We investigate the gravitational wave (GW) signal generated by a population of double neutron-star binaries (DNS) with eccentric orbits caused by kicks during supernova collapse and binary evolution. The DNS population of a standard Milky-Way type galaxy has been studied as a function of star formation history, initial mass function (IMF) and metallicity and of the binary-star common-envelope ejection process. The model provides birth rates, merger rates and total numbers of DNS as a function of time. The GW signal produced by this population has been computed and expressed in terms of a hypothetical space GW detector (eLISA) by calculating the number of discrete GW signals at different confidence levels, where `signal refers to detectable GW strain in a given frequency-resolution element. In terms of the parameter space explored, the number of DNS-originating GW signals is greatest in regions of recent star formation, and is significantly increased if metallicity is reduced from 0.02 to 0.001, consistent with Belczynski10a. Increasing the IMF power-law index (from --2.5 to --1.5) increases the number of GW signals by a large factor. This number is also much higher for models where the common-envelope ejection is treated using the $alpha-$mechanism (energy conservation) than when using the $gamma-$mechanism (angular-momentum conservation). We have estimated the total number of detectable DNS GW signals from the Galaxy by combining contributions from thin disc, thick disc, bulge and halo. The most probable numbers for an eLISA-type experiment are 0-1600 signals per year at S/N$geqslant$1, 0-900 signals per year at S/N$geqslant$3, and 0-570 at S/N$geqslant$5, coming from about 0-65, 0-60 and 0-50 resolved DNS respectively.
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