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 $4\min$ 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

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,
$\begin{aligned} & L={{V}_{S}}{{t}_{S}} \\\ & L={{V}_{P}}{{t}_{P}} \\\ \end{aligned}$
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,

& {{V}_{S}}=4km{{s}^{-1}} \\\ & {{V}_{P}}=8km{{s}^{-1}} \\\ \end{aligned}$$ From the equations of the distance between the epicentre and seismograph, We can write that, $${{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}}=4\min $$ Comparing both the equations will give, $$2\times {{t}_{P}}-{{t}_{P}}=4\min =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.