That is called passive sonar. I suspect that it is kinda hard to hear from the missile. Anything behind you is certainly not hearable, since the missile goes faster than sound. I have no idea if you could measure the sound coming from the front. You also have to take into account that you are chasing an after image, since the plane is also faster than sound. But torpedoes use this, so the idea is valid.
That is called passive sonar. I suspect that it is kinda hard to hear from the missile. Anything behind you is certainly not hearable, since the missile goes faster than sound. I have no idea if you could measure the sound coming from the front. You also have to take into account that you are chasing an after image, since the plane is also faster than sound. But torpedoes use this, so the idea is valid.
Those are likely quite valid points. This being noncredible defense they have no place here.