waves and optics - lunds universitetww2.sljus.lu.se/staff/johan/fysa01/spring14/ch15.pdf · johan...

Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15

Mechanical Waves

• Waves can occur whenever a system is disturbed from

equilibrium and when the disturbance can travel, or

propagate, from one region of the system to another.

• Mechanical waves travel in a medium.

• In chapter 15 – strings.

• Not all kinds of waves need a medium – light.


Mechanical Waves

• The disturbance is propagating in all kinds of waves.

• The speed of the wave is 𝑣. (The speed of the particles

are for instance 𝑣𝑦.)

• The medium does not move along the wave.

• Waves transport energy – not matter.


Periodic Waves

• An end of a string is moved up and down as in SHM.

• Symmetric pattern of crests and troughs


Periodic Waves

• The wave inherits 𝐴, 𝑓, 𝜔 = 2𝜋𝑓 and 𝑇 =1

𝑓from the SHM.

• Any kind of periodic wave can be expressed as a sum of

sinusoidal waves.

• The waves moves a distance 𝜆 in the time 𝑇.

• 𝜆: Wavelength = distance between two crests.

• Wave speed 𝑣 =𝜆

𝑇= 𝜆𝑓.

• All particles along the string moves with

the same frequency


Periodic Waves

• Here we will study one-dimensional waves, but the same ideas

are valid for 2- and 3- dimensional waves as well.

• 𝑣 is most often determined by the properties of the medium.

– 𝑣 = 𝜆𝑓

– 𝜆 decreases if 𝑓 increases and vice versa.


Periodic Longitudinal Waves

• Can be desscribed in the same way.


Mathematical Description of a Wave

• Every particle along the string moves according to

SHM.

• Described by the ”wave function” 𝑦(𝑥, 𝑡).

• Point A and B are not in phase.

• Left end: 𝑥 = 0

– 𝑦 𝑥 = 0, 𝑡 = 𝐴 cos𝜔𝑡

• It takes a time 𝑥

𝑣for the wave to move from 𝑥 = 0 to 𝑥.

A particle at 𝑥 moves as a particle in 𝑥 = 0 did at the

time 𝑡 −𝑥

𝑣.

• Exchange 𝑡 by 𝑡 −𝑥

𝑣:

– 𝑦 𝑥, 𝑡 = 𝐴 cos 𝜔 𝑡 −𝑥

𝑣


Mathematical Description of a Wave

• 𝑦 𝑥, 𝑡 = 𝐴 cos 𝜔 𝑡 −𝑥

𝑣

• Since cos 𝜃 = cos(−𝜃), we can write

– 𝑦 𝑥, 𝑡 = 𝐴 cos 𝜔𝑥

𝑣− 𝑡 - Cosine wave moving along 𝑥.

•𝜔

𝑣=2𝜋𝑓

𝜆𝑓= 2𝜋

1

𝜆; 𝜔 =

2𝜋

𝑇

– 𝑦 𝑥, 𝑡 = 𝐴 cos 2𝜋𝑥

𝜆−𝑡

𝑇


Wave Number

• Define the wave number as

𝑘 =2𝜋

𝜆(rad/m)

• Compare with angular frequency:

𝜔 =2𝜋

𝑇(rad/s)

• 𝑘 is the period in space.

• 𝑇 is the period in time.

𝑦 𝑥, 𝑡 = 𝐴 cos 2𝜋𝑥

𝜆−𝑡

𝑇

𝑦(𝑥, 𝑡) = 𝐴 cos(𝑘𝑥 − 𝜔𝑡)


Graphing the Wave Function

• The wave at a fixed time 𝑡 = 0.𝑦 𝑥, 0 = 𝐴 cos 𝑘𝑥

• The wave at a fixed point 𝑥 = 0.𝑦 0, 𝑡 = 𝐴 cos𝜔𝑡


Wave Along the −𝑥 Axis

• A wave moving in the opposite direction reaches the point

𝑥 a time 𝑡/𝑣 earlier than it reaches 𝑥 = 0.

• Previously we exchanged 𝑡 with 𝑡 −𝑥

𝑣for a wave going in

positive x direction. For the opposite direction we instead

exchange 𝑡 with 𝑡 +𝑥

𝑣.

𝑦 𝑥, 𝑡 = 𝐴 cos 2𝜋𝑥

𝑣+ 𝑡 = 𝐴 cos 2𝜋

𝑥

𝜆+𝑡

𝑇= 𝐴 cos(𝑘𝑥 + 𝜔𝑡)

• In general:

𝑦 𝑥, 𝑡 = 𝐴 cos(𝑘𝑥 ± 𝜔𝑡)


Phase

• 𝑦 𝑥, 𝑡 = 𝐴 cos(𝑘𝑥 ± 𝜔𝑡)

• The quantity (𝑘𝑥 ± 𝜔𝑡) is called the phase and is always

expressed in radians. The phase determines where on

the sinusoidal cycle a certain point 𝑥 is at the time 𝑡.

