {\omega t + \dfrac{\pi }{2}} \right) \cr
  & \Rightarrow {\phi _1} = (\omega t) \cr} $
Also, we know that:
$\Delta \phi = {\phi _2} - {\phi _1}$
Now, by substituting the value of two wave equations, in the above equation, we get:
$\eqalign{& \Delta \phi = \left( {\omega t + \dfrac{\pi }{2}} \right) - (\omega t) \cr
  & \therefore \Delta \phi = \dfrac{\pi }{2} \cr} $
Therefore, we get the required path difference of the two waves.

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.
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.
Phase of a wave specifies the location of a point within a wave cycle of a repetitive waveform. Generally, the phase differences between two or more sound waves are important.

Note: We should remember that two waves are said to be coherent only if they are moving with the same frequency and have constant phase difference. Also, a sine wave starts from zero, whereas the cosine wave starts from one. A wave which has the same amplitude but opposite orientation will cancel out each other and thereby give zero output.