Question

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

Answer 1

Amplitude`"_(10cm)=4/sqrt2=2.8cm`, rounded to first place of decimal.

Explanation:

For simplicity 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.

Let the energy of the ripple be `E_(cen)`.
At a distance of `5cm` from the point of creation the amplitude is `4cm`
The energy has been distributed in a circle of radius `5cm` or of circumference`=2pixx5cm`

Hence per unit energy, `E_(5cm)=E_(cen)/(2pixx5)`
When the ripple reaches a distance of `10cm` from the point of creation; similarly,
`E_(10cm)=E_(cen)/(2pixx10)`
Dividing the two expressions we obtain
`E_(10cm)=E_(5cm)/2`
We observe that per unit energy is reduced by `1/2`

We know that energy is directly proportional to the square of the Amplitude of the ripple.

`EpropA^2`

Since damping is to be ignored and assuming ripple to be perfect, the amplitude should decrease by a factor of `1/sqrt 2`
Hence requisite amplitude `=4/sqrt2=2.8cm`, rounded to first place of decimal.