Wavefronts bunch up, sound does not speed up

The picture that settles the mechanism: mark where the source was when each crest left, and draw each crest as a circle expanding at the medium's speed.

1. The circles are all expanding at the same rate, because the medium sets that rate.
2. The source has moved between one emission and the next, so the circle centres are strung out along its path.
3. Ahead of the source the circles crowd together: shorter arriving wavelength, more crests per second, higher pitch.
4. Behind the source they spread out: longer wavelength, fewer crests per second, lower pitch.

Adding the source's velocity to the wave speed and reporting sound at $343 + 30$ m/s gets the direction of the pitch shift right for the wrong reason. That surfaces immediately when a question asks about the speed or the wavelength rather than the pitch, and it is worth catching before then.

For a source moving at constant velocity, the observed frequency is steady and higher during the approach, then steady and lower after it passes, with the drop happening at the moment of passing.

It does not climb as the source gets closer. The shift depends on the relative velocity along the line joining source and observer, and while a train drives straight at you that component is constant, so the received frequency is constant too.

The recorded car whose sound sweeps down as it goes by is a different situation: there the observer stands beside the road rather than on it, so the angle between the motion and the line of sight changes, which changes the component along that line. Generalising that sweep into a continuous climb during a head-on approach is the error.

The related slip is forgetting the null case: a source and an observer moving together hear no shift at all, however fast both are going. The relative velocity along the line joining them is zero.

Keeping the three quantities separate makes the whole effect checkable.

1. Wave speed: unchanged. Set by the medium's temperature and composition, and by nothing else.
2. Emitted frequency: unchanged. The siren is doing exactly what it always does.
3. Arriving wavelength: shortened ahead, lengthened behind. This is the quantity the source's motion actually alters.
4. Received frequency: follows from $f = v/\lambda$ with $v$ fixed and $\lambda$ altered.

Faster relative motion widens the gap between the observed and the rest frequency. It does not make that gap grow over time, which is the distinction between "bigger shift" and "rising pitch".

A moving observer produces a shift too, by a slightly different route: the observer runs into wavefronts more often, or lets them catch up more slowly. The wavelength in the medium is unchanged in that case, and the rate of encounter changes.

Either way, the rule for the direction is the same and is worth stating as the one thing to remember: closing the gap raises the received frequency, opening it lowers the received frequency.

The effect is not confined to sirens. Light from galaxies moving away from us is shifted toward longer wavelengths, which is the redshift that established cosmic expansion. Radar and ultrasound both read a speed off the shift in the returned signal. In each case the underlying account is the same: the source or the observer moved, the wavefronts arrived with a different spacing, and the wave's own speed was never the thing that changed.