
In a standing wave formed as a result of reflection from a surface, the ratio of the amplitude at an antinode to that at node is x. The fraction of energy that is reflected is:
A. \[{\left[ {\dfrac{{x - 1}}{x}} \right]^2}\]
B. \[{\left[ {\dfrac{x}{{x + 1}}} \right]^2}\]
C. \[{\left[ {\dfrac{{x - 1}}{{x + 1}}} \right]^2}\]
D. \[{\left[ {\dfrac{1}{x}} \right]^2}\]
Answer
139.5k+ views
Hint: When two waves interfere then a stationary or standing wave is formed. In the question relation between the amplitude and x is given. As we know that the energy transported by any wave is directly proportional to the square of the amplitude. By using this concept, we can easily find the value for energy reflected.
Complete answer:
It is given that the ratio of the amplitude at an antinode to that at node is x.
\[\dfrac{{{A_i} + {A_r}}}{{{A_i} - {A_r}}} = x\]
Where \[{A_i}\] is the amplitude of incident wave and \[{A_r}\] is the amplitude of reflected waves.
By applying componendo and dividendo on both the sides, we get
\[\dfrac{{{A_r}}}{{{A_i}}} = \dfrac{{x - 1}}{{x + 1}}\]
As we know that energy that is reflected is directly proportional to the square of the amplitude.
\[E \propto {A^2}\]
\[\dfrac{{{E_r}}}{{{E_i}}} = {\left( {\dfrac{{{A_r}}}{{{A_i}}}} \right)^2} = {\left( {\dfrac{{x - 1}}{{x + 1}}} \right)^2}\]
Or \[\dfrac{{{E_r}}}{{{E_i}}} = {\left( {\dfrac{{x - 1}}{{x + 1}}} \right)^2}\]
Therefore, the fraction of energy that is reflected is \[{\left( {\dfrac{{x - 1}}{{x + 1}}} \right)^2}\]
Hence option C is the correct answer
Note: The energy (E) transported by a wave is directly proportional to the square of the amplitude (A) that is \[E \propto {A^2}\] . So whenever change occurs in the amplitude the square of that effect impacts the energy. This means that a doubling of the amplitude results in a quadrupling of the energy. The amplitude of a wave is defined as the distance from the centre lines to the top of a crest to the bottom of a trough.
Complete answer:
It is given that the ratio of the amplitude at an antinode to that at node is x.
\[\dfrac{{{A_i} + {A_r}}}{{{A_i} - {A_r}}} = x\]
Where \[{A_i}\] is the amplitude of incident wave and \[{A_r}\] is the amplitude of reflected waves.
By applying componendo and dividendo on both the sides, we get
\[\dfrac{{{A_r}}}{{{A_i}}} = \dfrac{{x - 1}}{{x + 1}}\]
As we know that energy that is reflected is directly proportional to the square of the amplitude.
\[E \propto {A^2}\]
\[\dfrac{{{E_r}}}{{{E_i}}} = {\left( {\dfrac{{{A_r}}}{{{A_i}}}} \right)^2} = {\left( {\dfrac{{x - 1}}{{x + 1}}} \right)^2}\]
Or \[\dfrac{{{E_r}}}{{{E_i}}} = {\left( {\dfrac{{x - 1}}{{x + 1}}} \right)^2}\]
Therefore, the fraction of energy that is reflected is \[{\left( {\dfrac{{x - 1}}{{x + 1}}} \right)^2}\]
Hence option C is the correct answer
Note: The energy (E) transported by a wave is directly proportional to the square of the amplitude (A) that is \[E \propto {A^2}\] . So whenever change occurs in the amplitude the square of that effect impacts the energy. This means that a doubling of the amplitude results in a quadrupling of the energy. The amplitude of a wave is defined as the distance from the centre lines to the top of a crest to the bottom of a trough.
Recently Updated Pages
How to find Oxidation Number - Important Concepts for JEE

How Electromagnetic Waves are Formed - Important Concepts for JEE

Electrical Resistance - Important Concepts and Tips for JEE

Average Atomic Mass - Important Concepts and Tips for JEE

Chemical Equation - Important Concepts and Tips for JEE

Concept of CP and CV of Gas - Important Concepts and Tips for JEE

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics

Degree of Dissociation and Its Formula With Solved Example for JEE

Other Pages
Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions for Class 11 Physics Chapter 1 Units and Measurements

Motion in a Straight Line Class 11 Notes: CBSE Physics Chapter 2

Important Questions for CBSE Class 11 Physics Chapter 1 - Units and Measurement

NCERT Solutions for Class 11 Physics Chapter 2 Motion In A Straight Line
