Two points A and B on the surface of the Earth are at the...

Two points $A$ and $B$ on the surface of the Earth are at the same longitude and $60.0^{\circ}$ apart in latitude. Suppose that an earthquake at point $A$ creates a $P$ wave that reaches point $B$ by traveling straight through the body of the Earth at a constant speed of $7.80 \mathrm{km} / \mathrm{s} .$ The earthquake also radiates a Rayleigh wave, which travels across the surface of the Earth in an analogous way to a surface wave on water, at $4.50 \mathrm{km} / \mathrm{s}$. (a) Which of these two seismic waves arrives at $B$ first? (b) What is the time difference between the arrivals of the two waves at $B ?$ Take the radius of the Earth to be $6370 \mathrm{km}$.

Two points $A$ and $B$ on the surface of the Earth are at the same longitude and $60.0^{\circ}$ apart in latitude. Suppose that an earthquake at point $A$ creates a $P$ wave that reaches point $B$ by traveling straight through the body of the Earth at a constant speed of $7.80 \mathrm{km} / \mathrm{s} .$ The earthquake also radiates a Rayleigh wave, which travels across the surface of the Earth in an analogous way to a surface wave on water, at $4.50 \mathrm{km} / \mathrm{s}$.
(a) Which of these two seismic waves arrives at $B$ first?
(b) What is the time difference between the arrivals of the two waves at $B ?$ Take the radius of the Earth to be $6370 \mathrm{km}$.

Key Concepts

Seismic Wave Types

This concept refers to the different kinds of seismic waves generated by earthquakes. Body waves, such as P waves (primary waves), travel through the interior of the Earth and typically have higher speeds, while surface waves, like Rayleigh waves, propagate along the Earth’s exterior and tend to move slower. Understanding these wave types is crucial for interpreting their arrival sequences at different locations and for inferring the Earth’s internal structure.

Spherical Geometry and Distance Calculations

In problems involving points on the Earth’s surface, it is important to differentiate between distances along the surface (arc lengths) and the straight-line distances through the Earth’s interior (chord lengths). Using spherical geometry, the arc length can be determined by multiplying the Earth's radius by the angular separation (in radians), while the chord length is calculated using trigonometric relationships. This distinction is essential when comparing the travel paths of waves that follow the curvature of the Earth versus those traveling in straight lines through its interior.

Wave Speed and Travel Time Calculations

The travel time of any wave is determined by dividing the distance it covers by its speed. This fundamental principle of wave propagation is used to compare the arrival times of different types of seismic waves. By calculating the time each wave takes—one following the chord path and the other following the arc length on the surface—it is possible to determine which wave arrives first and to quantify the time difference between them.

Transcript

00:01 In this problem, it is given that a and b, these are the two points on the same longitude, whereas they are separated by 60 degrees along these latitudes.

00:15 Okay? and at a point, if due to earthquake, two waves are formed, p wave and r wave, p wave, those are the body seismic waves we have, and r waves, and r waves, those are the seamic waves we have, and r waves, those are the sea.

00:30 Surface waves which travel along the surface up to this point b.

00:36 Now here it is given to us the velocity of this p wave and r wave.

00:41 The velocity of p wave is given to us which is 7 .80 kilometer per second which we can write in meter per second as kilometer means thousand meter so i can write this as 7 .80 into 10 rise to 3 meter per second and the velocity of a relays wave that is given to us 4 .50 km per second which travel along the surface of that earth.

01:12 So here i can write this as 4 .50 into 10 rise to 3 meter per second and the radius of earth here it is said to take 6 ,370 kilometers which i can write again in meter as 6 .370 into 10 to the power 6 meter.

01:36 So the first question here it is asked that out of this p wave and relase wave, which wave will reach at this point b first of all? so looking at this particular values, you see the p waves that has velocity greater than a relase wave.

01:56 And as this p wave, it is said in the problem that it travels directly to the point.

02:02 Point b and through the body of this earth.

02:05 So definitely this p wave, those are the compressional waves or we can say those are longitudinal in nature we have.

02:14 P waves, these longitudinal waves, that will reach.

02:19 First of all, reach to point b before that of release wave.

02:27 So this is our first answer.

02:29 Then second question here it is asked that what is the time difference between those two waves reaching at point b? so to find out that time difference, let's say delta t, what we have to find out? we have to find out time that required for t wave to reach at point b.

02:48 This we have to find it first.

02:50 Then time that is required for relish wave to reach at point b, that also i have to find...

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