Which wave reaches point B first?

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In summary, the earthquake at point A created a P wave that reached point B by traveling straight through the Earth at a constant speed of 7.80m/s. The earthquake also created a Rayleigh wave that traveled along the surface of the Earth at 4.50km/s. The time difference in the arrival of these two waves at point B is 0.5 seconds.
  • #1
~christina~
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Homework Statement


Two points A and B on the surface of the Earth are at the same longitude and 60 deg apart in latitude. Suppose an earthquake at point A creates a P wave that reaches B by traveling straight throught he body of the Earth at constant speed of 7.80m/s. The earthquake also creates a Rayleigh wave that travels along the surface of the Earth at 4.50km/s.

a) which of these 2 waves arives at B first?
b) What is the time difference in the arrival of these 2 waves at B? Take the Earth's radius to be 6,370km.




Homework Equations


well I found this online for the distance
D= R arc cos[sin(lat 1) x sin(lat 2) + cos (lat 1) x cos (lat 2) x cos [ lon 2- lon 1]]

y(x,t)= f(x- vt)

The Attempt at a Solution



well I first calculated the distance between the 2 points with the equation I found online.

D= R arc cos[sin(lat 1) x sin(lat 2) + cos (lat 1) x cos (lat 2) x cos [ lon 2- lon 1]]

R of earth= 6,370km
lat 1 = 0
lat 2= 60 deg => [tex] \pi / 3[/tex]
lon 1= 0
lon 2= 0

D= 6,370km arc cos[sin(0) x sin([tex] \pi / 3[/tex]) + cos (0) x cos ([tex] \pi / 3[/tex]) x cos [ 0]]

D= 6,370km arc cos [0 + 0.5]

D= 6,670.64km

x= 6,670.64km

But as to how to find the time has me confused a bit.

I think I use this equation where I just solve for time then subtract the 2 times for the different waves but I'm not 100% certain. (that's where I need help)

y(x,t)= f(x- vt)

for the 1st wave

y(6,670,640m,t)= f(6,670,640m - 7,800m/s (t))


same for the Raleigh wave

y(6,670,640m,t)= f(6,670,640m - 4,500km/s (t))

Or do I just use V= d/t and solve for t??

can someone help me out with which equation to use to find the time it takes for a wave to reach a certain distance?


Thanks
Well I'm not sure how to solve for t since isn't y the transverse position of the wave which I just don't have in the information?

Thanks a lot :smile:
 
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  • #2
~christina~ said:
Or do I just use V= d/t and solve for t??

can someone help me out with which equation to use to find the time it takes for a wave to reach a certain distance?


Thanks
Well I'm not sure how to solve for t since isn't y the transverse position of the wave which I just don't have in the information?

Thanks a lot :smile:

The same eqn as always, which you have written down.

A and B are two points on a circle of radius R. There are two paths to be considered here; one is a straight line from A to B, and the other is the arc from A to B. If you know the angle subtended at the centre, then the dist along the paths are known, and the time is got from dist/speed.

There is absolutely no reason for considering the wave eqn here.
 
  • #3
Shooting Star said:
The same eqn as always, which you have written down.

A and B are two points on a circle of radius R. There are two paths to be considered here; one is a straight line from A to B, and the other is the arc from A to B. If you know the angle subtended at the centre, then the dist along the paths are known, and the time is got from dist/speed.

There is absolutely no reason for considering the wave eqn here.

It's interesting since I got the answer incorrect from what the book has. I think the distance is what is causing this.
You didn't say whether the way I got the distance is correct or not... Below

The Attempt at a Solution



well I first calculated the distance between the 2 points with the equation I found online.

D= R arc cos[sin(lat 1) x sin(lat 2) + cos (lat 1) x cos (lat 2) x cos [ lon 2- lon 1]]

R of earth= 6,370km
lat 1 = 0
lat 2= 60 deg => [tex] \pi / 3[/tex]
lon 1= 0
lon 2= 0

D= 6,370km arc cos[sin(0) x sin([tex] \pi / 3[/tex]) + cos (0) x cos ([tex] \pi / 3[/tex]) x cos [ 0]]

D= 6,370km arc cos [0 + 0.5]

D= 6,670.64km

If the above way I got the distance from A=> B is incorrect then I'm not sure how I'd find it. I don't understand what you said about finding the distance.

Thanks
 
  • #4
Since the two points are on the same longitude, simply think of the circle you'll get get if you slice it through that longitude. You now have to deal with just a 2-d circle, whose radius is that of the earth. The difference of the latitudes being 60 deg means that the radii from the two points subtend an angle of 60 deg at the centre.

Suppose the centre of the Earth is O. Angle AOB is given. Draw a simple diagram. Can you find segment AB? Can you find arc AB? Big calculations are not necessary, I think. :smile:

After you find the distances, use v=d/t.
 

Related to Which wave reaches point B first?

1. What factors affect the speed at which a wave reaches point B first?

The speed at which a wave reaches point B first is affected by several factors, including the medium through which the wave travels, the amplitude of the wave, and the wavelength of the wave. In general, waves travel fastest through solids, followed by liquids, and then gases. Waves with larger amplitudes and shorter wavelengths also tend to travel faster.

2. Can two different types of waves reach point B at the same time?

Yes, it is possible for two different types of waves to reach point B at the same time. However, this depends on the specific properties of the waves and the medium through which they are traveling. In some cases, one type of wave may travel faster or be more dominant, resulting in it reaching point B first.

3. How does the distance between point A and point B affect which wave reaches point B first?

The distance between point A and point B does not have a direct impact on which wave reaches point B first. However, the longer the distance, the more likely it is that other factors such as reflections and interference may occur, which can affect the speed at which a wave reaches point B.

4. What is the difference between a transverse wave and a longitudinal wave?

A transverse wave is a type of wave where the particles of the medium move perpendicular to the direction of the wave. Examples of transverse waves include water waves and electromagnetic waves. On the other hand, a longitudinal wave is a type of wave where the particles of the medium move parallel to the direction of the wave. Examples of longitudinal waves include sound waves and seismic waves.

5. Is it possible for a wave to travel faster than the speed of light?

No, according to current scientific understanding, it is not possible for a wave to travel faster than the speed of light. The speed of light, which is approximately 299,792,458 meters per second, is considered to be the absolute maximum speed at which anything can travel in the universe.

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