• For a crest following a wave, the phase is constant:

𝑘𝑥 − 𝜔𝑡 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑥 =𝜔𝑡

𝑘+ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑑𝑥

𝑑𝑡=𝜔

𝑘= 𝑣

• 𝑣 is the wave speed or phase velocity.


Particle Velocity and Acceleration

• 𝑦 𝑥, 𝑡 = 𝐴 cos(𝑘𝑥 − 𝜔𝑡)

• 𝑣𝑦 =𝑑𝑦

𝑑𝑡= 𝜔𝐴 sin(𝑘𝑥 − 𝜔𝑡)

• 𝑎𝑦 =𝑑2𝑦

𝑑𝑡2= −𝜔2𝐴 cos 𝑘𝑥 − 𝜔𝑡 = −𝜔2𝑦(𝑥, 𝑡)

•𝑑𝑦

𝑑𝑥= −𝑘𝐴 sin 𝑘𝑥 − 𝜔𝑡 – The slope the string at time 𝑡.

•𝑑2𝑦

𝑑𝑥2= −𝑘2𝐴 cos 𝑘𝑥 − 𝜔𝑡 = −𝑘2𝑦(𝑥, 𝑡) – curvature at 𝑡.

•𝑑2𝑦

𝑑𝑥2=𝑘2

𝜔2d2y

dt2𝜔

𝑘= 𝑣

𝑑2𝑦

𝑑𝑥2=1

𝑣2𝑑2𝑦

𝑑𝑡2

Wave equation

Valid also in the −𝑥 direction


Wave Speed – Dimension Analysis

• What might affect the wave speed, v [LT-1]?

– 𝐹 tension [MLT-2]

– 𝜇 mass per length unit [ML-1]

– 𝑓 frequency [T-1]

• 𝑣 = 𝑐𝑜𝑛𝑠𝑡 ⋅ 𝐹𝑥𝜇𝑦𝑓𝑧

𝐿𝑇−1 = 𝑀𝑥𝐿𝑥𝑇−2𝑥𝑀𝑦𝐿−𝑦𝑇−𝑧

𝑀: 𝑥 + 𝑦 = 0

𝐿: 𝑥 − 𝑦 = 1

𝑇:−2𝑥 − 𝑧 = −1

𝑦 = −𝑥𝑥 − −𝑥 = 12𝑥 = 1

𝑥 =1

2

𝑦 = −1

2

𝑧 = 1 − 2𝑥 = 1 − 21

2= 0

𝑣 = 𝑐𝑜𝑛𝑠𝑡𝐹

𝜇


Wave Speed

• Constant force 𝐹𝑦

• Impulse 𝐹𝑦𝑡

• Momentum 𝑚𝑣𝑦

• Constant speed – 𝑃 moves with constant speed

• 𝑚 = 𝑣𝑡𝜇 increases linearly with time

• Total force on string end 𝐹𝑡𝑜𝑡 = 𝐹2 + 𝐹𝑦2 along the string

•𝐹𝑦

𝐹=𝑣𝑦𝑡

𝑣𝑡 𝐹𝑦𝑡 = 𝐹

𝑣𝑦𝑡

𝑣Transverse impulse

• Transverse momentum: 𝑚𝑣𝑦 = 𝑣𝑡𝜇𝑣𝑦

• Impulse=momentum: 𝐹𝑣𝑦𝑡

𝑣= 𝑣𝑡𝜇𝑣𝑦

𝐹

𝜇= 𝑣2 𝑣 =

𝐹

𝜇


Energy in Wave Motion

• Point 𝑎 on a string carrying a wave from left to right.

– Slope = Δ𝑦

Δ𝑥 𝜕𝑦

𝜕𝑥

– Negative slope = 𝐹𝑦

𝐹

– 𝐹𝑦 𝑥, 𝑡 = −𝐹𝜕𝑦(𝑥,𝑡)

𝜕𝑥

• Power on point 𝑎:

𝑃 𝑥, 𝑡 = 𝐹𝑦 𝑥, 𝑡 𝑣𝑦 𝑥, 𝑡

= −𝐹𝜕𝑦 𝑥,𝑡

𝜕𝑥

𝜕𝑦 𝑥,𝑡

𝜕𝑡

Instantaneous rate of energy transfer

Valid for any wave on a string

Note that at zero slope there is no energy transfer


Energy in Sinusoidial Wave

• 𝑦 𝑥, 𝑡 = 𝐴 cos 𝑘𝑥 − 𝜔𝑡

•𝜕𝑦 𝑥,𝑡

𝜕𝑥= −𝑘𝐴 sin 𝑘𝑥 − 𝜔𝑡

•𝜕𝑦 𝑥,𝑡

𝜕𝑡= 𝜔𝐴 sin 𝑘𝑥 − 𝜔𝑡

• 𝑃 𝑥, 𝑡 = 𝐹𝑘𝜔𝐴2 sin2(𝑘𝑥 − 𝜔𝑡)

