Wave Superposition and Reflections

8 January 2016

PHYC 1290 Department of Physics and Atmospheric Science

Demo

Apparatus

Fixed end Free end

Reflections from fixed and free ends

Observations

1. Reflections from a fixed end are inverted.

2. Reflections from a free end are upright.

How can we explain / model this behaviour?

Principle of Superposition

The “principle of superposition” holds that the total wave displacement is the sum of the individual displacements.

What happens where two waves overlap?

e.g., For two waves with displacements y1(x,t) and y2(x,t), the total displacement is given by

y(x,t) = y1(x,t)+ y2(x,t)

Unlike objects, when waves meet they pass through each other.

y1

y2

... superpose...

Eg: Two identical pulses, one invertedy1

y2

Two waves approach...

y1

y2

... and continue on after.

Three Questions

Q: Is there an instant at which the string is completely straight?A: Yes!

Consider the same two identical pulses, with one inverted:

(superposition of thepulses gives 0 displacement)

Q: When the string is completely straight, is its kinetic energy momentarily zero?

Although the string is momentarily straight, it is still in motion, and so KE ≠ 0 .

A: No!

Recall that KE = ½mvy2 for each particle.

Superposition of the velocity vectors when the pulses overlap reveals that the string is in motion:

This can be understood by drawing particle velocity vectors for the leading and trailing edge of each pulse.

Q: As the pulses pass through one another, is there a point on the string that does not move?

A: Yes!

The red dot never moves, and is always on the string.

Reflection from a fixed end

If the point doesn’t move, we can clamp it in place without affecting the pulses.

Once the string is clamped, the right side of the string cannot exert any influence on the left side.

We can cut the string!

Clamp

We can use this last idea to construct a model of wave reflection.

➞ Waves that reflect from a fixed end are inverted.

Conclusion: A reflection from a fixed end is equivalent to the superposition of a “real” pulse with an identical, but inverted

“virtual” pulse approaching from the other direction.

wall

The “real” pulse passes into the wall and becomes virtual, whereas the “virtual” pulse comes out of the wall and becomes real.

wall

String end fixed to wall

Reflections from a free endA reflection from a free end is equivalent to the superposition

of a “real” pulse with an identical, upright “virtual” pulse approaching from the other direction.

Free end is able to move

(we cut the string)

Notice the big kick to y=2 cm at the free end

Application: Coax SignalsCoaxial cable is widely used for electrical signal transmission.

e.g., Cable TV, audio/video cables, scientific equipment

BNC coax connectors

Coax cross-section

Coax DemoElectrical pulse generator

V

tOscilloscope: history graphs

Open circuit (R = ∞)

Short circuit(R = 0)

Variable resistor

Try three ends:

75 m

Screenshot for the open-circuit reflection (it is just like a free-end reflection).

Q: What is the speed of a wave pulse on coaxial cable?

We know it takes 668 ns for a pulse to travel to the end of the cable and back again (150 m).

v = 150 m/ 668×10-9 s = 2.25×108 m/s

The speed of light is 3.00×108 m/s.

➞ Electrical signals travel at about 2/3 the speed of light!!

Media Credits

http://commons.wikimedia.org/wiki/File:Coaxial_cable_cutaway.svgAuthor: Tkgd2007

http://commons.wikimedia.org/wiki/File:BNC_connector_20050720_001.jpgAuthor: Jonas Bergsten

http://commons.wikimedia.org/wiki/File:RG-59.jpg

http://commons.wikimedia.org/wiki/File:BNC_connector_%28male%29.jpgAuthor: Krzysztof Burghardt

http://commons.wikimedia.org/wiki/File:10base2_terminator.png

Extra Material

http://youtu.be/31DRnzStmL4http://youtu.be/oQwHp5Z6y_g

Wave Reflection VideosSimon has made some fantastic slow-motion videos of our wave

reflection demonstrations available on YouTube.

Fixed endOpen end

Fixed end Free end

Loads and terminators

Coaxial cables have a characteristic “impedance” (resistance) which tells you what kind of terminator to use.

e.g.: RG-59: 75 ΩRG-58: 50 Ω

Similarly, the “loads” (i.e., devices) that are connected by coax must have the same impedance to prevent reflections. The load impedance is usually specified by the manufacturer.

A coax terminator

Unconnected coaxial cables can cause reflections and troublesome interference for electronic systems. Such cables should be

appropriately “terminated”.

Time Domain ReflectometersTime Domain Reflectometers (TDRs) detect electrical

transmission line breaks by sending out electrical pulses. For properly terminated systems there should be no return pulse.

Line breaks cause reflected pulses, and the delay time is used to locate the fault. The technique is particularly useful for finding

faults in underground wires which are not easily inspected.

An industrial TDR system

Image source:http://commons.wikimedia.org/wiki/File:Megger_Time-Domain_Reflectometer_MTDR1.jpg