WAVE problem; Earthquakes produce two kinds of seismic waves: the longitudinal primary waves...?

Earthquakes produce two kinds of seismic
waves: he longitudinal primary waves (called
P waves) and the transverse secondary waves
(called S waves). Both S waves and P waves
travel through Earth’s crust and mantle, but
at different speeds; the P waves are always
faster than the S waves, but their exact speeds
depend on depth and location. For the purpose
of this exercise, we assume the P wave’s
speed to be 8740 m/s while the S waves travel
at a slower speed of 4100 m/s.
If a seismic station detects a P wave and
then 47.4 s later detects an S wave, how far away is the earthquake center?
Answer in units of km.

1 Answer
Apr 8, 2018

#366 km#

Explanation:

#d = st#

distance = speed * time

here, the distance is from the earthquake centre to the seismic station.

both the P and S waves travel from the earthquake centre before being detected by the seismic station, so the distance is the same for both.

the speeds are given as #8740 m//s# for the P wave and #4100 m//s# for the S wave.

we also know that the P wave arrives #47.4# seconds before the S wave.

we do not know the time that the P wave takes to travel, but we can denote it as #t_P#.

the time that the S wave takes to travel can be denoted as #t_P + 47.4#, where time is in seconds.

for the S wave, speed * time is #4100 * (t_P + 47.4)#
for the P wave, speed * time is #8740 * t_P#.

since the distances that they travel are the same, the two expressions for speed * time are equal.

#4100 * (t_P + 47.4) = 8740 * t_P#

if you expand the brackets, you can find that

#4100t_P + 194340 = 8740 t_P#

then you can subtract #4100t_P:#

#4640t_P = 194340#

and divide by #4640# to find #t_P#, which is the time that P takes to travel:

#t_P = 41.883... # seconds

since distance = speed * time, the distance that the P wave travels is #t_P *# the speed of P.

this is #41.883s * 8740 m//s#, which gives #366057.42 m#.

in kilometres, this is #366 km# to #3# significant figures.