## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

(a) We know that $f = \frac{v}{\lambda}$. If the length of the string doesn't change, then $\lambda$ doesn't change. To play twice the frequency, the speed of the wave along the string must double. $v \propto \sqrt{T}$ To double the speed, we need to increase the tension $T$ by a factor of 4. (b) We know that $f = \frac{v}{\lambda}$. If the tension of the string doesn't change, then $v$ doesn't change. To play twice the frequency, the wavelength must be half the original wavelength. The wavelength of the fundamental frequency is $2L$. To have a wavelength of $L$, the length of the string must be $\frac{L}{2}$ which is half the original length. We need to decrease the length by a factor of 2.