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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:
#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).
#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 &Delta;T, beginning the transmission at t=0 and terminating it at t=&Delta;T.  The receiver receives the first cycle after a delay time of t=t<sub>DELAY</sub> and the last cycle at time t=t<sub>DELAY</sub>+t=&Delta;T.  However the duration over which the N cycles are observed is &Delta;T.  Both the receiver and the sender perceive the same frequency, which is f=N/&Delta;T.

We also need two equations:

*<math>v_s =\sqrt{   \frac{F}{\mu}    } = f\lambda</math> is the speed of a wave in a stretched string, where <math>F</math> is the tension and <math>\mu</math> is the linear mass density, <math>f</math> is frequency, and <math>\lambda</math> is wavelength.

==How speed changes if density changes==
13. <!--b_waves_PC_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:

<math>v = \sqrt{   \frac{F}{\mu}    } = f\lambda</math> 

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. <!--b_waves_PC_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. <!--b_waves_PC_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.