What is the maximum depth to which a P-wave ray can travel?

[peeballs]

Homework Statement

For a spherical Earth (see Table 1 for velocity and depth, linear gradient),
what is the maximum depth a P-wave ray with p = 0:05 can travel to? If we
increase the ray parameter to p = 0:055 or p = 0:06, what are the maximum
depths these rays can travel to?

Relevant Equations

zf = -a*ln(r/a), (r/a)*vf = (a/r)*vs(r), 20/6371 = (vs/r)

I've created an excel spreadsheet with the given model in addition to calculating radius of the layer by subtracting depth from 6371. I've calculated Zf.

I've also found what I think is vs by doing alpha * (radius column/6371), but that could be wrong. I know I need to find where 20/6371 = Vs/r, but I think I made a mistake along the way because my solution doesn't include any answers that work.

 

[BvU]

HI,

Could you be a bit more complete ? Some explanation of meaning and units with the variables, which ones are known, which equations you use to calculate which unknowns, etc. etc.

Legibility can be improved if you use $\alpha$ instead of a, make use of subscripts, etc. etc.

Why doesn't p occur in your equations ?

This is about shock wave propagation in Earth's crust ?

 

[peeballs]

HI,

Could you be a bit more complete ? Some explanation of meaning and units with the variables, which ones are known, which equations you use to calculate which unknowns, etc. etc.

Legibility can be improved if you use $\alpha$ instead of a, make use of subscripts, etc. etc.

Why doesn't p occur in your equations ?

This is about shock wave propagation in Earth's crust ?

@BvU Here's another screengrab of the spreadsheet with more detailed units and formulas. Still confused. I don't use p because 1/p (i.e., velocity) is the relevant value I need - 1/p is the value I am trying to find depth for.

 

[haruspex]

@BvU Here's another screengrab of the spreadsheet with more detailed units and formulas. Still confused. I don't use p because 1/p (i.e., velocity) is the relevant value I need - 1/p is the value I am trying to find depth for.

What is meant by α?
Table 1 merits some explanation. Why do some depths occur twice, and some alpha values occur twice, but staggered?
What does p=0:05 mean? Do you mean 0.05, and would that mean the velocity at max depth is 20km/s? I don't see any alpha values that high.

 

[peeballs]

@haruspex
alpha is the p-wave velocity in km/s.

Some depths and velocities occur twice because they're the depths to the layer and the thickness of layers can be reoccurring, and since the velocities of p-waves in these layers depends on many factors, they may repeat as well. Table 1 is non-negotiable as it's given as part of the task.

The turning point (value I'm told to find) is where 1/v = 0.05 - so yes, the spherical velocity of the max depth is 20 km/s. I'm assuming that means I did the conversion wrong somewhere, which I'm asking about. I don't know where, though.

 

[haruspex]

@haruspex
alpha is the p-wave velocity in km/s.

Some depths and velocities occur twice because they're the depths to the layer and the thickness of layers can be reoccurring, and since the velocities of p-waves in these layers depends on many factors, they may repeat as well. Table 1 is non-negotiable as it's given as part of the task.

The turning point (value I'm told to find) is where 1/v = 0.05 - so yes, the spherical velocity of the max depth is 20 km/s. I'm assuming that means I did the conversion wrong somewhere, which I'm asking about. I don't know where, though.

I think the key item in the table is the last one. Note how the velocity drops. You know why that is, right? What will happen to a wave that gets that far?

 

[peeballs]

I think the key item in the table is the last one. Note how the velocity drops. You know why that is, right? What will happen to a wave that gets that far?

@haruspex once it slows down enough won't it just turn around? It still doesn't help me find the equivalent spherical depth where that occurs

 

[haruspex]

@haruspex once it slows down enough won't it just turn around? It still doesn't help me find the equivalent spherical depth where that occurs

You have it backward. Waves tend to curve back towards the surface because the velocity mostly increases with depth.
Think of a light ray through glass. Total internal reflection occurs because the light is slower in glass than in air.
Reaching a place where the P wave velocity is lower is not going to cause reflection. So what will happen?