Final answer:
To decrease the wavelength in a rope, a student could increase the frequency of shaking, decrease the length of the string, or increase its tension. Decreasing tension would, in fact, increase the wavelength, which is not the intended effect. The relationship is described by the wave equation and the tension-mass per unit length relationship.
Step-by-step explanation:
A student is shaking a flexible string attached to a wooden board in a rhythmic manner, and is interested in decreasing the wavelength within the rope. To decrease the wavelength, the student could increase the frequency of the shaking, decrease the length of the string, or increase the tension in the string.
The relationship between these factors is described by the wave equation v = f × λ, where v is the speed of the wave, f is the frequency, and λ (lambda) is the wavelength. Since speed is also determined by the tension and the mass per unit length (μ) of the string, as in the equation v = √(T/μ), decreasing the tension would actually increase the wavelength, contrary to the goal.
Therefore, to decrease the wavelength without changing the speed, one must increase the frequency of the shaking or decrease the length of the string.