# 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 ?

Feb 28, 2016

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 ${E}_{c e n}$.
At a distance of $5 c m$ from the point of creation the amplitude is $4 c m$
The energy has been distributed in a circle of radius $5 c m$ or of circumference$= 2 \pi \times 5 c m$

Hence per unit energy, ${E}_{5 c m} = {E}_{c e n} / \left(2 \pi \times 5\right)$
When the ripple reaches a distance of $10 c m$ from the point of creation; similarly,
${E}_{10 c m} = {E}_{c e n} / \left(2 \pi \times 10\right)$
Dividing the two expressions we obtain
${E}_{10 c m} = {E}_{5 c m} / 2$
We observe that per unit energy is reduced by $\frac{1}{2}$

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

$E \propto {A}^{2}$

Since damping is to be ignored and assuming ripple to be perfect, the amplitude should decrease by a factor of $\frac{1}{\sqrt{2}}$
Hence requisite amplitude $= \frac{4}{\sqrt{2}} = 2.8 c m$, rounded to first place of decimal.