Theme 3 Waves

Theme 3Waves

Module T3M1:Traveling Waves

1Copyright Phys 1A03, McMaster University

Periodic Sinusoidal Waves

A – amplitude (m)

f – frequency (Hz or cycles/second)

ω – angular frequency (radians/second)

T – period (seconds)

λ – wavelength (metres)

k – wavenumber (radians per meter)

c – speed of propagation

A

-A

y

x𝜆

𝜆


So many variables!!!• but it makes sense!

3

y(x, t) = A sin(kx± !t) y(x, t) = A sin[k(x± !k t)]

wavefunction: F (x± vt)

velocity (v or c) ! c = !/k

why ! and k instead of just � and T?

y(x, t) = A sin( 2⇡x� ± 2⇡tT )

Copyright Phys 1A03, McMaster University

Periodic wave functionThe graphs below describe a wave propagating in positive x-direction. What is the wave function y(x, t) for the wave?

Amplitude A:

Wavelength λ:

Wave vector k:

History (at x = 1)Snapshot (t = 0)

Period T:

Ang. Freq ω:

‘–’ for +x-directionCheck pts: (x,t) = (1,0), (x,t) = (1,1.5)


Quiz 9.5• What is the frequency of this traveling wave?

A. 10 HzB. 5 HzC. 2 HzD. 0.2 Hz

6

c = f�


Quiz 9.5• What is the frequency of this traveling wave?

A. 10 HzB. 5 HzC. 2 HzD. 0.2 Hz

7

c = f� f = c/�

f = 50/10 = 5 Hz

c = f�


How can you tell the direction of travel from the wave function?

)sin(),( tkxAtxy w-=

x

y

)cos(),( tkxAtxy w-=

y(x, t) = Asin(−kx +ωt)

)sin(),( tkxAtxy w+=

)cos(),( tkxAtxy w+=

y(x, t) = Acos(−kx −ωt)If kx and ωt term have opp. signs, then wave moves in +x dir.

remember sin(-h) = -sin(h) and cos(-h) = cos(h)8Copyright Phys 1A03, McMaster University

Traveling waves in a medium

• In order for an oscillation to propagate (travel), there must be interactions within the medium

1. The properties of the medium dictate the speed of the wave

2. The traveling wave carries energy with it

3. If the wave encounters a change in the medium, then at the boundary• It may be reflected and transmitted by the boundary• The wave characteristics (A, ƒ, λ, c) may change


Wave speed on a ropeThe tension of a rope and its mass influence the wave speed

Example: When the tension in a cord is 75.0 N, the wave speed is 140 m/s. What is the linear mass density of the cord?

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T – tension in the rope

(CAREFUL: Tension & Period!!!)

µ - mass density (mass per length)


Traveling wave ExampleA 2.0 m long string with a mass of 4.0 g is tied to a wall at one end, stretched horizontally to a pulley 1.5 m away, then tied to a physics book of mass M that hangs from the string. Experiments find that a wave pulse travels along the stretched string at 40 m/s. What is the mass of the book?

• What is the tension in the rope?• What is the mass density?

• Calculate mass of the book: Mg = T = μc2


Quiz 9.6A rope of length L and mass M hangs from a ceiling. If the bottom of the rope is given a gentle wiggle, a wave will travel to the top of the rope. As the wave travels upward, its speed…

A. …increases. B. …decreases. C. …stay the same.


Quiz 9.6A rope of length L and mass M hangs from a ceiling. If the bottom of the rope is given a gentle wiggle, a wave will travel to the top of the rope. As the wave travels upward, its speed…

A. …increases. B. …decreases. C. …stay the same.

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The mass density is constant, but the tension changes. The tension at any point in the rope is equal to the weight of the rope below that point.

c =

sT

µ

Increasing tension


Energy transported by wavesThe traveling wave carries energy with it• Kinetic energy through motion of the medium• Elastic potential energy in solid materials (eg. rope, slinky)• Gravitational potential energy (transverse water waves)• Potential energy associated with pressure/density (sound waves

in fluids)


Energy of periodic waves on a rope

• The vertical displacement of the rope → kinetic energy• The rope stretches from its rest length → elastic potential energyConceptually:• Larger amplitude (A):

→ more stretching→ rope must be moving up/down faster

• Heavier rope (µ):→ for a given A, more mass is being moved

• Greater frequency, speed (ω, c):→Vertical displacement happens faster→More waves pass a point per time

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Rate of energy transfer


Quiz 9.7Physics Girl and Einswine are creating periodic waves using identical ropes. Physics Girl is holding two ropes that are twisted together. Einswine is holding one rope but oscillates with twice the amplitude of Physics Girl. The frequencies and wave speeds are identical. Who transfers the most energy per unit time?

A. Physics GirlB. EinswineC. The energy per unit time is identicalD. The power is identical


Waves and boundaries• How do waves behave when they encounter a barrier?

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Fixed Boundary Loose Boundary


Quiz 9.8A transverse wave on a string is described by

It arrives at the point x = 0 where the string is fixed in place. Which function describes the reflected wave?


End Lecture 9B


Theme 3 Waves

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