Chapter 11, Problem 16Q

To determine

why the frequency does not change (although the wavelength and velocity do change)when a sinusoidal wave crosses the boundary between two sections of cord.

Answer to Problem 16Q

Solution:

The frequency must stay the same because the media is continuous and it is determined by the time between vector reversals for the force causing the wave motion.

Explanation of Solution

The frequency must stay the same because the media is continuous - the end of one section of cord is physically tied to the other section of cord. If the end of the first section of cord is vibrating up and down with a given frequency, then since it is attached to the other section of cord, the other section must vibrate at the same frequency. If the two pieces of cord did not move at the same frequency, they would not stay connected, and then the waves would not pass from one section to another.

The wavelength and velocity of a wave are functions of the medium, in this case the rope. A sigmoidal wave can be thought of as a series of forces acting on the rope perpendicular to its length, and the force is constantly reversing direction (probably as a consequence of someone's hand whipping one end of the rope back and forth). Given the same amount of force, a lighter rope will accelerate more quickly in the direction of the amplitude and create a greater wavelength and faster velocity. A heavier rope will have a smaller amplitude, wavelength and velocity due to the inertia caused by its greater mass. The frequency, however, is determined by the time between vector reversals for the force causing the wave motion (in other words, how fast the guy at the end of the rope is waving it back and forth). The source of the wave motion (i.e. the guy's hand)is what determines the frequency; the nature of the rope is irrelevant.

Conclusion:

The source of the wave motion is what determines the frequency; the nature of the rope is irrelevant.