PHY138 – Waves Lecture 9Quarter Review, including:

• Error Propagation• Simple Harmonic Motion: Force, Energy• Mass on spring / Pendulum• Damped Oscillations, Resonance• Traveling Waves, Power and Intensity• Standing Waves, Interference, Beats• Ray Model of Light, Ray-Tracing• Reflection, Refraction of Light• Thin Lens equation; Combination of Lenses

Tomorrow evening, 6:00 PM It is mandatory that you go to the room

assigned to your tutorial group. You should have no communication device

(phone, pager, etc.) within your reach or field of vision during the test.

The test has nine equally weighted multiple-choice questions (7 marks each).

The test has one multi-part problem counting for 37 marks; you must show your work.

Don’t forget… Your student card. A non-programmable calculator without

text storage and communication capability. A single original, handwritten 22 × 28

cm sheet of paper on which you have written anything you wish. We will supply any numerical constants you might need.

A dark-black, soft-lead 2B or 2HB pencil with an eraser.

Some more words to the wise… A good aid-sheet is well organized, easy to

read, and contains all the major equations from the assigned sections from the reading.

Copies of detailed specific problem solutions are unlikely to help.

Be ready to think; get a good night’s sleep tonight.

Keep in mind: Your best 3 out of 4 tests will count for 30% of your mark in the course.

The Eye

Propagation of Errors: good to have on aid sheet!!

z = A x Δz = A Δx

Mass on Spring versus Pendulum Mass on a

SpringPendulum

Condition for S.H.M.

Small oscillations

(spring obeys Hooke’s Law)

Small angles

(sinθ ≈ θ)

Natural frequency [rad/s]

Period

L

g

m

k

k

mT 2

g

LT 2

14.7 Damped Oscillations

Snapshot Graph

History Graph

Sinusoidal Wave Snapshot Graph

k = 2π/λ is the wave number

Sinusoidal Wave History Graph

ω=2π/T is the angular frequency

Sound Waves can be described either by the longitudinal displacement of the individual particles, or by the air or fluid pressure.

)sin(),( 0 tkxAxxtxD eqeqeq

)cos(),( 0max tkxPPPtxD atm

Electric and Magnetic fields, when oscillated, can create waves which carry energy. At certain frequencies, we see electro-magnetic waves as Light.

)sin(),( 00 tkxEtxE

)sin(),( 00 tkxBtxB

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 small source, the intensity is I=P/(4πr2)

Doppler Effect

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!

Standing Wave:

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

Harmonic frequencies of Standing WavesTransverse 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

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

favg

The Law of Reflection

11

Snell’s Law of Refraction

2211 sinsin nn

Total Internal Reflection

Can only occur when n2<n1

θc = critical angle.

When θ1 ≥ θc, no light is transmitted through the boundary; 100% reflection

1

2sinn

nc

Virtual Image Formation by Reflection

ss '

Virtual Image Formation by Refraction

sn

ns

1

2'

Real Image Formation with a Converging Lens

Focal length, f

Object

Real Image (inverted)