Why does a water fountain appear to pulse?

If it is a pump driven feature either the impeller blade of the pump are damaged or more likely the pump is cavitating, that is to say it has air trapped inside it causing variations in the outflow of the pump - this normally occurs with externally mounted water pumps (i.e. not submersible pumps).

If it is an external pump and this is the case, there is normally a bleed screw that can be removed to allow the air to escape. If it persists the consider installing a non-return valve between the pump and the fountain to prevent air being sucked back into the pump.