In a standing wave on a string, the wavelength is determined by the tension, frequency, and the linear mass density of the string. If we double the tension but hold the frequency and separation distance between the boundaries fixed, the wavelength of the standing waves will change.
According to the equation for the wavelength of a standing wave on a string:
wavelength = 2L/n
where L is the length of the string and n is the harmonic number. The tension is not directly involved in this equation.
Therefore, doubling the tension while keeping the frequency and separation distance fixed will not affect the wavelengths of the standing waves. The wavelengths will remain the same as they were before the tension was doubled.