𝜔 = 𝑣𝑘 and 𝑣2 =𝑓

𝜇

• 𝑃 𝑥, 𝑡 = 𝜇𝐹𝑤2𝐴2 sin2(𝑘𝑥 − 𝜔𝑡)

• 𝑃𝑚𝑎𝑥 = 𝜇𝐹𝑤2𝐴2

• 𝑃𝑎𝑣 =1

2𝜇𝐹𝑤2𝐴2 Average power sinusoidal wave on a string

0 T/2 T 3T/2 2T-1

-0.5

0

0.5

1

1.5

2

y

P

v


Wave Intensity

• 𝐼 = the time average rate at which energy is transported

by the wave, per unit area perpendicular to the direction

of the direction of propagation.

• Inverse square law

𝐼1 =𝑃

4𝜋𝑟12

4𝜋𝑟12𝐼1 = 4𝜋𝑟2

2𝐼2

𝐼1𝐼2=𝑟22

𝑟11


Wave Interference, Boundary

Conditions and Superposition


Wave Interference, Boundary

Conditions and Superposition

𝑦 𝑥, 𝑡 = 𝑦1 𝑥, 𝑡 + 𝑦2(𝑥, 𝑡)

Principle of superposition

Constructive interference Destructive interference


Standing Waves


Standing Waves

𝜆/2


Standing Waves


Standing Wave

on a string fix fixed end at x=0

𝑦1 𝑥, 𝑡 = −𝐴 cos(𝑘𝑥 + 𝜔𝑡)Wave in negative x direction

𝑦2 𝑥, 𝑡 = 𝐴 cos 𝑘𝑥 − 𝜔𝑡 Wave in positive x direction

𝑦 𝑥, 𝑡 = 𝑦1 𝑥, 𝑡 + 𝑦2 𝑥, 𝑡 = 𝐴 cos 𝑘𝑥 − 𝜔𝑡 − cos 𝑘𝑥 + 𝜔𝑡

cos(𝑎 ± 𝑏) = cos 𝑎 cos 𝑏 ∓ sin 𝑎 sin 𝑏𝑦 𝑥, 𝑡 = 𝐴[cos 𝑘𝑥 𝑐𝑜𝑠𝜔𝑡 + sin 𝑘𝑥 sin𝜔𝑡 − cos 𝑘𝑥 𝑐𝑜𝑠𝜔𝑡 + sin 𝑘𝑥 sin𝜔𝑡]= 2𝐴 sin 𝑘𝑥 sin𝜔𝑡

𝐴𝑆𝑊 = 2𝐴


Standing Wave

on a string fix fixed end at x=0

• 𝑦 𝑥, 𝑡 = 2𝐴 sin 𝑘𝑥 sin𝜔𝑡

• 2𝐴 sin 𝑘𝑥 can be considered an amplitude that varies with x

• Nodes when sin 𝑘𝑥 = 0, i.e.

𝑥 = 0,𝜋

𝑘,2𝜋

𝑘,3𝜋

𝑘,… = 0,

𝜆

2, 𝜆,3𝜆

2,…

• Antinodes when sin 𝑘𝑥 = 1, i.e.

𝑥 =𝜋

2𝑘,3𝜋

2𝑘,5𝜋

2𝑘,… =

𝜆

4,3𝜆

4,5𝜆

4,…


Normal Modes

• String of length 𝐿.

• Nodes in both ends

• Distance between 2 nodes = 𝜆/2.

• 𝐿 = 𝑛𝜆

2(n = number of antinodes)

• 𝜆𝑛 =2𝐿

𝑛

• 𝑓 =𝑣

𝜆

• 𝑓𝑛 =𝑣𝑛

2𝐿

• 𝑓1 =𝑣

2𝐿– Fundamental frequency

• 𝑓𝑛 = 𝑛𝑓1 – Harmonics


Normal Modes

• 𝑦𝑛 𝑥, 𝑡 = 𝐴𝑆𝑊 sin(𝑘𝑛𝑥 − 𝜔𝑛𝑡)

where 𝑘𝑛 =2𝜋

𝜆𝑛and 𝜔𝑛 = 2𝜋𝑓𝑛

• When a guitar string is plucked and released, a standing

wave results, which is a sum of all the normal modes.

• 𝑓1 =𝑣

2𝐿and 𝑣 =

𝐹

𝜇gives 𝑓1 =

1

2𝐿

𝐹

𝜇

waves and optics - lunds universitetww2.sljus.lu.se/staff/johan/fysa01/spring14/ch15.pdf · johan...

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