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 Superposition The 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: What does he hear now??

He moves again: 3.5 m 6 m 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: tiny for light larger for sound (better dispersion) wavelength size of obstacle

Huygen’s principle + math = ………… For a single slit (or doorway) of width D : D  Angle of 1st diffraction minimum Web Links: Diffraction of light Sun diffraction For a circular opening of diameter D : D  Angle of 1st diffraction minimum

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 If the frequency is just right, an integral number of these fit on the string, and we have Resonance Web Link: Transverse Standing Wave There are actually a number of different frequencies that will result in a standing wave

nodes (no vibration) antinodes (max. vibration)

In the previous example, the string was fastened to the wall: Hard Reflection: inverts the wave Soft Reflection: the wave returns upright If it had been loose instead: This creates an antinode at the end This creates a node 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: Web Link: String Harmonics

Longitudinal Standing Waves nodes (no vibration) antinodes (max. vibration) Web Link: Longitudinal standing wave

When air is blown over a bottle, it creates a standing longitudinal (sound) wave Remember, this is a longitudinal wave even though we draw it like this to visualize the shape. 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…… 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

Harmonic Frequencies for a tube open at both ends Web Link: Flute

Ex: 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.

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)

Now imagine starting with the complex sound wave, and trying to separate it into sine waves: f1 f2 f3 = Fourier Analysis Any periodic wave form can be represented as the sum of sine waves. Web Link: Fourier series