Two points A and B on the surface of the Earth are at the same longitude and = 4

Question

Two points A and B on the surface of the Earth are at the same longitude and = 45.0° apart in latitude as shown in the figure below. Suppose 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 6.25 km/s. The earthquake also radiates a Rayleigh wave that travels at 3.00 km/s. In addition to P and S waves, Rayleigh waves are a third type of seismic wave that travels along the surface of the Earth rather than through the bulk of the Earth.

(a) Which of these two seismic waves arrives at B first? the longitudinal P wave or the Rayleigh wave

(b) What is the time difference between the arrivals of these two waves at B in seconds? (The radius of the Earth is 6370 km.)

Explanation / Answer

Taking your radius figure, the earth's diameter is 12,740 km.
Therefore its circumference is 12,740 x Pi = 40,024 km.
Halve that, one longitude must be 20,012 km.
This is divided into 180 deg., so 1 deg latitude = 111.118 km.
So 45 deg. of latitude is 5,000.31 km.
The Rayleigh wave travels at 3 km/sec., so dividing, it arrives 1,666.77 secs. later.
The P wave travels along the straight line difference between the 2 points 45 deg. apart, So you need chord length.
2r sin (c/2) = (6370 x 2) x (sin 45/2).
= 12740 x 0.3827, = 4,875.387 km.
Dividing this by 6.25, = 780.062 secs.
1,666.77 - 780.062 = 886.71 secs. the Rayleigh wave arrives after the P wave.
About 14.778 minutes.

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