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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.
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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.
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
Periodic Waves
• An end of a string is moved up and down as in SHM.
• Symmetric pattern of crests and troughs
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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.
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
Periodic Longitudinal Waves
• Can be desscribed in the same way.
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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)
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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)
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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.
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
Wave Interference, Boundary
Conditions and Superposition
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
Wave Interference, Boundary
Conditions and Superposition
𝑦 𝑥, 𝑡 = 𝑦1 𝑥, 𝑡 + 𝑦2(𝑥, 𝑡)
Principle of superposition
Constructive interference Destructive interference
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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,…
Johan Gustafson Optics and Waves, FYSA01, Spring 2014 – Chapter 15
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