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We study numerically the transport and localization properties of waves inordered and disordered ladder-shaped lattices with local $\mathcal{PT}$symmetry. Using a transfer matrix method, we calculate the transmittance andthe reflectance for the individual channels and the Lyapunov exponent for thewhole system. In the absence of disorder, we find that when the gain/lossparameter $\rho$ is smaller than the interchain coupling parameter $t_{v}$, thetransmittance and the reflectance are periodic functions of the system size,whereas when $\rho$ is larger than $t_{v}$, the transmittance is found to be anexponentially-decaying function while the reflectance attains a saturationvalue in the thermodynamic limit. For a fixed system size, there appear perfecttransmission resonances in each individual channel at several values of thegain/loss strength smaller than $t_{v}$. A singular behavior of thetransmittance is also found to appear at various values of $\rho$ for a givensystem size. When disorder is inserted into the on-site potentials, thesebehaviors are changed substantially due to the interplay between disorder andthe gain/loss effect. When $\rho$ is smaller than $t_{v}$, we find that thepresence of locally $\mathcal{PT}$-symmetric potentials suppresses Andersonlocalization, as compared to the localization in the corresponding Hermitiansystem. When $\rho$ is larger than $t_{v}$, we find that localization becomesmore pronounced at higher gain/loss strengths. We also find that the phenomenonof anomalous localization occurs in disordered locally $\mathcal{PT}$-symmetricsystems precisely at the spectral positions $E=0$ and$E=\pm\sqrt{t_{v}^2-\rho^2}$. The anomaly at the band center manifests as asharp peak contrary to the conventional cases, whereas the anomalies at$E=\pm\sqrt{t_{v}^2-\rho^2}$ manifest as sharp dips.

Transport and localization of waves in ladder-shaped lattices with locally $\mathcal{PT}$-symmetric potentials