Welcome to Physics 7C!Welcome to Physics 7C!

Lecture 2 -- Winter Quarter -- 2005

Professor Robin Erbacher

343 Phy/Geo

erbacher@physics.ucdavis.edu

AnnouncementsAnnouncements• Course policy and regrade forms on the web: http://physics7.ucdavis.edu

• Quiz today! ~20 minutes long.

• Reminder: Friday is “Academic Monday”. Monday DL’s: please attend DL this Friday. Friday DL’s are canceled. See calendar on website.

• Final on Saturday, March 19 1:30pm. Let me know this week if you cannot make this.

• Turn off cell phones and pagers during lecture.

Simple Harmonic OscillatorsSimple Harmonic Oscillators

2

€

T = 2πm

k€

T = 2πl

gPendulum:

Mass/Spring:

€

y(t) = Asin(2π

Tt +φ) + BA generalized solution

is of the form:

Simple 1D WavesSimple 1D WavesWhat is a wave?A wave is a type of internal motion of a medium, in which the displaced portion returns to equilibrium. This disturbance propagates in space as well.

Particles of the medium oscillate about their equilibrium positions in both a spatial and a temporal way.

What kind of waves are there?

Transverse Waves

Longitudinal Waves

Combo Waves

(circular)

WaterWaves

We will focus on these

Wave ParametersWave ParametersCertain independent parameters characterize all waves:1) Amplitude: Controlled by the magnitude of the forces that

started the wave2) Speed: Determined by the properties of the medium.3) Direction: Determined by the direction of the forces starting

the wave 1) Longitudinal: Oscillations in direction of wave velocity v 2) Transverse: Oscillations are perpendicular to v

4) Frequency f of oscillations: controlled by forces starting the wave

Need y(x,t) !

Wave: disturbancepropagates in x…

Snapshot v. MovieSnapshot v. MovieSome waves are simply a pulse, and some are repetitive. These are harmonic (or sinusoidal), generated by SHOs. Harmonic waves have a dependent variable, wavelength , the distance at which the oscillation repeats.

wavelength: vwave/f

2

y(x)

2

y(t)

Snapshot: Hold time constant, see where we are in space.

Movie: Go forward in time, see how spatial points move in SHM.

The Wave RepresentationThe Wave RepresentationDescribing the behavior of harmonic (sinusoidal) waves is extremely important in our physical world.

Because there is both a time-dependence and a translation of the wave in space, we need to represent the wave using both t and x.

€

What are all these parameters?x: location in the medium (spatial)t: time (temporal)T,f, period, freq., wavelengthA: amplitude phase

€

Δy(x,t) = Asin[Φ(x,t)]

Too complicated? Think of the sin argument as one big phase (or angle)

€

y(x,t) - y0 = A sin(2π

Tt ±

2π

λx +φ) + BThe most general

solution is of the form:

Note: I swapped x and t term. Block notes differ from DL expression. Both ok. Use DL version

Total Phase of the WaveTotal Phase of the Wave

€

y(x,t) - y0 = A sin(2π

Tt ±

2π

λx +φ) + B

€

€

Δy(x,t) = Asin[Φ(x,t)]

Total phase

If we hold x constant, the wave will repeat in T seconds.If we hold t constant, the wave will repeat in meters.

T and play similar roles in the wave function, determininghow often the wave will repeat in time and space.

Wave: a displacement in space and time. The angle, , found from (T, , x, t, ), determines the total displacement y(x,t).

Period, Frequency, Wavelength, Wave Speed

Period, Frequency, Wavelength, Wave Speed

€

T = 1f (sec)period:

€

f = vwaveλ (1/sec)frequency:

wavelength:

€

= vwavef = vwaveT (meters)

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

€

Φ = 2π

Tt ±

2π

λx +φ = wave phase

Ride the wave: constant

€

± x = λ

2πΦ m

λ

Tt -

λ

2πφ

€

∴ v = dx

dt= m

λ

T

If we choose + in the wave function, the velocity is negative.

What’s the wave velocity?

Particle VelocityParticle Velocity

€

vwave = dxdt = mλ T

So, the velocity of the wave, or propagating disturbance, can be found by riding along the wave at constant :

What is the velocity of a particle (or length of string) on the wave?

As always:

€

v = dy(x,t)

dt Why y?

Transverse Waves:Particle moves as SHO!

€

vparticle = A2π

Tcos(

2π

Tt ±

2π

λx +φ)

€

y(x,t) - y0 = Δy(x,t) = A sin(2π

Tt ±

2π

λx +φ) + B

Note: I swapped x and t term. Block notes differ from DL expression. Both ok. Use DL version

Longitudinal Waves

Sound WavesSound Waves

€

P(x,t) - Patm = ΔP(x,t) = A sin(2π

Tt ±

2π

λx) ⇒ AsinΦ(x,t)

The sound vibrations in 1-Dimension, such as long, narrow tubes, trombone, flute, trumpet, follows harmonic oscillations. But how does one describe the vibrations of the air?

It’s all about pressure (density) fluctuations!

Equilibrium = Atmospheric (or surrounding) pressure

Power and IntensityPower and IntensitySound is a pressure fluctuation in a medium. Sound energy is transported through the medium via these fluctuations.

Power: sound energy time

emitted by a source

Intensity: Psource

area(area of wavefront)

How About Light?How About Light?What kind of wave is a light wave? It’s a transverse excitation, perpendicular to the direction of wave propagation.

What’s the medium that’s displaced as the wave propagates?

Nothing!

Light propagates via oscillating electric and magnetic fields (more on this later in the course!)

The Enigmatic Ether!

Light: Visible, and Invisible Light: Visible, and InvisibleThe light we see is a small portion of the radiation that exists!

Visible Light:4.3-7.5 x 1014 Hz

Ultra Violet (UV)X-rays/ rays

Infra Red IRwave, AM/FM, TV

frequency

wavelength

Wave Interference Wave InterferenceWhat happens when there is more than one wave?When two or more waves meet, they interfere with each other.Combining waves by adding them is known as superposition.

Consider two waves on a string. What’s the maximum displacement of the string from equilibrium?

Δy(wave1+wave2) = A1+A2

In Phase: 1 - 2 = n2 (n=integer)

(constructive interference)

Out of Phase: 1 - 2 = [(2n-1)/2]2 (n=integer)

(destructive interference)

Superposition of Waves Superposition of Waves

Adding 1D Waves Together:

€

Δytotal(x,t) = Δy1(x,t) +Δy2(x,t) = A1 sinΦ1 + A1 sinΦ2

€

Δytotal(x,t) = A1 sin(2πt

T1

± 2πx

λ1

+ϕ1) + A2 sin(2πt

T2

± 2πx

λ 2

+ϕ 2)

Using the Full Expressions:

What determines the total excursion of the medium at arbitrary time and position?

Phase angles and amplitudes!