Chapter 20, Problem 2P

Look at Figure 19-3. The earthquake occurred 7440 km from the seismograph. How fast did the P waves travel in km/s? How fast did the S waves travel? How long did the P waves and the S waves take to travel 100 km from the epicenter? Assume the wave speeds are constant.

To determine

The speed of $P$ waves and $S$ waves; and the time taken by $P$ waves and $S$ waves to travel $100\text{km}$ from the epicenter.

Answer to Problem 2P

The speed of $P$ waves is $11.3\text{km}/\text{s}$, the speed of $S$ waves is $6.2\text{km}/\text{s}$, the time taken by $P$ waves to travel $100\text{km}$ from the epicenter is $9\text{s}$ and by the $S$ wave is $16\text{s}$.

Explanation of Solution

Write the expression for speed of wave.

    $v=\frac{d}{t}$        (I)

Here, $t$ is time taken, $d$ is the distance travelled and $v$ is the speed of wave.

From the figure $19\text{-}3$, the first $P$ wave arrives at $11\text{min}$ and the $S$ wave arrives at $20\text{min}$.

Conclusion:

Substitute, $7440\text{km}$ for $d$ and $11\text{min}$ for $t$ in equation (I) to find speed of $P$ wave, $v_{\text{p}}$.

    $\begin{array}{c}{v}_{\text{P}}=\frac{\left(7440\text{km}\right)}{\left(11\text{min}\right)}\\ =\frac{\left(7440\text{km}\right)}{\left(11\text{min}\left(\frac{60\text{s}}{1\text{min}}\right)\right)}\\ =11.3\text{km}/\text{s}\end{array}$

Substitute $7440\text{km}$ for $d$ and $20\text{min}$ for $t$ in equation (I) to find speed of $S$ wave, $v_{\text{S}}$.

    $\begin{array}{c}{v}_{\text{S}}=\frac{\left(7440\text{km}\right)}{\left(20\text{min}\right)}\\ =\frac{\left(7440\text{km}\right)}{\left(20\text{min}\left(\frac{60\text{s}}{1\text{min}}\right)\right)}\\ =6.2\text{km}/\text{s}\end{array}$

The distance travelled from the epicenter is $100\text{km}$.

Substitute $100\text{km}$ for $d$ and $11.3\text{km}/\text{s}$ for $v$ in equation (I) to find $t_{\text{p}}$.

    $\begin{array}{c}11.3\text{km}/\text{s}=\frac{\left(100\text{km}\right)}{t_{\text{P}}}\\ t_{\text{P}}=\frac{\left(100\text{km}\right)}{\left(11.3\text{km}/\text{s}\right)}\\ \approx 9\text{s}\end{array}$

Substitute $100\text{km}$ for $d$ and $6.2\text{km}/\text{s}$ for $v$ in equation (I) to find $t_{\text{S}}$.

    $\begin{array}{c}6.2\text{km}/\text{s}=\frac{\left(100\text{km}\right)}{t_{\text{S}}}\\ t_{\text{S}}=\frac{\left(100\text{km}\right)}{\left(6.2\text{km}/\text{s}\right)}\\ \approx 16\text{s}\end{array}$

The speed of $P$ waves and $S$ waves are $11.3\text{km}/\text{s}$ and $6.2\text{km}/\text{s}$ respectively. The time taken by $P$ waves and $S$ waves to travel $100\text{km}$ from the epicenter is $9\text{s}$ and $16\text{s}$ respectively.