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A ripple is created in water. The amplitude at a distance of 5cm from the point where the sound ripple was created is 4cm. Ignoring damping, what will be the amplitude at the distance of 10cm.

A.$\sqrt {16} $​cm
B.$\sqrt 8 $​cm
C.$\sqrt 4 $​cm
D.$\sqrt 2 $​cm

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Last updated date: 13th Jun 2024
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Answer
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Hint: in order to solve the question, let's assume that the ripple was created in a pool of water at the center so that with time the ripple expands on all sides unhindered. The ripple was created in a pool of water at the center so that with time the ripple moves away.

Complete step by step answer:
Given:
Amplitude = 5cm
The amplitude in the form of Sound ripple = 4cm
Let us consider the energy of the ripple be E.
At a distance of 5 cm from the point of origin the amplitude is 4 cm.
The Energy has been distributed in the form of a circle of radius.
Therefore its circumference of the circle is taken as 2π×5 cm.
The Energy per unit length, ${E_1} = \dfrac{E}{{10\pi }}$​
When the ripple reaches the distance of 10 cm from the point of origin,
 Similarly,
 ${E_2} = \dfrac{E}{{2\pi \times 10}} = \dfrac{{{E_1}}}{2}$​
Now the energy is proportional to square of amplitude (A).
Thus if per unit energy decreases by 1/2, then amplitude will decrease by 1/$\sqrt {2} $.
The New amplitude, A1​=4/$\sqrt 2 $ ​= ​$\sqrt{8}$
Hence the correct option is B.

Note:Energy of sound wave is directly proportional to the amplitude of the sound. If the amplitude of a sound wave is large, then the loudness of sound will be more. If the amplitude is small, then the sound will be feeble.