Chapter 11

Vibrations and Waves

Driven Harmonic Motion and Resonance

RESONANCE Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. Resonance occurs when the frequency of the force matches a natural frequency, f0, at which the object will oscillate.

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f0 =12π

km

The Nature of Waves

1.  A wave is a traveling disturbance.

2.  A wave carries energy from place to place.

The Nature of Waves

Longitudinal Wave - the “disturbance” caused by the wave moves along the direction that the wave propagates, e.g., sound waves, “compressed slinky waves”……

The Nature of Waves

Transverse Wave - the “disturbance” caused by the wave moves perpendicular to the direction that the wave propagates, e.g., water waves, “shaken slinky waves”, electromagnetic waves….

The Nature of Waves

Water waves are partially transverse and partially longitudinal.

Periodic Waves

Periodic waves consist of cycles or patterns that are produced over and over again by the source. In the figures, every segment of the slinky vibrates in simple harmonic motion, provided the end of the slinky is moved in simple harmonic motion.

Periodic Waves

In the drawing, one cycle is shaded in color.

The amplitude A is the maximum excursion of a particle of the medium from the particles undisturbed position. The wavelength is the horizontal length of one cycle of the wave. The period is the time required for one complete cycle. The frequency is related to the period and has units of Hz, or s-1.

Tf 1=

Periodic Waves

λλ fT

v ==Since velocity is distance/time è

The propagation velocity of a periodic wave is related to its frequency and wavelength. Consider the motion of a long train as a periodic wave which repeats itself with the passing of each identical car:

Periodic Waves

Example: The Wavelengths of Radio Waves AM and FM radio waves are transverse waves consisting of electric and magnetic field disturbances traveling at a speed of 3.00x108m/s. A station broadcasts AM radio waves whose frequency is 1230x103Hz and an FM radio wave whose frequency is 91.9x106Hz. Find the distance between adjacent crests in each wave.

λλ fT

v ==

fv

=λ

Periodic Waves

AM m 244Hz101230

sm1000.33

8

=×

×==

fv

λ

FM m 26.3Hz1091.9

sm1000.36

8

=×

×==

fv

λ

The Speed of a Wave on a String

The speed at which the wave moves to the right depends on how quickly one particle of the string is accelerated upward in response to the net pulling force.

LmFv =

tension

linear density

The Speed of a Wave on a String

Example: Waves Traveling on Guitar Strings Transverse waves travel on each string of an electric guitar after the string is plucked. The length of each string between its two fixed ends is 0.628 m, and the mass is 0.208 g for the highest pitched E string and 3.32 g for the lowest pitched E string. Each string is under a tension of 226 N. Find the speeds of the waves on the two strings.

The Speed of a Wave on a String

High E

( ) ( )sm826

m 0.628kg100.208N 226

3- =×

==LmFv

Low E

( ) ( )sm207

m 0.628kg103.32N 226

3- =×

==LmFv

The Speed of a Wave on a String

Conceptual Example: Wave Speed Versus Particle Speed Is the speed of a transverse wave on a string the same as the speed at which a particle on the string moves?

Speed of longitudinal waves

LIQUIDS SOLID BARS

v = Bρ

B ≡ bulk modulus

v = Yρ

Y ≡ Young's modulus

Example: Compare the speed of a wave in water to a wave in a bar of aluminum. Water: Aluminum:

B = 2.2×109 N/m2 ρ =1000 kg/m3

v = Bρ=

2.2×109

1000=1500 m/s

Y = 6.9×1010 N/m2 ρ = 2700 kg/m3

v = Yρ=

6.9×1010

2700= 5100 m/s

Intensity

Waves carry energy that can be used to do work.

The amount of energy transported per second is called the power of the wave. The intensity is defined as the power that passes perpendicularly through a surface divided by the area of that surface.

API =

Intensity

Example: Sound Intensities 12x10-5 W of sound power passed through the surfaces labeled 1 and 2. The areas of these surfaces are 4.0 m2 and 12 m2. Determine the sound intensity at each surface.

Intensity

252

5

11 mW100.3

4.0mW1012 −

−

×=×

==API

252

5

22 mW100.1

12mW1012 −

−

×=×

==API

Intensity

24 rPIπ

=

power of wave source

area of sphere

If the source emits the wave uniformly in all directions, the intensity depends on the distance from the source in a simple way.

Intensity

I = P4π r2 ⇒ I ∝ 1

r2

⇒I1I2

=r2

2

r12

For SHM E = 12 kA

2 ⇒ I ∝P∝E∝ A2

∴I1I2

=r2

2

r12 =

A12

A22 ⇒

r2r1=A1

A2

Example: The intensity of an earthquake wave passing through the Earth is measured to be 2.5×106 J/(m2s) at a distance of 44 km from the source. What was its intensity when it passed a point only 1.0 km from the source? At what rate did energy pass through an area of 7.0 m2 at 1.0 km?

I1I2

=r2

2

r12

⇒ I1 = I2r2

2

r12 = 2.5×106( ) 442

12 = 4.8×109 J/(m2s)

P = I1A1 = 4.8×109( ) 7.0( ) = 3.4×1010 J/s

Wave Fronts and Rays

Defining wave fronts and rays. Consider a sound wave since it is easier to visualize. Shown is a hemispherical view of a sound wave emitted by a pulsating sphere. The rays are perpendicular to the wave fronts (e.g. crests) which are separated from each other by the wavelength of the wave, λ.

Wave Fronts and Rays

The positions of two spherical wave fronts are shown in (a) with their diverging rays. At large distances from the source, the wave fronts become less and less curved and approach the limiting case of a plane wave shown in (b). A plane wave has flat wave fronts and rays parallel to each other.

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Reflection of string pulses at boundaries and interfaces

} }

From less dense medium to more dense medium

From more dense medium to less dense medium

Reflection inverted Reflection not inverted

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Law of reflection of waves at a boundary or interface:

θi =θr