Would an aircraft which could go through sound barrier at an over 25 degree climb avoid the sonic boom problem?

Question

I read that the "cone" of the sonic boom is a cone of angle 25 degrees extending from the nose of the aircraft, centred on the axis of the aircraft.

I also read that the English Electric Lightning (and no doubt many another jet since) could break the sound barrier while climbing near-vertically.

If Concorde could have climbed at 45 degrees (or less) when it was accelerating through Mach 1 would that have solved the problems associated with sonic boom over land? The idea being that at that angle the cone would usually not intercept the ground (subject to mountainous terrain).

Edit

Following Bianfable's comment I now see the problem: you have to be at this angle from the point you are just about to hit Mach 1 to the point you get to the ocean ... and climbing at that angle would probably reach your operational ceiling fairly fast (and no doubt the thinner the air the more difficult it is to maintain).

So the obvious thing would be to dip down to, ooh, 2,000 feet as you approach Mach 0.95 and then pivot upwards! (To clarify, I do realise Mach 0.95 is trickier at lower altitudes... and also that it's quite noisy...).

And/or use rocket motors to get to a more satisfactory ceiling, perhaps.

Answer 1

When flying at Mach 2, the Mach cone will look like this:

The opening angle of the cone $\mu$ at a given Mach number $M$ is given by

$$ \mu = \arcsin \left( \frac{1}{M} \right) $$

I added the red arrows to indicate the direction in which the wavefronts forming the cone will expand as time passes. The lower one will eventually hit the ground and can then be heard as a sonic boom.

When flying upwards at an angle higher than $90^\circ - \mu$, the cone will never hit the ground. For Mach 2 this would be a climb angle of more than 60°:

The individual waves (grey circles) will still eventually hit the ground, but never more than one circle at a time. The wavefronts forming the cone do not hit the ground anymore, therefore no sonic boom can be heard. Note that the cone only extends backwards up to the moment where the aircraft first reached supersonic speeds and therefore does not continue backwards into the ground (because the aircraft obviously wasn't in the ground when flying).

It goes without saying that Concorde was not nearly capable of climbing at 60°, especially not while flying at Mach 2. The required angle is a bit less when going slower (about 45° at Mach 1.4), but this does not seem to be realistic for any supersonic airliner.