Superposition and Wave Interference

IB Physics

Superposition and

Wave Interference

What happens when two waves are present at the same place at the same time?

Web Link: Wave Interference

The Principle of SuperpositionThe net effect = The sum of the individual effects

For waves:The resulting wave = the sum of the individual waves

This applies to all waves: water, light, sound, etc.

Interference of Sound Waves

Imagine two speakers, each playing a pure tone of wavelength 1 meter:

3 m 3 m

This is called Constructive Interference

We also say that these two waves are In Phase

Now suppose the listener moves:

3 m

5 m

What does he hear now??

3.5 m

6 m

He moves again:

Path length difference = 2.5 m = 2.5

off by ½ wavelength

This is called Destructive Interference

We also say that these two waves are Exactly Out of Phase

Ex: Noise canceling headphones

If you’re standing in a place where destructive interference is occurring, where did the energy of the sound waves go? Is energy still conserved in this case??

Web Link: Interference patterns

Interference Summary

path 1 path 2

If the difference in path lengths is………

0, 1, 2, 3, etc…… Constructive

½ , 1½ , 2½ , etc…… Destructive

Ex:

If these two speakers are each playing a 412 Hz tone, and the listener is standing 3.75 m away from one and 5.00 m away from the other, what does he hear?

Diffraction –The bending of a wave around an obstacle

with diffraction without diffraction

Web Link: Diffraction

Why does a wave bend??

Huygen’s Principle – Every point on a wavefront acts as a new spherical source

Web Link: Huygen’s Principle

All waves exhibit diffraction, including light

So why can’t you see around corners?

The extent of diffraction is determined by this ratio:

D

wavelength

size of obstacle

tiny for lightlarger for sound (better dispersion)

Huygen’s principle + math = …………

For a single slit (or doorway) of width D :

s in

D

Angle of 1st diffraction minimum

D

For a circular opening of diameter D :

s in .

1 22D

Angle of 1st diffraction minimum

Web Links: Diffraction of lightSun diffraction

D

Remember Constructive and Destructive Interference?

So far, we’ve only looked at interference between waves of the same frequency. What if the frequencies are slightly different?

We can still use Superposition to add them

fbeat = f1 – f2

The beat frequency of an additional loudness wave

Web Links: Sound Beats, Beats

Ex: Piano Tuning

Transverse Standing Waves

Hits the wall and bounces back

Web Link: Transverse Standing Wave

There are actually a number of different frequencies that will result in a standing wave

If the frequency is just right, an integral number of these fit on the string, and we have Resonance

nodes (no vibration)

antinodes (max. vibration)

In the previous example, the string was fastened to the wall:

If it had been loose instead:

Hard Reflection: inverts the wave

This creates a node at the end

Soft Reflection: the wave returns upright

This creates an antinode at the end

Web Link: Hard & soft reflections

back to…… Harmonics-

Natural frequencies of the system(f1, f2, f3, etc.)

fundamental frequency

Ex: The Cello

The C-string on a cello plays a fundamental frequency of 65.4 Hz. If the tension in the string is 171 N, and the linear density of the string is 1.56 x 10-2 kg/m, find the length of the string.

We can derive a formula to calculate all of the harmonic frequencies for any string:

f n v2Ln

n = 1,2,3,4,…

Web Link: String Harmonics

Longitudinal Standing Waves

antinodes (max. vibration) nodes (no vibration)

Web Link: Longitudinal standing wave

Remember, this is a longitudinal wave even though we draw it like this to visualize the shape.

When air is blown over a bottle, it creates a standing longitudinal (sound) wave

open end: antinode

vibrating air molecules

closed end: node

You can also ring a tuning fork over a bottle or tube, and if it creates wavelengths of just the right length, you’ll get a standing wave (loud sound).

Just like we did for strings, we can also derive a formula to calculate……

f n v4Ln

n = 1,3,5,…

The Harmonic Frequencies for a tube open at one end

speed of sound

odd harmonics only

Standing waves can also occur in a tube that is open at both ends

f n v2Ln

n = 1,2,3,4,…

Harmonic Frequencies for a tube open at both ends

Web Link: Flute

Find the range in length of organ pipes that play all frequencies humans can hear. Assume that the organ pipes are open at both ends, and they each play their fundamental frequency.

Ex:

Complex Sound Waves

Musical instruments play different harmonics at the same time

Web Links: String Harmonics, Flute

f1

f2 f3

=

Shape identifies the instrument

The shape of a vocal sound wave tells us who’s singing (or who’s on the other end of the phone)

Fourier AnalysisAny periodic wave form can be represented as the sum of sine waves.

f1

f2 f3

=

Now imagine starting with the complex sound wave, and trying to separate it into sine waves:

Web Link: Fourier series