1 Wave phenomena

Waves show reflection, refraction, diffraction and interference.

The reflection, refraction and dispersion of waves can be explained by Huygen’s principle.

Huygen’s principle (Essay)Every point on a wavefrontEvery point on a wavefront may be regarded as a sourcesource of secondary spherical wavelets which spread out with the wave velocity.

The new wavefrontnew wavefront is the envelopeenvelope of these secondary wavelets.

Constructed wavefront

First position of wavefront

Secondary source

First position of wavefront

Constructed wavefront

Secondary wavelet simulationsimulation

Explanation of law of reflection by Huygen’s principle

Consider AA’B’ and B’BA.AA’ = BB’ (provided)AA’B’ = ABB’ = 90o AB’ = AB’ (common side)AA’B’ B’BA (R.H.S.)a = b (corr. , )∵ i = a, a = b and b = r∴ i = r (law of reflection)

IncidentIncident wavefront

Secondary wavelet from A

ReflectedReflected wavefront

ia b

r

Explanation of law of refraction by Huygen’s principle

Secondary wavelet from A

Refracted wavefront

Incident wavefront

a

i

b

r

Medium 1

Medium 2

Consider AA’B’ and ABB’

2

1

2

1

'

''

'

sin

sin

v

v

tv

tv

AA

BB

AB

AAAB

BB

b

a

∵ i = a and r = b

∴ 2

1

sin

sin

v

v

r

i (constant)

the constant is called the refractive index, 1n2 for waves passing from medium 1 to medium 2.

(Snell’s law)

Explanation of dispersion by Huygen’s principle

If the speed of wavesspeed of waves in a given medium depends on the frequency of the waves, the medium is called a dispersive mediumdispersive medium.

Vacuum is a non-dispersive medium since the velocity of light for different colours (or frequencies) in vacuum is the same.

Glass is a dispersive medium because when white light enters glass, the velocity is not the same for different colours (or frequencies).

Dispersion

The secondary wavelet of blue light travels slower than that of red light in glass. Blue light is refracted more than red light and the refracted waves travel in slightly different direction.

This phenomenon is called dispersion

Waves of frequency f1 and f2 travelling with same speed

Medium 1

Medium 2

White light

Blue wavefront

Red wavefront

Reflection of a longitudinal pulsecompression rarefaction

rarefactioncompression

Explanation

rarefaction compressioncompression

Equilibrium positions

+ ve slope - ve slope

- ve slope

displacement

distance

Direction of wave

displacement

distance

Direction of wave

With a phase change of

(compression compression)

Explanation

rarefaction compressioncompression

Equilibrium positions

+ ve slope - ve slope

- ve slope

displacement

distance

Direction of wave

displacement

distance

Direction of wave

No phase change (compression rarefaction)

Application of reflection

Radar (radio detection and ranging)Employs microwaves (e.g. 3 cm microwaves )The distance d of the object can be calculated from the time lag t between the transmitted pulse P1 and the reflected pulse P2 by the equation d = 2ct where c is the speed of light.Distance of the object is determined form the time lag tSize of the object is determined by the strength of reflected waves.

Radar aerial

P1 P2

t

Sonar (sound navigation and ranging)

Employs ultrasonic waves. i.e. waves with f > f audible (20kHZ)

Submarines use sonar to keep track of water depth.

Fishing vessels use sonar to spot shoals of fish.

transmitter

ultrasound waves produced by a sonar

echo

Reasons for using ultrasound rather than audible sound

Less diffraction so that the wave is more concentrated and can penetrate to a greater depth.

Not be interfered by the audible sound in the sea.

Smaller objects can be located.

Reflection of transverse waves

• video

RefractionExample 2

• If i = 70o, find the angle of deviation d.

• By symmetry, i = r, a = b.• By geometry, d = a + b --- (1) and • (i – a) + (r – b) = 60o --- (2)• By Snell’s law,

sin i = n sin (i – a) --- (3)• Sub (1), (2) and (3)• sin i = n sin 30o

• sin 70o = n sin 30o

• n = sin 70o / sin 30o = 1.879• a + 30o = 70o

• a = 40o

• angle of deviation d = 2a = 80o

i r

60o

60o 60o

da b

60o

Real depth and apparent depth• Suppose a fish is at A

but it is image is at B which is nearer to water surface.

• AC is called the real depth and BC is the apparent depth.

i

i

r

r

A

B

C O

object

image

real depth D

apparent depth D’

)1(tan D

COi

)2('

tan D

COr

'tan

tan

)1(

)2(tan

tan

sin

sin

D

D

i

ri

r

i

rnwater

'Ddepthapparent

Ddepthrealnwater

SuperpositionTwo pulses on a string approaching each other.

The resultant displacement of the string is the sum of the individual displacements. i.e. the pulses superpose(疊置 ) .

A large pulse is produced.

After crossing, each pulse travels along the string as if nothing had happened and it has its original shape.

Principle of Superposition

Pulses (and waves), unlike particles, pass through each other unaffected.

The resultant displacement is the vector sum of individual displacements due to each pulse at that point

• Superposition can be used to find the resultant (solid line) of two waves (dotted line) of different wavelength and amplitude.

-1.5

-1

-0.5

0

0.5

1

1.5

A

BC

D

P

Q

R

S

distance

displacement

Example 2Two pulses are traveling toward each other, each at 10 cm s-1 on a long string. Sketch the shape of the string in the following at t = 0.6 s.

Solution:

Distance travelled by each pulse = vt = (10)(0.6) = 6 cm

1 cm