Your expectation "does that not mean that any echos should die down more quickly" is largely inaccurate for that portion of the explosive energy directed vertically, or near vertically.
Precisely because sound travels so much faster in water than air, and that water is much denser than air, the transmission coefficient from water to air for sound is very close to zero even at an incident angle of zero degrees (vertical).
- c is the speed of sound for the medium
- ρ is the density of the medium
- subscript l represents the second medium (air), no subscript the origin medium (water)
Typical values for seawater are:
ρ = 1020 kg/m3
c = 1500 m/s;
and for air:
ρ = 1.225 kg/m3
c = 340 m/s
so from Eqn. 1.28 above
m ~ 1.225 / 1020 = 0.0012
n ~ 1500 / 340 = 4.41
Then even at an incident angle of 0 degrees (so cos θ == 1 and sin θ == 0), from Eqn. 1.30 above we obtain as transmission coefficient:
T ~ ( 2 . 0.0012 ) / ( 0.0012 + 4.41 . 1 )
~ 0.0024 / 4.4112
A similar analysis will show that a very small transmission coefficient also occurs with respect to the sea floor.
Therefore in the immediate vicinity of sub and destroyer/corvette, where the incident angle of the sound blast is very close to zero in respect to both sea surface and sea floor, the echo of the sound blast reverberates in a nearly vertical pattern until that echo slowly migrates off of the vertical or is absorbed by the water through heating.
Most of the Atlantic is between 3000 and 4000 metres deep, averaging about 3600 metres. This means that the echos of a depth charge explosion are racing back and forth between sea surface and seafloor in 2 -3 seconds each way. With near zero transmission to both air and bedrock, a loud reverberation would echo around the vicinity of the explosion, typically the immediate vicinity of both sub and hunter. Yelling out in a cave system with strong echos, and then waiting for enough silence to hear a pin drop, would be an analogous situation.
Typical frequency for sonar in WW2 was 20kHz to 30kHz. The wavelength for a 25kHz signal would be ~ 6 cm given the sea water properties above, so all features of sea surface or sea floor < ~3 cm would be invisible.
The claim is made in another answer:
For ASDIC to work best the water should be homogeneous, with laminar flow only.
While true, this is never the case. Rather there are always three thermoclines affecting the transmission, except in waters so shallow that one or more are squeezed out.
Therefore the notion that waiting 15 minutes or so for a return to homogeneity of the sea is absurd. What actually happens is that any great disturbance of the water will disturb the "lamination" of the ocean that facilitates the sonar shadow effect illustrated here, that occurs when a positive thermocline gradient closest to the surface lays above a negative one:
Other more complex propagation patterns must also be allowed for by the experienced sonar operator, as described in the reference.