Axial sinusoidal pulser

A mud pulse telemetry system, comprising:

• a drill string;
• a pulser positioned within the drill string;
• and
• a piston disposed within the drill string and connected to the pulser, the piston configured to generate axial movement of the pulser within the drill string, wherein a portion of the pulser is defined by a transcendental function, wherein the transcendental function is an exponential equation, a logarithmic equation, a trigonometric equation or any combination thereof, wherein axial movement of the pulser within the drill string generates a sinusoidal pressure wave to the surface via mud within the drill string.

A mud pulse telemetry system, comprising:

• a drill string;
• a pulser supported for axial motion within the drill string between an uphole position and a downhole position;
• a restriction positioned within the drill string and configured to generate a pressure differential therein between a high pressure region adjacent one side of the restriction and a low pressure region on another side of the restriction;
• a conduit extending through the pulser, the conduit including a first communication passage facilitating fluid communication between a first piston chamber and the high pressure region and a second communication passage facilitating fluid communication between a second piston chamber and the high pressure region;
• a first outlet pipe facilitating fluid communication between the first piston chamber and the low pressure region, a first valve configured to selectively open and close the first outlet pipe;
• and
• a second outlet pipe facilitating fluid communication between the second piston chamber and the low pressure region, a second valve configured to selectively open and close the second outlet pipe, cyclical axial motion of the pulser within the drill string generating a sinusoidal pressure wave in an uphole direction.