
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
124.2k+ 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
Difference Between Circuit Switching and Packet Switching

Difference Between Mass and Weight

JEE Main Participating Colleges 2024 - A Complete List of Top Colleges

JEE Main Maths Paper Pattern 2025 – Marking, Sections & Tips

Sign up for JEE Main 2025 Live Classes - Vedantu

JEE Main 2025 Helpline Numbers - Center Contact, Phone Number, Address

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

Class 11 JEE Main Physics Mock Test 2025

JEE Main Exam Marking Scheme: Detailed Breakdown of Marks and Negative Marking

JEE Main 2023 January 24 Shift 2 Question Paper with Answer Keys & Solutions

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

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

Other Pages
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

NCERT Solutions for Class 11 Physics Chapter 9 Mechanical Properties of Fluids

Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

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

NCERT Solutions for Class 11 Physics Chapter 7 Gravitation
