Difference between sound echo and mirror reflection

[BOYLANATOR]

Homework Statement

Why is is that when we look in a mirror we expect to see a reflection equally as bright as the real image, but when we stand in front of a cliff and clap our hands we don't hear an echo as loud as the original sound?

Homework Equations

R = (z₂ - z₁)/(z₂ + z₁)

The Attempt at a Solution

In a mirror, R is close to -1. For the cliff, R should still be pretty close to 1. For the echo spherical divergence must lead to a decrease in the volume heard after reflection but this does not seem to apply to the mirror. (Is this because we compare the mirrored image we see to the everyday image which has the same spherical divergence?). In reality, any cliff we have done this in front of is finite in size so not all sound energy is reflected.

 

[Svein]

How loud would the sound be if you had no cliff, but somebody who listens at twice the distance to the cliff?

 

[haruspex]

I like this question, quite subtle.
Think about the relationship between the brightness of an object you look at and the light power arriving at your eyes.
Then think about the relationship between sound power arriving at your ears and the loudness you experience.

 

[BOYLANATOR]

Svein: Ok, obviously the energy has decreased according to the square of the distance, and the amplitude falls of as the distance. So you hear it as if the source was at 2x. This is identical to seeing yourself in the mirror as you would see yourself at 2x.

Haruspex: The difference must arrive from our experience. We have ears that can detect the loudness of our echo at effectively x=0. Then how bright would it be to see ourselves at x=0? Perhaps this isn't the right question, better to think of our line of sight eventually 'hitting' something at all angles and therefore that the brightness of an object is something to do with the angular area it occupies in our line of sight.

 

[BOYLANATOR]

Afterthought. If I held a bright lamp to the side of my face it would obviously be very bright, but it wouldn't be as painfully bright if I looked at its reflection in a mirror some distance away. I think this adds to what I was trying to say above

 

[haruspex]

the brightness of an object is something to do with the angular area it occupies in our line of sight.

That's exactly what it isn't, and that's the key here. The power it sends to your eye depends on the angular area, but the brightness doesn't. Contrast that with the loudness of a sound.

 

[BOYLANATOR]

Ha, I should have italicized brightness. What the eye sees is mainly just color then? The effect of the lamp I mentioned though is related to the power.

 

[haruspex]

Ha, I should have italicized brightness. What the eye sees is mainly just color then? The effect of the lamp I mentioned though is related to the power.

No, that's not what I'm saying.
Consider a sheet of paper lying on a desk in front of you. Make it larger by putting a second sheet next to it. The brightness is the same, but what about the power?

 

[BOYLANATOR]

The total power received by the eye from paper is doubled. But the power per unit solid angle is the same.

 

[haruspex]

The total power received by the eye from paper is doubled. But the power per unit solid angle is the same.

Exactly. So now consider looking at one of these pieces of paper from twice the distance. What happens to the brightness?

 

[BOYLANATOR]

It has the same power per unit solid angle but less total power to the eye. But the paper looks the same, so brightness is related to power per unit solid angle?

 

[haruspex]

It has the same power per unit solid angle but less total power to the eye. But the paper looks the same, so brightness is related to power per unit solid angle?

Yes. Now contrast that with perceived loudness.

 

[BOYLANATOR]

A single ear does not have any directionality to its recording. We hear the total energy from all of our surroundings. This answers my question! Thanks.