  We explore the impact of corrections to the propagation on the waveforms of
gravitationally lensed gravitational waves under the geometrical optics
approximation, focusing on both uniform cosmological modifications and local
modifications localized around lensing objects. By adopting a model-independent
phenomenological approach, we systematically investigate the effects of these
modifications in strong lensing scenarios, where detection of multiple images
is expected. Our analysis reveals that cosmological modifications can yield
corrections to the time delay that remain to be minor compared with the effects
that accumulate over the whole propagation process, which are present also in
the unlensed waveform. By contrast, local modifications around lensing objects
can alter the image position and also the magnification factor, which is
potentially polarization-selective and frequency-dependent. In some case we can
have image disappearance as well as signal amplification. Furthermore, we
demonstrate that such modifications can cause degradation of waveform match
with the templates based on general relativity. This study highlights the
importance of considering waveform modifications to search for the signature of
modified propagation or the existence of extra polarization modes, and proposes
potential observational targets.
