  In this work, analyzing the propagation of electromagnetic waves in the field
of gravitational waves, we show the presence and significance of the so called
surfing effect for pulsar timing measurements. It is shown that, due to the
transverse nature of gravitational waves, the surfing effect leads to enormous
pulsar timing residuals if the speed of gravitational waves is smaller than
speed of light. This fact allows to place significant constraints on parameter
$\epsilon$, which characterizes the relative deviation of the speed of
gravitational waves from the speed of light. We show that the existing
constraints from pulsar timing measurements already place stringent limits on
$\epsilon$ and consequently on the mass of graviton $m_g$. These limits on
$m_g$ are three orders of magnitude stronger than the current constraints from
Solar System tests. The current constraints also allow to rule out massive
gravitons as possible candidates for cold dark matter in galactic halo. In the
near future, the gravitational wave background from extragalactic super massive
black hole binaries, along with the expected sub-microsecond pulsar timing
accuracy, will allow to achieve constrains of $\epsilon\lesssim 0.4\%$ and
possibly stronger.
