# Wright State University Lake Campus/2016-1/Phy2400/log/Guy vandegrift

Solutions to questions 13, 14, 15 on How things work college course/Waves (Physics Classroom)

This is demonstrated at http://www.physicsclassroom.com/mmedia/waves/ltm.cfm

To do these problems, we need formulas relating how wave speed, wavelength and frequency change when a ropes density changes.

First we need to understand what cannot change along a rope:

1. Unless the rope is massive and held vertically, the tension remains constant. (If the rope is massive and held vertically, the weight of the rope causes the tension to change; we shall neglect this effect).
2. Excluding exotic effects associated with General relativity, the frequency remains constant whenever the source and reciever remain at a constant distance from each other. To understand why, suppose the transmitter creates N cycles in a time ΔT, beginning the transmission at t=0 and terminating it at t=ΔT. The receiver receives the first cycle after a delay time of t=tDELAY and the last cycle at time t=tDELAY+t=ΔT. However the duration over which the N cycles are observed is ΔT. Both the receiver and the sender perceive the same frequency, which is f=N/ΔT.

We also need two equations:

• ${\displaystyle v_{s}={\sqrt {\frac {F}{\mu }}}=f\lambda }$ is the speed of a wave in a stretched string, where ${\displaystyle F}$ is the tension and ${\displaystyle \mu }$ is the linear mass density, ${\displaystyle f}$ is frequency, and ${\displaystyle \lambda }$ is wavelength.

## How speed changes if density changes

13. A dense rope is connected to a rope with less density (i.e. fewer kilograms per meter). If the rope is stretched and a wave is sent along high density rope,

___ a) the low density rope supports a wave with a higher speed
___ b) the low density rope supports a wave with a lower frequency
___ c) the low density rope supports a wave with a higher frequency
___ d) the low density rope supports a wave with a lower speed

We already know that both waves have the same frequency, which eliminates two of these answers. Now use:

${\displaystyle v={\sqrt {\frac {F}{\mu }}}=f\lambda }$

Since speed is inversely porportional to the square root of density, we see that the lower density rope supports a wave of higher speed, so the correct answer is "a".

## How wavelength changes if rope density changes

14. What happens to the wavelength on a wave on a stretched string if the wave passes from lightweight (low density) region of the rope to a heavy (high density) rope?

___ a) the wavelength gets longer
___ b) the wavelength stays the same
___ c) the wavelength gets shorter

Recall that f is constant, which implies that lambda is also inversely proportional to the square root of density. At high density, the wavelength gets smaller, so the correct answer is "c". Note: The answer key is wrong here! (This question will therefore be removed from the exam).

## How frequency changes

15. When a wave is reflected off a stationary barrier, the reflected wave

___ a) has higher frequency than the incident wave
___ b) has lower amplitude than the incident wave
___ c) both of these are true

This question is somewhat misleading. Under ideal conditions, the reflected wave is equal, unless part of the wave is transmitted. I suppose the "least wrong" answer is (b), on the grounds that something always gets through. The question needs to be changed from reflection off a barrier to reflection due to a suddent change of density. In that case, the reflected wave has a lower amplitude. It is also lower if the "barrier" is a mass attached to the string.