During trials of a new submarine, its speed is measured with an underwater sonar gun anchored to the ocean floor. The gun emits sound waves at a frequency \(f\), and these waves reflect off the surface of the submarine and return to the source at a frequency \(f^{\prime} eq f\). If the submarine is receding from the sonar gun at speed \(v\), derive an expression for \(f^{\prime}\) in terms of f, v, and c (the speed of the waves through water).