Question

Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse (S) and longitudinal (P) sound waves. Typically the speed of S wave is about 4.0 km $s^{-1}$ and that of P wave is 8.0 km $s^{-1}$. A seismograph records P and S waves from an earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, at what distance does the earthquake occur ?

Answer 1

Let $v_S$and $v_P$ be the velocities of S and P waves respectively.
Let $L$ be the distance between the epicentre and the seismograph.
We have:

$L=v_S{t}_S$ ...(i)
$L=v_P{t}_P$ ...(ii)

Where,
$t_S$ and $t_P$ are the respective times taken by the S and P waves to reach the seismograph from the epicentre
It is given that:

$v_P=$ 8 km/s
$v_S=$ 4 km/s
From equations (i) and )ii), we have:

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

$4t_S=8t_P$

$t_S=2t_P$ ...(iii)
It is also given that:

$t_S-t_P=4min=240s$

$2t_P-t_P=240$

$t_P=240$

And $t_S=2\times 240=480s$
From equation (ii), we get:

$L=8\times 240$

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