Waves - Case Studies - Questions 12 Help

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The speed of a wave in a rope is affected by the properties of the rope - mainly, how tight it is pulled (tension) and its mass per length (linear density). Changes in either of these properties changes the wave speed. For a fixed frequency, a change in the wave speed will also change the wavelength of the wave.

There are three similar versions of this question. Here is one of the versions:

Version 1:

The diagram at the right is a snapshot in time of a wave moving along a rope. Which diagram below represents a wave with the same frequency but moving through a different rope having one-fourth the linear density and pulled to the same tension? Tap on an image to select or deselect it.

You have to make some decisions to be successful with this question. First you have to decide on how quartering the linear density affects the speed. Then you have to decide the affect of this speed change on wavelength. And finally, you have to decide which diagram displays such a wavelength.

How Does Density Affect Speed?
In this question, the linear density of the rope is quartered (made four times smaller). The speed of a wave depends upon the linear density of the rope through which it moves. Decreasing the linear density increases the speed. The relatonship is that speed is inversely proportional to the square root of linear density. So a quartering of the linear density causes the speed to double or be twice as fast.

The Effect of Speed on Wavelength
In this question, the frequency at which the rope vibrates is not changed. For a constant frequency, a change in speed will affect the wavelength. Speed and wavelength are directly proportional to each other. So a doubling of speed causes a doubling of wavelength. You need to select the wave pattern that shows double or twice the wavelength.

Deciding on a Diagram
You can use the background grid to determine the wavelength of the wave. You can count the squares. It can be thought of as the distance between one crest and the next adjacent crest. Once you determine the wavelength in the given diagram, it's time to select the answer. The previous section on this Help page describe what type of wavelength you are looking for. You can do the square counting again to determine which pattern displays such a wavelength.

Try these links to The Physics Classroom Tutorial for more help with understanding wave properties:

The Anatomy of a Wave (wavelength and amplitude)

The Speed of a Wave

The Wave Equation