Answer:
The ratio of the amplitudes of the two waves is [tex]\sqrt{2}[/tex] .
Explanation:
The intensity of the spherical wave has the formula as:
[tex]I= \frac{P}{4\pi r^{2} }[/tex]
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,
[tex]\frac{I_{1} }{I_{2} }[/tex] = [tex]\frac{E_{1} }{E_{2} }[/tex]
Here, [tex]I_{1}[/tex] and [tex]I_{2}[/tex] are the intensities of the two waves, [tex]E_{1}[/tex] and [tex]E_{2}[/tex] are the amplitudes of the two waves
The total energy of the sinusoidal wave can be calculated by using the formula,
[tex]E\\[/tex] = [tex]\frac{1}{2}[/tex] [tex]K\\[/tex][tex]A^{2}[/tex]
Here, k is wave number and A is the amplitude of wave.
From the above equation, energy is directly proportional to the squared amplitude,
[tex]\frac{E_{1} }{E_{2} }[/tex] = [tex](\frac{A_{1} }{A_{2} }) ^{2}[/tex]
Since the energy of one earth quake is twice the energy of other earth quake. Thus,
[tex](\frac{A_{1} }{A_{2} }) ^{2}[/tex] = 2
Rearrange the equation for [tex]\frac{A_{1} }{A_{2} }[/tex],
[tex]\frac{A_{1} }{A_{2} }[/tex] = [tex]\sqrt{2}[/tex]
The ratio of amplitudes is [tex]\sqrt{2}[/tex] .
