Wave Interference

• Superposition Principle – when two or more waves encounter each other while traveling through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point.

Sound Wave Interference

When the path difference is an integer multiple of the wavelength of the sound, there is constructive interference. In this case a maximum in sound intensity is detected at the receiver.

When the path difference is a half-integer multiple of the wavelength of the sound, there is destructive interference. In the case of completely destructive interference, no sound is detected at the receiver.

Path difference r = |r2 - r1|

Mathematics of Interference

• Two waves, traveling in the same direction, with a phase difference

tkxAy sin1 tkxAy sin2

tkxAtkxAy sinsin

we can rewrite this using the following trigonometry rule:

2

cos2

sin2sinsin BABABA

2sin

2cos2 tkxAy

Mathematics of Interference (cont.)

• Two identical waves, traveling in opposite directions

tkxAy sin1 tkxAy sin2

tkxAtkxAy sinsin

using the same trig rule as before we get

tkxAy cossin2

Standing Waves

tkxAy cossin2

Standing Waves

• Standing waves can occur mathematically, as seen in the previous slides, and they can also occur in ropes and strings.

• This is how stringed musical instruments (violins, guitars, harps) produce sound. This effect also explains how you can change the sound produced by a string by tightening or loosening the string.

Standing Waves on a String• There are only certain stable patterns that will

occur on a particular string. These are called the normal modes of oscillation. The properties of the string (, L, T) determine the normal modes of oscillation.

• The motion of the string at one of these normal modes of oscillation is a standing wave. If the string is driven at a frequency that is not one of the normal frequencies, then the string will not exhibit a stable pattern (it will not produce a pleasant sound)

Standing Waves - String

Standing Wave - string terminology

n = 1 Fundamental 1st harmonic

n = 2 First overtone 2nd harmonic

n = 3 Second overtone

3rd harmonic

n = 4 Third overtone 4th harmonic

Standing Waves - string

• We can look at the standing wave patterns to determine a relationship between L and .

Fundamental

2nd harmonic

3rd harmonic

nth harmonic

21L

23L

L

2nL

Standing Waves - string

Rearrange that last equation:and since v=f

Replace the velocity of a wave on a string with the equation from last chapter,

nL

n2

Lnvvf

nn 2

121

2nfT

LnT

Lnfn

Standing Waves in a Pipe• Just like how standing waves are formed on a

string by the interference between two oppositely directed transverse waves, standing waves in pipes are the result of interference between two longitudinal sound waves traveling in opposite directions

• The interference is between the original wave sent into the pipe and its reflection.

• This is how musical instruments like flutes and pipe organs produce sound.

Standing Waves - pipe

• There are particular harmonics for sound waves in pipes. At each of the harmonics, the pipes produce a “clean” sound.

• The harmonics are dependent on the length of the pipe.

Pipe - open at both ends

Standing Wave – open pipe terminology

n = 1 Fundamental 1st harmonic

n = 2 First overtone 2nd harmonic

n = 3 Second overtone

3rd harmonic

n = 4 Third overtone 4th harmonic

Pipe - open at one end, closed at the other end

Standing Wave – closed pipe terminology

n = 1 Fundamental 1st harmonic

n = 3 First overtone 3rd harmonic

n = 5 Second overtone

5th harmonic

n = 7 Third overtone 7th harmonic

Resonance• All of the possible harmonic

frequencies are also called resonance frequencies.

• If you add energy to the system at a frequency equal to one of the resonance frequencies, you will continually add to the amplitude of the vibration (motion) of the system. Eventually, the system will break.

• Exs: Tacoma Narrows Bridge, a shattered wine glass (from a high note), a building that collapses during an earthquake

Beats

tfAy 11 2cos tfAy 22 2cos

tftfAtfAtfAyy 212121 2cos2cos2cos2cos

tfftffAy

2

2cos2

2cos2 2121

Beats (cont.)• There are a number of frequencies present in the

previous equation. One is the frequency of the resultant sound wave, one is the frequency of the amplitude. Another frequency we can pull out of that equation is the beat frequency.

• The beat frequency is the number of beats you hear per second. The maximum our ears can detect is about 20 beats per second. If the beat frequency is larger than that, the two sounds blend together.

21 fffb