Wave Propagation, Huygens’ Principle, and

Interference

Physics 101 LO

By Elaine Lee

Huygens’s Principle

Any point on a wave front can be considered to be point

source producing spherical secondary wavelets. The tangential surface of the

secondary wavelets predict the new position of the wave

front over time.

Christiaan Huygens. Digital image. Molecular Expressions. N.p., n.d. Web. 12 Mar. 2015.

This principle can be applied to all waves.

Plane Waves: -have wave fronts that are parallel to each other

Spherical Waves:- have spherical wave fronts that are centered on the point source

To Apply Huygens’ Principle:1. Draw a set of equally spaced points on the wave front.2. Using each point as the center of a secondary wavelet, draw a set of

spherical wavelets with the same radius. 3. Predict the resulting wave front by drawing tangents to the

spherical waves. Huygens' Principle. Digital image. Oocities. N.p., n.d. Web. 12 Mar. 2015. Huygens' Principle. Digital image. Cliffsnotes. N.p., n.d. Web. 12 Mar. 2015.

INTERFERENCE-separated by intervals of space and time

INTERFERENCE- separated by intervals of space

COMPARING THE PHASE DIFFERENCE OF ONE DIMENSIONAL WAVES TO 3D WAVES Consider waves with the same frequency and wavelengthOne Dimensional Waves:(waves propagate in one direction)

-have fixed phase differencesthat are independent of time and position(depends on the difference between phase constants of the two waves)

Three Dimensional Waves:(waves propagate in different directions)

-have relative phases that vary with position

E.g. phase constant difference of pi/3 rad

positions where the waves are perfectly out of phase positions where the waves are perfectly in phase

Crests of waves

Crests of waves

Hawkes, Iqbal, Mansour, Milner-Boloton, and Williams. Wave Interference Diagrams. Digital image. N.P., n.d. Web. 12 Mar 2015.

Interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lower amplitude

Two Point Source Interference Pattern

Crests

Troughs

Constructive Interference

Destructive Interference

Line of Constructive Interference

Two Point Source Interference Pattern. Digital image. Physicsclassroom. N.p., n.d. Web. 12 Mar. 2015.

DESTRUCTIVE INTERFERENCE:The path difference (d) between the two sources must be a half-integer multiple of the wavelength (i.e. an odd number of half wavelengths)

NOTE: Path difference is the difference in distance travelled by the two waves from their respective sources to a given point

NOTE: destructive interference can be observed when 2 waves out of phase by pi

General Condition For:

E.g. d = 4λ – 3.5λ = 0.5λ

Two Point Source Interference. Digital image. Physicsclassroom. N.p., n.d. Web. 12 Mar. 2015.

General Condition For:CONSTRUCTIVE INTERFERENCE:The path difference between the two sources must be an integer multiple of the wavelength

NOTE: the paths individually do not have to be integer multiples of the wavelengthNOTE: constructive interference can be observed when 2 waves are in phase

The function of a spherical wave is given by

We can neglect the phase constant if both waves are in phase

\\

NOTE: if d1 = d2 = d, simply add the two waves to find the resultant wave

E.g. d = 7λ – 6λ = 1λ

E.g. d = 7.5λ – 6.5λ = 1λ Two Point Source Interference. Digital image. Physicsclassroom. N.p., n.d.

Web. 12 Mar. 2015

QUESTION 1:

What is the path difference between the two sources to point A? Is there constructive interference or destructive interference at Point A?

Two Point Source Interference Pattern. Digital image. Physicsclassroom. N.p., n.d. Web. 12 Mar. 2015.

QUESTION 1 ANSWER:

The red lines indicate the paths travelled by the two waves from their respective sources to a Point A. The distance between each red dot is one wavelength (crest to crest). d = d2 – d1 = 6λ – 5λ = 1λThe path difference between the two sources to Point A is an integer multiple of the wavelength, thus Point A is a point of constructive interference.

QUESTION 2:At which points would constructive interference occur? How many of the labeled points represent nodes?

Two Point Source Interference Pattern. Digital image. Physicsclassroom. N.p., n.d. Web. 12 Mar. 2015.

QUESTION 2 ANSWER: Constructive interference would occur at

Point A and Point B because both points are at locations where a crest meets a crest.

Four out of the six points represent nodes. Points C, D, E and F are points where crests and troughs meet.

Interference- separated by intervals of time Beats are periodic fluctuations heard in the

intensity of a sound when two sound waves of very similar frequencies interfere with one another

The rate at which amplitude increases and decreases as a function of time is proportional to the frequency difference

Beat Frequency: the rate at which the volume is heard to be

oscillating from high to low volume

E.g. If three complete cycles of high and low volumes are heard every second, the beat frequency is 3 Hz.

The beat frequency is equal to the difference in frequency of the two tones that interfere to produce beats.

QUESTION 3:A guitarist plays a 110 Hz tone while his friend simultaneously plays a tone with a frequency of 115 Hz. How many beats will be heard over a period of 15 seconds?

QUESTION 3 ANSWER:The beat frequency is equal to the difference in frequency of the two tones that interfere to produce beats.

The beat frequency will be 5 Hz. ( 115 Hz – 110 Hz = 5 Hz )

Thus, in 15 seconds, there should be 75 beats.( 5 Hz x 15s = 75 beats)