  Gravitational waves are messengers carrying valuable information about their
sources. For sources at cosmological distances, the waves will contain also the
imprint left by the intervening matter. The situation is in close analogy with
cosmic microwave photons, for which the large-scale structures the photons
traverse contribute to the observed temperature anisotropies, in a process
known as the integrated Sachs-Wolfe effect. We derive the gravitational wave
counterpart of this effect for waves propagating on a Friedman-Robertson-Walker
background with scalar perturbations. We find that the phase, frequency and
amplitude of the gravitational waves experience Sachs-Wolfe type integrated
effects, this in addition to the magnification effects on the amplitude from
gravitational lensing. We show that for supermassive black hole binaries, the
integrated effects could account for measurable changes on the frequency, chirp
mass and luminosity distance of the binary, thus unveiling the presence of
inhomogeneities, and potentially dark energy, in the Universe.
