The constant impedance (with time) allows the wave shape of the steady-state current to follow the wave shape of the applied voltage (excepting a phase shift).
Linear loads can be resistive, capacitive, or inductive, but the impedance must remain fixed throughout the full power cycle. A linear load does not change its impedance even if the source’s output changes.
With this linear relationship, the source’s voltage waveform, whatever it is, will not be distorted in shape by the linear load. A sine wave voltage source applied to a linear load will result in a sine wave current at the same frequency, with no other distortion or waveform changes.
The voltage waveform’s amplitude and phase angle may change, but it will remain a pure sine wave of the original frequency. Similarly, if the voltage waveform is already distorted with the addition of harmonics, the current from a linear load will also have each of those harmonics present, with no additional harmonics added.
Each current harmonic may be shifted in phase from the voltage but no other frequencies will be added.