More Waves!

Intensity

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I ≡1

A

ΔE

Δt=P

A

Intensity – the rate at which energy flows through a surface

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I[ ] =W

m2

Point source outputs a fixed power, say P0 Watts

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Ir1 =P0

Ar1=P0

4πr12

Ir2 =P0

Ar2=P0

4πr22

Ir1Ir2

=r2

2

r12

Example

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I =1

AP

ΔE

Δt

ΔE = IAPΔt

I =P0

AS=P0

4πr2

ΔE =P0

4πr2

⎛

⎝ ⎜

⎞

⎠ ⎟APΔt

=60

4π22

⎛

⎝ ⎜

⎞

⎠ ⎟ .1* .2( )60 =1.43 J

ΔE =60

4π 42

⎛

⎝ ⎜

⎞

⎠ ⎟ .1* .2( )60 = 0.36 J

A 60 W light bulb emits light waves in every direction. If you stand 2 m away and hold up a black sheet of paper that is 10 cm x 20 cm, how much energy does that paper absorb in 1 minute? What if you move to 4 m away?

Doppler Effect

Stationary source emits sound waves at a given frequency, fs, and wavelength, λs.

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fO = fS +vOλ S

= fS +fSvOv

= fSv + vOv

⎛

⎝ ⎜

⎞

⎠ ⎟

€

vOt

λ S

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fextra =vOλ S

Approaching observer detects extra wavefronts each second.

Perceived frequency is the sum of the emitted frequency and the extra wavefronts detected per second.

Doppler Effect

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fO = fSv + vOv − vS

⎛

⎝ ⎜

⎞

⎠ ⎟

For a stationary observer and a moving source, the equation becomes:

If both the observer and the source are moving (along a line), the equations combine to yield:

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fO = fSv

v − vS

⎛

⎝ ⎜

⎞

⎠ ⎟

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fA = fSv

v − +vS( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

fB = fSv

v − −vS( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Choose signs so as to yield correct result!

Shock waves

Wave fronts “bunch up” on forward side of moving source.

If vs > v, wavefronts fall on a single, ultra-high intensity line.

Shock waves

Wave fronts “bunch up” on forward side of moving source.

If vs > v, wavefronts fall on a single, ultra-high intensity line.

Standing WavesStanding Wave – a wave pattern that does not appear to move

These are the ONLY types of waves that can persist on a string.

If string is fixed at each end, then the end points must not move (obvious).

For a given length of string, L, there are only a few specific wavelengths that will satisfy this condition.

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f =1

2L

T

μ

fn = nf0

Fundamental frequency

Harmonics, or overtones

n = 5

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λ =2L

n, n =1, 2, 3K

f =v

λ=n

2L

T

μ

Guitar, piano, cello…

Standing WavesStanding Wave – a wave pattern that does not appear to move

These are the ONLY types of waves that can persist on a string.

If string is free at each end, then the end points must be points of maximum displacement (less obvious than fixed case, but still true).

For a given length of string, L, there are only a few specific wavelengths that will satisfy this condition.

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λ =2L

n, n =1, 2, 3K

f =v

λ=n

2L

T

μ

n = 3

Same as for fixed ends!Pan flute, clarinet, pipe organ

Standing WavesStanding Wave – a wave pattern that does not appear to move

These are the ONLY types of waves that can persist on a string.

If string is fixed at one end and free at the other end…

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λ =4L

2n −1, n =1, 2, 3K

f =v

λ=

2n −1

4L

T

μ

n = 2

Pipe organ, marimba…

Sound “Beats”

f1 = 50 Hz

f2 = 60 Hz

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fBeat = Δf

Doppler Use – Radar Guns

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fO = fSv + vOv − vS

⎛

⎝ ⎜

⎞

⎠ ⎟

= 36 ×109 3.0 ×108 + 0

3.0 ×108 − 25

⎛

⎝ ⎜

⎞

⎠ ⎟

= 36 ×109 1

1− 253.0×108

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

= 36 ×109 1

1− 8.3 ×10−8

⎛

⎝ ⎜

⎞

⎠ ⎟

= 35,999,997,000 Hz

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fO − fS = fSv + vOv − vS

⎛

⎝ ⎜

⎞

⎠ ⎟−1

⎛

⎝ ⎜

⎞

⎠ ⎟

Δf = fSv + vO − v + vS

v − vS

⎛

⎝ ⎜

⎞

⎠ ⎟

= fSvSv − vS

⎛

⎝ ⎜

⎞

⎠ ⎟

= fS1

vvS

−1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

≈ fS1vvS

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

=fSvSv

=36 ×109 * 25

3 ×108 = 3000

A police radar gun emits electromagnetic waves at a frequency of 36 GHz (36x109 Hz). If these waves reflect off of an oncoming car traveling at 50 mph (~25 m/s), what return frequency does the radar gun detect?

The speed of electromagnetic waves in air is 3.0x108 m/s.

There’s a better way!

Resonance

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ω 0 =ωd

Resonance occurs when a system with a natural oscillation frequency is driven by a force with the same frequency.

When this condition is met, the system oscillates with an increasingly large amplitude.