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Ordinarily a transverse wave does not transmit momentum, as it is the case of the well-known sea wave phenomenon. But the electromagnetic radiation has both fields transversal to the movement of the wave. Anyway, the electromagnetic wave transmits momentum in the direction of the wave.

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    $\begingroup$ @lemon: EM waves carry momentum classically. $\endgroup$ – Jerry Schirmer Feb 16 '15 at 23:17
  • $\begingroup$ As can classical waves in a medium. $\endgroup$ – dmckee Feb 17 '15 at 1:14
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It's possible to prove that while the fields $E$ and $B$ oscillations are transverse to the direction of propagation of the e.m. wave, the momentum carried by the electromagnetic field moves along the direction of propagation of the field(in vacuum) since it is defined in units of area and time by the relation $$\vec{S}=\frac{\vec{E}\times\vec{B}}{\mu_0}$$ where $\vec{S}$ is the Poynting vector and the transferred momentum is $$\vec{P}=\frac{\vec{S}}{c}$$ that's a mathematical property of classical electrodynamics. Obviously a more precise explanation would require the concept of electromagnetic quanta, the photon which carries momentum $|p|=\frac{E=h\nu}{c }$ (from the special relativistic dispersion relation for massless particles $E=|p|c$) clearly along the direction of it's propagation and it's polarisation is transverse respect to it's direction of propagation since it is a massless particle.

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