Chapter 11, Problem 46P

(a)

To determine

To Find: The ratio of the intensities of an earthquake of P wave.

Answer to Problem 46P

The ratio of the intensities of an earthquake P wave passing through the Earth and detected at two points from the source is $0.25$ .

Explanation of Solution

Given:

Radius of point 1, $r_1=10\text{ km}$

Radius of point 2, $r_2=20\text{ km}$

Formula used:

Intensity is inversely proportional to the square of the distance of source of wave, it is described by the following formula,

Intensity, $I=\frac{1}{r^2}$

Here,

  $r$ is the distance of source of wave.

Calculation:

The intensities are determined at both points by using the above formula,

Intensity at point 1, $I_1=\frac{1}{r_1{}^2}$

Intensity at point 2, $I_2=\frac{1}{r_2{}^2}$

Taking ratio of both the intensities,

  $\begin{array}{c}\frac{I_2}{I_1}=\frac{\frac{1}{r_2{}^2}}{\frac{1}{r_1{}^2}}\\ =\frac{r_1{}^2}{r_2{}^2}\\ ={\left(\frac{r_1}{r_2}\right)}^2\\ ={\left(\frac{\text{10}\text{km}}{\text{20 km}}\right)}^2\\ =\frac{1}{4}\\ \frac{I_2}{I_1}=0.25\end{array}$

Conclusion:

Thus, the intensity of P wave which is passed through the Earth and detected at two points is $0.25$ .

(b)

To determine

To Find: The ratio of the amplitudes of an earthquake P wave.

Answer to Problem 46P

The ratio of the amplitudes of an earthquake P wave passing through the Earth and detected at two points from the source is $0.5$ .

Explanation of Solution

Given:

Radius of point 1, $r_1=10\text{ km}$

Radius of point 2, $r_2=20\text{ km}$

Formula used:

Amplitude is inversely proportional to the distance from the source of wave, it is described by the following formula,

Amplitude

  $A=\frac{1}{r}$

Here,

  $r$ is the distance of source of wave.

Calculation:

The amplitudes is determined at both points by using the above formula,

Amplitude at point 1, $A_1=\frac{1}{r_1}$

Amplitude at point 2, $A_2=\frac{1}{r_2}$

  $\begin{array}{c}\frac{A_2}{A_1}=\frac{\frac{1}{r_2}}{\frac{1}{r_1}}=\frac{r_1}{r_2}\\ =\frac{\text{10}\text{km}}{\text{20 km}}=\frac{1}{2}\\ \frac{A_2}{A_1}=0.5\end{array}$

Conclusion:

Thus, the amplitude of P wave which is passed through the Earth and detected at two points is $0.5$ .