In lower hybrid current drive (LHCD) experiments on tokamaks, the parallel wavenumber of lower hybrid waves is an important physics parameter that governs the wave propagation and absorption physics. However, this parameter has not been experimentally well-characterized in the present-day high density tokamaks, despite the advances in the wave physics modeling. In this paper, we present the first measurement of the dominant parallel wavenumber of lower hybrid waves in the scrape-off layer (SOL) of the Alcator C-Mod tokamak with an array of magnetic loop probes. The electric field strength measured with the probe in typical C-Mod plasmas is about one-fifth of that of the electric field at the mouth of the grill antenna. The amplitude and phase responses of the measured signals on the applied power spectrum are consistent with the expected wave energy propagation. At higher density, the observed k|| increases for the fixed launched k||, and the wave amplitude decreases rapidly. This decrease is correlated with the loss of LHCD efficiency at high density, suggesting the presence of loss mechanisms. Evidence of the spectral broadening mechanisms is observed in the frequency spectra. However, no clear modifications in the dominant k|| are observed in the spectrally broadened wave components, as compared to the measured k|| at the applied frequency. It could be due to (1) the probe being in the SOL and (2) the limited k|| resolution of the diagnostic. Future experiments are planned to investigate the roles of the observed spectral broadening mechanisms on the LH density limit problem in the strong single pass damping regime.