  Using an asymptotic phase representation of the particle density operator
$\hat{\rho}(z)$ in the one-dimensional harmonic trap, the part $\delta
\hat{\rho}_F(z)$ which describes the Friedel oscillations is extracted. The
expectation value $<\delta \hat{\rho}_F(z)>$ with respect to the interacting
ground state requires the calculation of the mean square average of a properly
defined phase operator. This calculation is performed analytically for the
Tomonaga-Luttinger model with harmonic confinement. It is found that the
envelope of the Friedel oscillations at zero temperature decays with the
boundary exponent $\nu = (K+1)/2$ away from the classical boundaries. This
value differs from that known for open boundary conditions or strong pinning
impurities. The soft boundary in the present case thus modifies the decay of
Friedel oscillations. The case of two components is also discussed.
