We propose an effective method to transmit only the non-dispersive lowest torsional wave mode at a high frequency range even above the cutoff frequency of the third torsional mode. Unlike existing methods that tune the wavelength or phase of the target wave mode, the proposed method is based on the thickness change and the cutoff phenomenon. A specially configured necked waveguide, consisting of three regions of which the middle region is thinner than the so-called cutoff thickness, is put in end-to-end contact with a test pipe to transmit only the first torsional wave mode to a test pipe. After explaining the underlying role of the proposed necked waveguide, we propose a technique to mainly transmit the lowest torsional wave mode at a frequency where higher modes can also propagate. Numerical simulations and damage detection experiments were carried out to show the effectiveness of the proposed method.