Copyright

Suppose two linear waves of equal amplitude and frequency have a phase difference \phi as they...

Question:

Suppose two linear waves of equal amplitude and frequency have a phase difference {eq}\phi {/eq} as they travel in the same medium. They can be represented by:

{eq}D_1 = A \sin (kx - \omega t) {/eq}

{eq}D_2 = A \sin (kx - \omega t + \phi) {/eq}

(a) Use trigonometric identity {eq}\sin\theta_1 + \sin\theta_2=2\cdot \sin (\frac{1}{2}(\theta_1 + \theta_2)) \cdot \cos (\frac{1}{2}(\theta_1 - \theta_2)) {/eq} to show that the resultant wave is given by:

{eq}D = 2\cdot A\cdot \cos \frac{\phi}{2}\cdot \sin (kx - \omega t + \frac{\phi}{2}) {/eq}

(b) What is the amplitude of this resultant wave? Is the wave purely sinusoidal, or not?

(c) Show that constructive interference occurs if {eq}\phi = 0, 2\pi, 4\pi, {/eq} and so on. Show that destructive interference occurs if {eq}\phi = \pi, 3\pi, 5\pi, {/eq} and so on.

(d) Describe the resultant wave, by equation and in words, if {eq}\phi = \frac{\pi}{2}. {/eq}

Superposition of Waves:

The principle of superposition of waves states that if there 2 or more waves traveling in a region of space, then the effect of each wave at any point in space and time, simply add up like vectors.

{eq}y=y_1+y_2+y_3... {/eq}

Here,

  • {eq}y_1 {/eq} is the displacement caused by wave 1.
  • {eq}y_2 {/eq} is the displacement caused by wave 2.
  • {eq}y_3 {/eq} is the displacement caused by wave 3 and so on.

Answer and Explanation:

The equations of individual waves are:

{eq}d_1=A\sin\left ( kx-\omega t \right )\\ d_2=A\sin\left ( kx-\omega t+\phi \right ) {/eq}

According to the principle of superposition, the net displacement due to two or more waves is equal to the vector sum of the individual displacements caused by the waves.

{eq}d=d_1+d_2 {/eq}

Here,

  • {eq}d_1 {/eq} is the displacement of the first wave.
  • {eq}d_2 {/eq} is the displacement of the second wave.


Therefore,

{eq}\begin{align*} d&=d_1+d_2\\ &=A\sin\left ( kx-\omega t \right )+A\sin\left ( kx-\omega t+\phi \right )\\ &=A\left ( \sin\left ( kx-\omega t \right )+\sin\left ( kx-\omega t+\phi \right ) \right )\\ &=A\left (2\sin\left (\dfrac{ kx-\omega t+kx-\omega t+\phi}{2} \right )\cos\left (\dfrac{ kx-\omega t-kx+\omega t-\phi}{2} \right ) \right )\\ &=2A\sin\left ( kx-\omega t+\dfrac{\phi}{2} \right )\cos\left (\dfrac{ -\phi}{2} \right ) \\ &=2A\sin\left ( kx-\omega t+\dfrac{\phi}{2} \right )\cos\left (\dfrac{ \phi}{2} \right ) \end{align*} {/eq}


b)

The amplitude of the resultant wave is:

{eq}A'=2A\cos\left (\dfrac{ \phi}{2} \right ) {/eq}

In terms of new amplitude, the resultant wave becomes:

{eq}d=A'\sin\left ( kx-\omega t+\dfrac{\phi}{2} \right ) {/eq}

which is purely sinusoidal.


c)

Constructive interference occurs when the amplitude of the resultant wave is maximum i.e

{eq}A'=2A\cos\left (\dfrac{ \phi}{2} \right ) {/eq}

is maximum.

This occurs when {eq}\cos\left (\dfrac{ \phi}{2} \right ) {/eq} is maximum.

The maximum values of {eq}\cos\,\theta {/eq} occurs when {eq}\theta=0,\pi,2\pi,3\pi... {/eq}

Therefore,

{eq}\phi/2=0,\pi,2\pi,3\pi...\\ \therefore \phi=0,2\pi,4\pi,6\pi ... {/eq}


Similarly, desctructive interference occurs when {eq}A' {/eq} is minimum. This occurs when {eq}\cos\left (\dfrac{ \phi}{2} \right ) {/eq} is zero.

Since {eq}\cos\,\theta=0 {/eq} for {eq}\theta=\pi/2,3\pi/2,5\pi/2...\\ \therefore \phi/2=\pi/2,3\pi/2,5\pi/2...\\ \therefore \phi=\pi, 3\pi, 5\pi {/eq}


d)

For {eq}\phi=\dfrac{\pi}{2} {/eq}, the amplitude of the resultant wave becomes:

{eq}A'=2A\cos\left (\dfrac{ \pi}{4} \right ) =\dfrac{A}{\sqrt 2} {/eq}

The equation of the resultant wave becomes:

{eq}d=\dfrac{A}{\sqrt 2}\sin\left ( kx-\omega t+\dfrac{ \pi}{4} \right ) {/eq}

Therefore, the resultant wave has an amplitude lower than either of the original waves. The angular frequency of the resultant wave is the same as that of the original waves.



Learn more about this topic:

Loading...
The Resultant Amplitude of Two Superposed Waves

from MEGA Physics: Practice & Study Guide

Chapter 15 / Lesson 10
23K

Related to this Question

Explore our homework questions and answers library