A rotary acoustic horn imparts energy at a selected wavelength, frequency, and amplitude. The horn includes a cylindrical shaft and a cylindrical welding portion mounted on and coaxial with the shaft. The diameter of the welding portion is greater than the diameter of the shaft. The welding portion has a cylindrical weld face having a diameter that expands and contracts with the application of ultrasonic energy, and first and second opposing end portions. At least one end portion is formed with a first undercut; the radial width and the depth of the first undercut is selected to affect the amplitude of vibration at the weld face to provide a desired amplitude profile along the axial length of the weld face.