Question

Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse $\left(S\right)$ and longitudinal $\left(P\right)$ sound waves. Typically, the speed of $S$ wave is about$4.0km/s,$ and that of 𝑃 wave is $8.0km/s.$ A seismograph records $P$ and $S$ waves from an earthquake. The first $P$ wave arrives $4min$ ⁡before the first $S$ wave. Assuming the waves travel in straight line, at what distance does the earthquake occur?

Answer 1

Step1: Find the relation of time for $S$ and $P$ waves

Let,$v_S$=velocity of$S$ waves

$v_P=$velocity of $P$ waves

$L=$distance between the epicenter and the seismograph.

We have

$L=v_S{t}_S..\left(i\right)$

$L=v_P{t}_P\dots \left(ii\right)$

Where, $t_s$ and $t_p$ are the respective times taken by the $S$ and $P$ waves to teach the seismograph from the epicenter. We are given,$v_P=8km/s$

$v_S=4km/s$

$t_S-t_P=4min=240s$

$\Rightarrow t_S=t_P+240s$

Step 2: Calculate the time taken by 𝑆 waves

From equations $\left(i\right)$ and $\left(ii\right)$, we have:

$v_S{t}_S=v_P{t}_P$

$\Rightarrow 4\times (t_P+240)=8\times t_P$

$\Rightarrow 4t_P+960=8t_P$

$\Rightarrow 4t_P=960$

$\Rightarrow t_P=\frac{960}{4}=240s=4min$

$\Rightarrow t_S=t_P+240s$

$=240+240=480s$

Step 3: Find the distance at the earthquake occur

From equation $\left(ii\right)$, we get: $(L=v_P{t}_P....(ii\left)\right)$

$L=8\times 240=1920km$

Hence, the earthquake occurs at a distance of $1920km$ from the seismograph.