Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Lecture 22

Chapter 21

Interference

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII

Lecture Capture: http://echo360.uml.edu/danylov201415/physics2spring.html

04.14.2015Physics II

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Standing Waves

Shown are the first four possible standing waves on a string of fixed length L.

These possible standing waves are called the normal modes of the string.

Each mode, numbered by the integer m, has a unique wavelength and frequency.

The lowest allowed frequency is called the fundamental frequency: f1 = v/2L.

Standing waves occur when both ends of a string are fixed.

The fundamental frequency f1 can be found as the difference between the frequencies of any two adjacent modes: f1 = Δf = fm+1 – fm.

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Standing Wave Generation (Demo)There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value.

What is the mode number of this standing wave? A) 4

B) 5C) 6D) Can’t say without knowing what kind of wave it is.

Number of antinodes = mode number

ConcepTest 1 Standing Wave

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

InterferenceA standing wave is the interference pattern produced when two waves of equal

frequency travel in opposite directions.

In this section we will look at the interference of two waves traveling in the same direction.

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Interference in One DimensionThe pattern resulting from the superposition of two waves is often called interference.

In this section we will look at the interference of two waves traveling in the same direction.

The resulting amplitude is A 0 for perfect destructive interference

The resulting amplitude is A 2a for maximum constructive interference.

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Let’s describe 1D interference mathematicallyConsider two traveling waves. They have:1. The same direction, +x direction2. The same amplitude, a3. The same frequency,

Let’s find a displacement at point P at time t:

P

, ,

The phase constant 0tells us what the source is doing at t 0.

+ Using a trig identity:0 0

cosΔ2 sin 2 2

sin

+

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Constructive/destructive interferenceIt is still a traveling wave with an amplitude:

where 1 - 2 is the phase difference between the two waves.

The amplitude has a maximum value A = 2a if

, , , , …

Conditions for constructive interference:

cos(/2) 1.

Similarly, the amplitude is zero, A=0 if

Conditions for destructive interference

cos(/2) 0. , , , , …

The amplitude:

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Let’s look deeper in Δ

So, there are two contributions to the phase difference:

2 - 1 is the phase difference between the two waves.

1. – pathlength difference2. -- inherent phase difference

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Inherent phase difference

These are identical sources: These are not identical sources: out of phase

/

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Pathlength difference for constructive interferenceAssume that the sources are identical 0. Let’s separate the sources with a pathlength ∆x

Conditions for constructive interference:

Thus, for a constructive interference of two identical sources with A = 2a, we need to separate them by an integer number of wavelength

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Pathlength difference for destructive interference

Conditions for destructive interference:

Thus, for a constructive interference of two identical sources with A =0, we need to separate them by an integer number of wavelength

/2

Assume that the sources are identical 0. Let’s separate the sources with a pathlength ∆x

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Applications

Thin transparent films, placed on glass surfaces, such as lenses, can control reflections from the glass.

Antireflection coatings on the lenses in cameras, microscopes, and other optical equipment are examples of thin-film coatings.

Noise-cancelling headphones

It allows reducing unwanted sound by the addition of a second sound specifically designed to cancel the first (destructive interference).

Two loudspeakers emit waves with .

What, if anything, can be done to cause constructive interference between the two waves?

A) Move speaker 1 forward by 0.5 mB) Move speaker 1 forward by 1.0 mC) Move speaker 1 forward by 2.0 mD) Do nothing

ConcepTest 3 1D interference

/

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Interferencein two and three

dimensions

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

A Circular or Spherical Wave

, ,

A circular (2D) or spherical (3D) wave can be written

A linear (1D) wave can be written

where r is the distance measured outward from the source.

The amplitude a of a circular or spherical wave diminishes as r increases.

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Transition from 1D to 2D/3D interference The mathematical description of interference in two or three dimensions is

very similar to that of one-dimensional interference. The conditions for constructive and destructive interference are:

Constructive:

Destructive:

one-dimensional two or three dimensions

where r is the path-length difference.

If the sources are identical ( ), the interference is

Constructive if

Destructive if

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Example of 2D interference

The figure shows two identical sources that are in phase.

The path-length difference rdetermines whether the interference at a particular point is constructive or destructive.

Two in-phase sources emit sound waves of equal wavelength and intensity. At the position of the dot,

A) The interference is constructive.B) The interference is destructive C) The interference is somewhere between constructive and destructive

D) There’s not enough information to tell about the interference.

ConcepTest 4 2D Interference

..

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

What you should readChapter 21 (Knight)

Sections 21.5 21.6 21.7

Department of Physics and Applied Physics95.144, Spring 2015, Lecture 22

Thank youSee you on Friday

At a football game, the “wave” might circulate through the stands and move around the stadium. In this wave motion, people stand up and sit down as the wave passes. What type of wave would this be characterized as?

A) polarized waveB) longitudinal waveC) lateral waveD) transverse waveE) soliton wave

The people are moving up and down, and the wave is traveling around the stadium. Thus, the motion of the wave is perpendicular to the oscillation direction of the people, and so this is a transverse wave.

ConcepTest 2 The Wave