
Two identical sounds \[{S_1}\] and ${S_2}$ reach at a point P in phase. The resultant loudness at point P is $n\,dB$ higher than the loudness of ${S_1}$ . The value of $n$ is
A. 2
B. 4
C. 5
D. 6
Answer
162.9k+ views
Hint: Start with finding the relation between loudness of two sound waves and the resultant loudness of the two sound waves that are reaching at a point P in phase then equate it with the information provided in the question about the resultant loudness. Use the logarithmic formula of resultant loudness and the intensity of the sound waves.
Formula used:
The intensity of sound is,
$I \propto {a^2}$
Where, $I$ is the intensity of the sound wave and $a$ is the amplitude.
Logarithmic formula of resultant loudness is,
$\Delta L = 10{\log _{10}}\left( {\dfrac{{{I_2}}}{{{I_1}}}} \right)$
Where, $I_1$ and $I_2$ is the intensity of sound.
Complete step by step solution:
Let $a$ be the amplitude sound ${S_1}$ and ${S_2}$. Now we know that;
Intensity of the sound is directly proportional to amplitude,
$I \propto {a^2}$
Hence the loudness of the two sound waves will be:
Loudness due to first sound,
${S_1} = {I_1} = K{a^2}$
Loudness due to second sound,
${S_2} = {I_2} = K(2{a^2}) = 4K{a^2} = 4{I_1}$
Resultant loudness will be
$\Delta L = 10{\log _{10}}\left( {\dfrac{{{I_2}}}{{{I_1}}}} \right) \\
\Rightarrow \Delta L= 10{\log _{10}}\left( {\dfrac{{4{I_1}}}{{{I_1}}}} \right)$........(by putting values from above two equation)
By solving, we get;
$\Delta L = 10{\log _{10}}(4) = 10 \times 0.6 = 6$
Here $\Delta L = n$ that is resultant loudness.
Hence the correct answer is option D.
Note: Here all the values of the intensity of the sound waves are in variables if the numerical values provided in the question put the same in the respective equations. Also note that the sound waves reaching at point P here is in phase if it is not then the answer will be different according to the phase.
Formula used:
The intensity of sound is,
$I \propto {a^2}$
Where, $I$ is the intensity of the sound wave and $a$ is the amplitude.
Logarithmic formula of resultant loudness is,
$\Delta L = 10{\log _{10}}\left( {\dfrac{{{I_2}}}{{{I_1}}}} \right)$
Where, $I_1$ and $I_2$ is the intensity of sound.
Complete step by step solution:
Let $a$ be the amplitude sound ${S_1}$ and ${S_2}$. Now we know that;
Intensity of the sound is directly proportional to amplitude,
$I \propto {a^2}$
Hence the loudness of the two sound waves will be:
Loudness due to first sound,
${S_1} = {I_1} = K{a^2}$
Loudness due to second sound,
${S_2} = {I_2} = K(2{a^2}) = 4K{a^2} = 4{I_1}$
Resultant loudness will be
$\Delta L = 10{\log _{10}}\left( {\dfrac{{{I_2}}}{{{I_1}}}} \right) \\
\Rightarrow \Delta L= 10{\log _{10}}\left( {\dfrac{{4{I_1}}}{{{I_1}}}} \right)$........(by putting values from above two equation)
By solving, we get;
$\Delta L = 10{\log _{10}}(4) = 10 \times 0.6 = 6$
Here $\Delta L = n$ that is resultant loudness.
Hence the correct answer is option D.
Note: Here all the values of the intensity of the sound waves are in variables if the numerical values provided in the question put the same in the respective equations. Also note that the sound waves reaching at point P here is in phase if it is not then the answer will be different according to the phase.
Recently Updated Pages
Fluid Pressure - Important Concepts and Tips for JEE

JEE Main 2023 (February 1st Shift 2) Physics Question Paper with Answer Key

Impulse Momentum Theorem Important Concepts and Tips for JEE

Graphical Methods of Vector Addition - Important Concepts for JEE

JEE Main 2022 (July 29th Shift 1) Chemistry Question Paper with Answer Key

JEE Main 2023 (February 1st Shift 1) Physics Question Paper with Answer Key

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

IIIT JEE Main Cutoff 2024

IIT Full Form

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

JEE Main Cut-Off for NIT Kurukshetra: All Important Details

JEE Main Cut-Off for VNIT Nagpur 2025: Check All Rounds Cutoff Ranks

Other Pages
NEET 2025: All Major Changes in Application Process, Pattern and More

Verb Forms Guide: V1, V2, V3, V4, V5 Explained

NEET Total Marks 2025: Important Information and Key Updates

1 Billion in Rupees - Conversion, Solved Examples and FAQs

NEET 2025 Syllabus PDF by NTA (Released)

Important Days In June: What Do You Need To Know
