Final answer:
If the tension on a string is doubled, the speed of the wave increases by a factor of √2, and consequently, the frequency required to achieve the fundamental mode should also be multiplied by √2.
Step-by-step explanation:
The question concerns the relationship between the tension on a string and the frequency of waves needed to produce the fundamental mode of vibration. In physics, particularly when discussing waves on strings such as on musical instruments, the speed of a wave v depends on the tension FT and the linear mass density μ of the string according to the equation v = √(FT/μ). When the tension is doubled, the speed of the wave also increases because the speed is proportional to the square root of the tension.
The frequency f of the fundamental mode is related to the speed v and the length L of the string by f = v/(2L). If you double the tension, the wave speed increases by a factor of √2, and so to achieve the same fundamental frequency, you should increase the frequency by a factor of √2 as well. So the correct answer to the question is that to get the fundamental mode back when the tension is doubled, the frequency must be multiplied by √2.