The speed of a wave traveling through a medium is determined by the properties of the medium itself. In the case of a transverse wave moving through a rope, the speed of the wave depends on the tension in the rope and the linear mass density (mass per unit length) of the rope.
When the rope is heavier (has a higher linear mass density), it means that more mass is distributed along its length. As a result, the rope becomes more resistant to motion, and the wave speed decreases. The formula that relates the wave speed (v), tension (T), and linear mass density (μ) of a rope is given by:
v = √(T / μ)
Since the wave speed is determined by the properties of the medium, changing the linear mass density of the rope will affect the wave speed.
The wavelength (λ) of a wave is defined as the distance between two consecutive points that are in phase with each other. The relationship between the wave speed (v), frequency (f), and wavelength (λ) is given by:
v = f * λ
Since the wave speed (v) decreases when the rope is heavier, and the frequency (f) remains constant, according to the above equation, the wavelength (λ) must also decrease. This is because the product of frequency and wavelength must remain constant when the wave speed changes.
In summary, a transverse wave moving through a heavier rope will have a smaller wavelength because the wave speed decreases due to the increased resistance of the medium. However, the frequency of the wave remains unaffected by the properties of the medium.