The wall that clapped back

In front of a big wall I clapped my hands once. Half a second later, the wall clapped back. I wanted to know how far away it was.

I clapped once at a big wall and it clapped back $t=0.5$ s later. How far is the wall?

Echo: $t=0.5$ s, distance $d=vt/2$.

Sound travels about three hundred and forty-three metres every second. In half a second that is about one hundred and seventy-two metres, but that is there and back, so the wall is half of it away: about eighty-six metres.

Sound: $v\approx 343$ m/s. In $0.5$ s: $343\times 0.5\approx 172$ m, there and back, so $d=vt/2\approx 86$ m.

$v\approx 343\text{m/s}$; $vt\approx 172$ and $d=vt/2\approx 86$ m.

I walked towards the wall and clapped again. The echo came sooner and sooner. At about seventeen metres it came a tenth of a second later and stuck to my clap: I could no longer hear it as a separate sound.

Closer to the wall, the echo came sooner. At $d\approx 17$ m: $t=2d/v=2\times 17/343\approx 0.1$ s, and the echo stuck to the clap.

$d\downarrow \Rightarrow t\downarrow$; $d=17\Rightarrow t=2d/v\approx 0.1$.

So a room smaller than that cannot echo; it only makes a sound last longer. That is why a bathroom makes your voice rich, while a big hall repeats every word.

A room with $d<17$ m cannot echo, it only blurs: $t<0.1$ s.

$d<17\Rightarrow t<0.1$: a blur.