Question

Assume two identical sinusoidal waves are moving through the same medium in the same direction. Under what condition will the amplitude of the resultant wave be greater than either of the two original waves?(a) in all cases(b) only if the waves have no difference in phase (c) only if the phase difference is less than 90°(d) only if the phase difference is less than 120° (e) only if the phase difference is less than 180°

Answer 1

The correct options are B and C

Let us assume the amplitude of the two sinusoidal waves be A1 and A2

As it is mentioned that the waves are identical, therefore A1=A2= a

The resultant wave of any two waves is given by the formula:

Resultant wave= $\sqrt{A1^2+A2^2+2A1A2cos\alpha }$

where A1 and A2 are amplitudes of waves 1 and 2 and $\alpha$ is the angle between the waves

Now as per the formula, it is the the factor 2cos$\alpha$ that will decide that amplitude is greater than original or not

Hence we know that maximum value of cos $\alpha$ is when $\alpha$ is 0 degrees

Now it is known that value of cos$\alpha$ varies from 1 to 0 for o degrees to π/2

and the value varies from 0 to -1 for angles π/2 to π

Case 1: when $\alpha$=0 degree

           Resultant wave R = $\sqrt{A1^2+A2^2+2cos0A1A2}= \sqrt{2A^2+2A^2}=\sqrt{4A^2} =2A$

Case 2: When $\alpha$ is less than 90

let us consider $\alpha = 45 degree$

 Resultant wave R =

$\sqrt{A1^2+A2^2+2cos45A1A2}= \sqrt{2A^2+2A^2/2}=\sqrt{3A^2} =\sqrt{3} A$

case 3:

When $\alpha$ is less than 120

$\sqrt{A1^2+A2^2+2cos120A1A2}= \sqrt{2A^2-2A^2/2}=A$

case 4: when $\alpha$ is less than 180 degrees

the value will be 0

Hence only option b and d are correct

For further reference:

brainly.com/question/14286615?referrer=searchResults

#SPJ4