In a stretched string, the speed of a wave is directly proportional to the square root of the tension in the string. Therefore, if the speed of a wave is doubled, the tension in the string will need to be quadrupled (increased by a factor of four) in order to maintain the doubled speed.
Mathematically, the relationship between wave speed (v), tension (T), and linear mass density (μ) of the string can be expressed as:
v = √(T/μ)
If we denote the initial tension as T₁ and the initial speed as v₁, and the new tension as T₂ (doubled tension) and the new speed as v₂ (doubled speed), we can set up the following equation:
v₂ = 2v₁ √(T₂/μ) = 2√(T₁/μ)
Squaring both sides of the equation:
(T₂/μ) = 4(T₁/μ)
Canceling out the linear mass density (μ) on both sides:
T₂ = 4T₁
Therefore, if the speed of the wave in a stretched string is doubled, the tension in the string should be quadrupled.