Superposition and Interference

Chapter 17

Expectations

After this chapter, students will: understand the principle of linear superposition apply the concept of interference to the interaction

of two or more waves use the concept of diffraction to understand what

happens when a wave passes through a limited opening

analyze situations that produce standing waves

Superposition

We defined a wave as a traveling condition or disturbance in a material.

We can also think of a wave as a traveling instruction to each particle of matter in the material: Be displaced upward! Be displaced downward! Be compressed! Re rarefied!

What happens when two waves give instructions to the same bit of material?

Superposition

What happens when two waves give instructions to the same bit of material?

The principle of linear superposition says that the two instructions add.

More formally: when two or more waves are present at the same place, the resultant disturbance is the sum of the individual disturbances due to the individual waves.

Superposition

Here, two positive-going pulses meet at a point. At the moment both pulses are centered at the same point, the displacement there is the sum of the individual pulse heights.

This is an example of constructive interference.

Superposition

Now, a positive-going pulse meets a negative-going one. The result, where they meet, is again the sum of the pulse heights, which in this case is zero.

This is an example of destructive interference.

Superposition: Interference

Suppose two sources of waves, in phase, send waves to the same location. The interference depends on the difference in path lengths.

d1

d2

Superposition: Interference

Constructive interference: path difference is an integer number of wavelengths.

Destructive interference: path difference is a half-integer number of wavelengths.

... 2, 1, ,0 21 mmdd

... 2, 1, ,0 2

121

mmdd

Huygens’ Construction

Christiaan Huygens 1629 – 1695

Dutch natural philosopher

Advanced the wave theory of light; constructed practical telescopes and microscopes, pendulum and spring-regulated clocks

Huygens’ Construction

Huygens said that each point on a wavefront can be thought of as a source of “wavelets:”

Huygens’ Construction

The wavelets interfere. If the wavefront is infinite in extent, for every point, in every direction except straight ahead, there is another point whose wavelets interfere destructively.

Huygens’ Construction

If the wavefront is limited, there are points near the edge which, for some directions, have no other point to serve as a “partner in destruction.” The wave spreads out at the edges: diffraction.

Diffraction

We can calculate directions for destructive interference:

D

D/2

(m + ½)

Dm

D

m

sin :0

2

21

sin

Diffraction

Note that this equation:

gives the first minimum in the diffraction pattern

(m = 0).

Applies to a “slit” opening: width (D) << height

D

sin

Diffraction

For a circular opening, we calculate:

This time, D is the diameter of the opening.

Notice that , the “spread angle,” is small as long as the opening size D is much larger than .

D

22.1sin

Interference: Different Frequencies

So far, we’ve been considering the interference of waves that have the same frequency.

What happens if two waves of different frequencies interfere at a location?

Interference: Different Frequencies

placeholder: beat frequency plot

Interference: Different Frequencies

The modulation or “beat” frequency in the superposition of the two waves is simply the difference between the interfering frequencies:

Note that the closer the two frequencies are to each other, the lower is the beat interference modulation frequency between them.

21 fffB

Interference: Standing Waves

Consider a transverse pulse on a string, encountering a rigidly-fixed end of the string.

The pulse is inverted

and reflected from that

end.

Interference: Standing WavesInstead of a pulse, consider a periodic wave that is so

reflected.

The “inverted” wave (phase-shifted by 180°, or half a cycle) interferes with the incident wave, producing a periodically-varying pattern in the string.

What if the length of the string is a multiple of half the wavelength?

Interference: Standing Waves

Interference: Standing Waves

Each half- section of the string becomes a “loop.”

nodes

antinodes

Interference: Standing WavesIf the string’s length is an integer multiple of half a

wavelength:

But: So,

The natural or standing-wave frequencies of a string depend on the velocity and string length.

And velocity depends on tension and linear mass density. This tells us how to tune a stringed instrument.

... 3, 2, 1, 2

nnL

f

v L

nvf

f

nvL

2or ,

2

Standing Longitudinal Waves

We can calculate natural standing-wave frequencies for sound waves in tubes from the same sort of analysis.

The results:

... 7, 5, 3, 1, 4

... 4, 3, 2, 1, 2

nL

nvf

nL

nvf

n

n

tube open at both ends

tube closed at one end