Question

Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can feel both transverse (S) and longitudinal (P) sound waves. Typically the speed of the S wave is about $4km{s}^{-1}$ and that of P wave is $8km{s}^{-1}$ . A seismograph records P and S waves from an earthquake. The first P wave arrives $4min$ before the first S wave. Let us assume that the waves are travelling in a straight line, at what distance will the earthquake occur?

Answer 1

Hint: The concepts of distance and displacement have been used here. Distance is given as the total path length traversed by a body or a wave during the motion. Time taken to travel is given as the velocity divided by the distance travelled. These concepts may help you to solve this question.

Complete step-by-step answer:
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 know that,
 $\begin{array}{rl}& L=V_S{t}_S\\ & L=V_P{t}_P\end{array}$
In which $t_S$ ​and $t_P$ are the respective times taken by the S and P waves in order to reach the seismograph from the epicentre.
It is already mentioned in the question that,
$$\begin{array}{rl}& V_S=4km{s}^{-1}\\ & V_P=8km{s}^{-1}\end{array}$$
From the equations of the distance between the epicentre and seismograph,
We can write that,
$$V_S{t}_S=V_P{t}_P$$
Substituting the values in it will give,
$$4\times t_S=8\times t_P$$
Therefore,
$$t_S=2\times t_P$$
And it is also mentioned in the question that,
$$t_S-t_P=4min$$
Comparing both the equations will give,
$$2\times t_P-t_P=4min=240s$$
That means,
$$t_P=240s$$
Thus
$$t_S=480s$$
Therefore from the equation regarding length cited above can be written as,
$$L=V_S{t}_S$$
Substituting the values in the equation,
$$L=4\times 480=1920km$$
Therefore the answer is obtained.

Note: Seismic waves are referred to as the waves of energy as a result of the sudden breaking of rock within the earth or by an explosion. They are the energy that can travel through the earth. These are recorded using a seismograph.