# Two waves, identical except for phase, are traveling in the same direction through the same...

## Question:

Two waves, identical except for phase, are traveling in the same direction through the same medium. Each of the waves has amplitude A. The amplitude of the combined (net) wave is {eq}1.14\ A {/eq}. What is the absolute value for the smallest possible phase difference between the two waves in the range 0 to {eq}\pi {/eq}?

## Interference

When two waves having same angular frequency {eq}\omega {/eq}, wave vector {eq}k {/eq} and amplitude A traveling in the same direction interfere each other to give constructive of destructive interference. If the waves are of the form {eq}y_1 = A \sin ( k x - \omega t ) {/eq} and {eq}y_2 = A \sin ( kx - \omega t + \phi ) {/eq}, the resulting wave will be of the form {eq}y = 2 A \cos ( \phi/2 ) \sin ( kx - \omega t + \phi/2 ) {/eq}. In these equations {eq}x, \ \ \phi, \ \ t {/eq} are the distance, initial phase angle, and time respectively. In the resulting wave first part gives the amplitude and the second part is the phase part of the wave. Depending on the value of the phase part the interference will be constructive or destructive.

## Answer and Explanation:

Given data

• Two identical waves except the phase difference interfere each other.
• Individual wave amplitude is A
• Resultant wave amplitude {eq}A_n = 1.14 A {/eq}

Let {eq}k, \ \ \omega, \ \ \phi {/eq} be the wave vector, angular frequency of the waves and phase difference between the waves.

Let the first wave be {eq}y_1 = A \sin ( kx - \omega t ) {/eq}

Second wave {eq}y_2 = A \sin ( kx - \omega t + \phi ) {/eq}

Then the resultant wave {eq}y = 2 A \cos ( \phi/2 ) \sin ( kx - \omega t + \phi/2 ) {/eq}

For constructive interference the phase part will be equal to one.

Then on equating the amplitude part with the amplitude given in the question we get {eq}A_n = 2 A \cos ( \phi/2 ) \\ \implies 1.14 A = 2 A \cos ( \phi/2 ) \\ \implies \dfrac { 1.14} { 2 }= 0.57 = \cos ( \phi/2 ) {/eq}

Therefore the phase difference between the waves {eq}\phi = 2 \cos^{-1} ( 0.57 ) \\ \phi = 110.5 ^o {/eq}