Question

An earthquake generates a sound wave 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 4km per sec and that of ‘p’ wave is 8 km per sec. A seismograph records p and s waves from an earthquake. The first p wave arrives 4 min before the first 5 waves. Assuming the wave travel in straight line the epicenter of earthquake is at $120\eta$ $($in km$)$, find $\eta$

Answer 1

In order to solve this problem first we have to calculate the time taken by transverse and longitudinal wave to arrive with the help of general relation between time, distance and velocity
i.e., Velocity $=$Distance $\div$Time
After then according to given situation make a equation between time ${t_s}$ and ${t_p}$ i.e., time taken by transverse wave and longitudinal wave respectively and we will get the value of $\eta$

Step by step answer:
Given that the speed of transverse wave ‘s is $=$ 4km per sec and the epicenter of earthquake is at distance $=$ 120$\eta$
So, the time taken by transverse wave is
${t_s} = \dfrac{{120\eta }}{4} = 30\eta$ second …..1
Also given that the speed of longitudinal wave ‘p’ is $=$ 8km per sec and the epicenter of earthquake is at distance $=$ 120$\eta$
So, time taken by longitudinal wave is-
${t_p} = \dfrac{{120\eta }}{8}$
${t_p} = 15\eta \,$second …..2
Since the first p wave arrives 4 min before the first S wave
${t_p} + 4 = {t_s}$
$\Rightarrow 15\eta + 4 \times 60 = 30\eta$
$\Rightarrow 30\eta + 15\eta = 240$
$\Rightarrow 15\eta = 240$
$\Rightarrow \eta = \dfrac{{240}}{{15}}$
$\therefore \eta = 16$
Hence $\eta = 16$ is correct answer

Note: An earthquake is caused by a sudden slip on a fault. When the stress on the edge overcomes the friction, there is an earthquake that releases energy in waves that travel through the earth’s crust and cause the shaking that we feel.