You know v = fλ. That equation links three quantities, but it does not say which one is in charge. The answer: the medium sets the speed, the source sets the frequency, and the wavelength is whatever fits, $\lambda = v/f$.
Key idea
On a stretched string with tension $F_T$ and mass per length $\mu = m/L$, $$v = \sqrt{\frac{F_T}{\mu}}.$$ Shaking faster raises $f$ and shortens $\lambda$, but leaves $v$ alone.
Why it works
A wave moves because each bit of string pulls on the next. Tension is the pull that snaps a displaced bit back, so more tension means a quicker response and a faster wave. Mass per length is the inertia that resists being moved, so a heavier string is slower. The square root comes from the units: $\frac{\text{N}}{\text{kg/m}} = \frac{\text{kg·m/s}^2}{\text{kg/m}} = \frac{\text{m}^2}{\text{s}^2}$, and only its square root is a speed. You found tension in Connected Objects and Tension; here it sets the speed.
Common mistake
Thinking that shaking a rope faster sends the waves along it faster. The speed is set by the rope (tension and mass per length). Shaking faster makes more waves per second, each one shorter, all moving at the same speed. Check: in $v = f\lambda$, a bigger $f$ with the same $v$ must give a smaller $\lambda$.
Worked example: Speed on a string
A string $0.80\ \text{m}$ long has a mass of $4.0\ \text{g}$ and a tension of $50\ \text{N}$. How fast do waves travel on it?
- Mass per length: $\mu = \dfrac{0.0040\ \text{kg}}{0.80\ \text{m}} = 0.0050\ \text{kg/m}$.
- $v = \sqrt{\dfrac{50}{0.0050}} = \sqrt{10{,}000} = 100\ \text{m/s}$.
Answer: $100\ \text{m/s}$
Worked example: Changing the tension
On the same string, what tension makes the waves go $200\ \text{m/s}$? What wavelength does a $250\ \text{Hz}$ wave then have?
- Square both sides: $F_T = \mu v^{2} = (0.0050)(200)^{2} = 200\ \text{N}$.
- Doubling the speed takes four times the tension, because $v \propto \sqrt{F_T}$.
- $\lambda = \dfrac{v}{f} = \dfrac{200}{250} = 0.80\ \text{m}$.
Answer: $200\ \text{N}$; $0.80\ \text{m}$
Sound follows the same rule: stiffness against inertia. It travels about $343\ \text{m/s}$ in air at $20^\circ\text{C}$, roughly $1500\ \text{m/s}$ in water and several thousand meters per second in steel.