A point isotropic sonic source of sound power 1 milliwatts emits sound of frequency in all directions.
A. Find the distance of a point from the source where the loudness level is . [Take v]
B. It is known that at some moment , the displacement of air particles at a certain point away from the source is . The amplitude of oscillation at this point is known to be . Find the displacement of air particles in terms of , at a point away from the source at the moment . Consider displacement of particles to be positive if it is away from the source.
Answer
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Hint: Power and intensity of a sound wave are related as . The loudness of sound is .
To calculate the displacement, use the general displacement formula .
Where, is the maximum displacement of air particles or amplitude.
is the initial phase
is the wavelength of the sound wave and it is defined as
Complete step by step solution:
(A). It is given that,
Power of the sound source is .
Frequency of the sound, .
Velocity of the sound,
Loudness level
We know that the threshold Intensity of sound for human hearing is .
Also, .
Substitute the values of and in the above formula. We got
Further simplifying,
Further calculating the logarithmic values, we got
We know that the intensity of sound is related to its power and it is given as,
.
Where, is the distance of a point from the source.
Substitute all the values of and in the above formula of intensity.
Further calculating, we got
Hence, the distance of the point from the source is .
(B) Let’s consider the displacement of air particles at a distance from the source at any time is given as,
…… (1)
Where,
is the maximum displacement of air particles or amplitude.
is the initial phase
is the wavelength of the sound wave and it is given by
Substitute the required values in the above formula
So,
It is given that for and , the displacement is and amplitude
From equation (1) we got,
Further calculating for ,
For a distance at and time at , the displacement of the particle is given as,
Further calculating
Or
Since it is asked to consider the displacement as positive.
Hence, the displacement of air particles is at a distance from the source.
Note:
The lowest possible sound that the typical human ear can detect has an intensity of . Hence, the loudness at this intensity is . The particles in air vibrate parallel to the direction of propagation of sound waves. So, the sound wave is known as longitudinal wave.
To calculate the displacement, use the general displacement formula
Where,
Complete step by step solution:
(A). It is given that,
Power of the sound source is
Frequency of the sound,
Velocity of the sound,
Loudness level
We know that the threshold Intensity of sound for human hearing is
Also,
Substitute the values of
Further simplifying,
Further calculating the logarithmic values, we got
We know that the intensity of sound is related to its power and it is given as,
Where,
Substitute all the values of
Further calculating, we got
Hence, the distance of the point from the source is
(B) Let’s consider the displacement of air particles at a distance
Where,
Substitute the required values in the above formula
So,
It is given that for
From equation (1) we got,
Further calculating for
For a distance at
Further calculating
Or
Since it is asked to consider the displacement as positive.
Hence, the displacement of air particles is
Note:
The lowest possible sound that the typical human ear can detect has an intensity of
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