Speed of Sound Waves: Echoes in a Canyon

[mickellowery]

Homework Statement

A cowboy stands on horizontal ground between two parallel vertical cliffs. He is not midway between the cliffs. He fires a shot and hears it echoes. The second echo arrives 1.92s after the first and 1.47s before the third. Consider only the sound traveling parallel to the ground and reflecting from the cliffs. Take the speed of sound as 340 m/s. What is the distance between the cliffs? If he could hear a fourth echo, how long after the third echo does it arrive?

Homework Equations

I was thinking that I could just try multiplying 340m/s by 1.92s, but this isn't correct. I thought it has something to do with the fact that he isn't standing in the center of the two cliffs. Since this didn't work I am now stumped how to get started.

The Attempt at a Solution

 

[Doc Al]

Attack it step by step. Name all the variables: Distance to closer cliff = d₁; further cliff = d₂. Write expressions for the time it takes to hear all three echoes. Then apply the given info.

 

[mickellowery]

OK I have been trying to come up with an expression for time, but I'm having trouble. Would it just be: t₁= \frac{340}{d₁} t₂= \frac{340}{d₂}?

 

[Doc Al]

OK I have been trying to come up with an expression for time, but I'm having trouble. Would it just be: t₁= \frac{340}{d₁} t₂= \frac{340}{d₂}?

What's the total distance the sound has to travel?

(FYI: Don't use 'sub' tags within Latex. Use d_1 and d_2 instead.)

 

[mickellowery]

OK so it's d₁+d₂= (340)t₁+(340)t₂?

 

[Doc Al]

OK so it's d₁+d₂= (340)t₁+(340)t₂?

No. Just consider the first echo, which bounces off the near cliff. What total distance does that sound travel? Use that distance to calculate the time for that echo's arrival.

Then do the same for the second and third echoes.

 

[mickellowery]

I think what I'm having trouble seeing is whether or not all of these echoes are bouncing off of the closer wall. I see that it should be 2d_1= 340t_1 but then would it be 2d_1= 340t_2?

 

[Doc Al]

I think what I'm having trouble seeing is whether or not all of these echoes are bouncing off of the closer wall.

They don't all bounce off the closer wall. The first one does, but the second one bounces off the far wall.

I see that it should be 2d_1= 340t_1

Right.

but then would it be 2d_1= 340t_2?

No, the second echo has nothing to do with the near wall.

Draw a diagram and trace the path of each echo.

(What about that third echo?)

 

[mickellowery]

OK so the first echo which has no time attached to it will be off the close wall and the second echo which is 1.92s is off the far wall. The third echo which is 1.47s should be off the close wall again.

 

[Doc Al]

OK so the first echo which has no time attached to it will be off the close wall and the second echo which is 1.92s is off the far wall. The third echo which is 1.47s should be off the close wall again.

Correct for the first two echoes. Note that those times are really differences in times, not what we're calling t₁, t₂, t₃.

What's the complete path of that third echo? How far does that sound travel?

 

[mickellowery]

Alright I think I've got it Doc. Would the second one be 2d₂= 340(1.92+t₁ ? I'm still thinking about the third one a little, but would it have bounced off both walls?

 

[Doc Al]

Would the second one be 2d₂= 340(1.92+t₁ ?

Sounds good to me.

I'm still thinking about the third one a little, but would it have bounced off both walls?

Exactly.

Tip: Don't think you have to combine these times in your head. I would first write expressions for all three times, and then worry about the time differences. For example:
t₁ = 2d₁/v
t₂ = 2d₂/v

Then you can add that t₂ - t₁ = 1.92, and so on.

 

[mickellowery]

Excellent! Thanks so much for all your help again Doc. You're my hero!