The process described involves a rapid heating of the conducting air channel by an electrical discharge, such as lightning. This intense heating causes the air to expand very quickly.
This rapid expansion creates a sudden increase in pressure, generating a disturbance that propagates outwards. These pressure disturbances traveling through the air are known as sound waves. The audible phenomenon associated with these waves from lightning is thunder.
Therefore, the shock waves that spread are sound waves.
The sequence of events in a cloud to ground electrical discharge is

The above cartoon exhibits a cloud during the mature stage of development of a simple convective thunderstorm. The simplest charge structure sequence in boxes. A, B, C, respectively, can be depicted as follows. Positive charges are shown as +++ and negative charges ---