A reference metallic alloy has a specific longitudinal wave speed. A newly developed variant of this alloy has a Young's modulus that is 4 % greater than that of the reference alloy. If the speed of longitudinal waves in this new variant is observed to be 1 % greater than in the reference alloy, the approximate percentage change in the density of the new variant compared to the reference is:

Step-by-step solution

The speed of a longitudinal wave in a solid is given by v = Y . Taking the natural logarithm and differentiating for small changes, we get: v v = 1 2 Y Y - 1 2 Given that the wave speed increases by 1 % ( v v = 1 % ) and Young's modulus increases by 4 % ( Y Y = 4 % ): 1 = 1 2 (4) - 1 2 ( ) 1 = 2 - 0.5 ( ) 0.5 ( ) = 1 = 2 % Since the value is positive, there is a 2 % increase in density. Answer: a 2 % increase