PHY II – Waves

The Wave Model Speed of Waves on a String Sinusoidal Waves Spherical Waves, Plane Waves

Transverse versus Longitudinal

Snapshot Graph

History Graph

A string of beads are connected by a set of tiny springs. At the instant the clock starts (t=0), a pulse is moving to the right on the beads and the snapshot graph looks like this:

Which of the following history graphs tracks the position for the bead marked with a red arrow as a function of time?

The graph at the top is the history graph at x = 4 m of a wave traveling to the right at a speed of 2 m/s. Which is the history graph of this wave at x = 0 m?

(A) (B) (C)

Sinusoidal Waves

What is the frequency of this traveling wave?

A. 10 Hz

B. 5 Hz

C. 2 Hz

D. 0.2 Hz

E. 0.1 Hz

Quiz 2

What is the phase difference between the maximum of a wave and the adjacent minimum? (crest to trough)

A. 0

B. π /4

C. π /2

D. 3 π /2

E. π

Quiz 3

Sinusoidal Wave Snapshot Graph

k = 2π/λ is the wave number

Sinusoidal Wave History Graph

ω=2π/T is the angular frequency

PHY II – Waves

• Sound and Light

• Medical Applications of Ultrasound

• Power and Intensity

• The Doppler Effect

Sound and Light• Sound is a pressure wave in a gas, liquid

or solid. Speed depends on material.

• Light is one type of electromagnetic wave.

• In a vacuum, all electromagnetic waves (including light) travel at c = 3×108 m/s.

• In transparent media, light slows down. Index of Refraction is n > 1.

• This reduces the wavelength, but does not change the frequency!

A light wave travels through three transparent materials of equal thickness. Rank is order, from the largest to smallest, the indices of refraction n1, n2, and n3.

A. n2 > n1 > n3 B. n3 > n1 > n2 C. n1 > n2 > n3

D. n3 > n2 > n1 E. n1 = n2 = n3

Reflection of Transverse Wave Pulse

• A pulse traveling to the right on a heavy string attached to a lighter string

• Speed suddenly increases

Reflection of Transverse Wave Pulse

• A pulse traveling to the right on a light string attached to a heavier string

• Speed suddenly decreases

Physics of Ultrasound• Speed of sound in bone, flesh and

blood are all different• When the speed of any wave suddenly

changes, there is a reflection and transmission

• Ultrasound images are formed from reflected high frequency sound

• Image resolution is set by wavelength, λ• λ=v/f, so higher frequency yields smaller λ, and better resolution

Speed of sound in humansTissue Sound Speed (m/s)

Air 350

Fat 1450

Brain 1540

Blood 1570

Bone 4080

Muscle 1585

Power and Intensity• The Power, P, of any wave source is how

much energy per second is radiated as waves [units = Watts]

• The Intensity, I, is the energy rate per area. This determines how loud (sound) or bright (light) the wave is.

• I=P/a, where a is an area perpendicular to the wave direction.

• At a distance r from a spherically symmetric source, the intensity is I=P/(4πr2)

Chapter 20, Problem 34• The sound intensity from a jack hammer

breaking concrete is 2 W/m2 at a distance of 2 m from the point of impact. This is sufficiently loud to cause permanent hearing damage if the operator doesn’t wear ear protection. What is the sound intensity for a person watching from 50 m away?

Doppler Effect

Valerie is standing in the middle of the road, as a police car approaches her at a constant speed, v. The siren on the police car emits a “rest frequency” of f0.

Which statement is true?

A. The frequency she hears rises steadily as the police car gets closer and closer.

B. The frequency she hears steadily decreases as the police car gets closer and closer.

C.The frequency she hears does not change as the police car gets closer.

Valerie is standing still as a police car approaches her at a constant speed, v. Daniel is in his car moving at the same constant speed, v, toward an identical police car which is standing still. Both hear a siren.

Which statement is true?

A. The frequency Daniel hears is lower than the frequency Valerie hears.

B. The frequency Daniel hears is higher than the frequency Valerie hears.

C.The frequencies that Daniel and Valerie hear are exactly the same.

(Almost correct)

Actual Answer! Check eq.20.38

and 20.39!

PHY II – Waves

• The Principle of Superposition• Standing Waves• The Guitar: Stringed

Instruments• The Trumpet: Wind Instruments

Principle of Superposition• If two or more waves combine at a

given point, the resulting disturbance is the sum of the disturbances of the individual waves.

• Two traveling waves can pass through each other without being destroyed or even altered!

Some Results of Superposition:• Two waves, same wavelength and

frequency, opposite direction:

Standing Wave

• Two waves, same wavelength and frequency, similar direction, different phase:

Interference

• Two waves, same direction, slightly different frequency and wavelength:

Beats!

Standing Wave:The superposition of two 1-D sinusoidal waves traveling in opposite directions.

Standing Waves• Are a form of “resonance”• There are multiple resonant frequencies

called harmonics• The boundary conditions and speed of

waves determine which frequencies are allowed.

• The ends of the resonant cavity have forced nodes or antinodes

• With a wave on a string, it is possible to force an intermediate node

OMGWTFBBQOMGWTFBBQStanding Wave in PressureStanding Wave in Pressure

Forces the Flames higher at Forces the Flames higher at antinodesantinodes!!

Harmonic frequenciesTransverse standing wave on a string

clamped at both ends: there are nodes in displacement at both ends.

,...)3,2,1(2

mL

vmfm

Standing sound wave in a tube open at both ends: there are nodes in pressure both ends.

,...)3,2,1(2

mL

vmfm

Harmonic frequenciesStanding sound wave in a tube closed at one end: there is a node in pressure at the open end, and an anti-node at the closed end.

,...)5,3,1(4

mL

vmfm

PHY II – Waves

•Constructive and Destructive Interference

•Interference Patterns•Beats

Wave Interference• Two waves moving in the same direction

with the same amplitude and same frequency form a new wave with amplitude:

2

cos2

aA

where a is the amplitude of either of the individual waves, and is their phase difference.

Beat frequency• Beats are loud sounds separated by soft sounds• The beat frequency is the difference of the

frequencies of the two waves that are being added:

21mod2 ffffbeat • The frequency of the actual sound is the

average of the frequencies of the two waves that are being added:

221 ff

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