Question

* 48, (I) Two earthquake waves of the same frequency travel

through the same portion of the Earth, but one is carrying
twice the energy. What is the ratio of the amplitudes of
the two waves?

Answer 1

The ratio of the amplitudes of the two waves is
`√(2)` .

Step-by-step explanation:

The intensity of the spherical wave has the formula as:

`I= (P)/(4\pi r^(2) )`

Here, P is power and r is the radius of sphere.

Power is equivalent to the ratio of energy and time and intensity is directly proportional to energy. So,

`(I_(1) )/(I_(2) )` =
`(E_(1) )/(E_(2) )`

Here,
`I_(1)` and
`I_(2)` are the intensities of the two waves,
`E_(1)` and
`E_(2)` are the amplitudes of the two waves

The total energy of the sinusoidal wave can be calculated by using the formula,

`E\\` =
`(1)/(2)`
`K\\`
`A^(2)`

Here, k is wave number and A is the amplitude of wave.

From the above equation, energy is directly proportional to the squared amplitude,

`(E_(1) )/(E_(2) )` =
`((A_(1) )/(A_(2) )) ^(2)`

Since the energy of one earth quake is twice the energy of other earth quake. Thus,

`((A_(1) )/(A_(2) )) ^(2)` = 2

Rearrange the equation for
`(A_(1) )/(A_(2) )`,

`(A_(1) )/(A_(2) )` =
`√(2)`

The ratio of amplitudes is
`√(2)` .