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From Wikipedia, "In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport." But Einstein said that energy equals mass so if a wave transfers energy therefore it transfer mass?

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But Einstein said that energy equals mass

In fact, 'Einstein said' that the total energy of a particle of invariant mass $m$ is (see Energy-momentum relation)

$$E = \sqrt{(pc)^2 + (mc^2)^2}$$

and so, for $mc^2 \ll pc$,

$$E \approx pc$$

which is exact for massless particles. Thus, e.g, there can be electromagnetic waves that transport energy and momentum without transporting mass.

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You're very nearly correct: mass and energy are equivalent through $E=mc^2$ (assuming $p$ is insignificant), yet they are not the same, light does not have mass yet it does have energy and the same is true for waves

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A better formulation would be to replace mass with matter:

In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated transport of matter.

Indeed, energy is heavy and behaves in that way as mass. The equation $E = mc^2$ really says that energy and mass are equivalent. But many physicists want to reserve the notation of mass for what is sometimes called rest mass. You can see this in the other answers, like "light does not have mass" (in answer by Cursed).

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