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Honors Physics 1Lecture 21 - F2013

Waves– Terms– Travelling waves– Waves on a string

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Waves Overview

A wave is a disturbance that moves through a medium.

– manifestations– common language of waves

» amplitude, period, frequency, wavelength, speed

– special case: waves on a string» speed, energy transmission, interference, standing waves and

resonance

– sound waves» mechanism, speed, intensity, beats

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Mechanical wavese.g. - water, sound, seismic, string, drumheadtransmitting effect: forces between atoms and

moleculesmoves through: matterElectromagnetic wavese.g. – radio; microwave; infrared,visible and

ultraviolet light; x-rays; gamma raysmoves through: vacuumMatter wavesAll the stuff around you - observable on very

short length scales and for very light particlesmoves through: vacuum

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Classifications of waves

Transverse wave – the local displacement or field amplitude is perpendicular to the direction of energy flow– examples: string, drumhead, water surface

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Classifications

Longitudinal waves – the local displacement or field amplitude is parallel to the direction of energy flow– examples: sound, Slinky™, seismic p-wave

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Terms: Periodic Waves

amplitude

period - T

time

If you’re sitting at one point* and watch the waves go by…

f=1/T

*e.g.- on the dock of the bay

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Terms: Periodic Waves

amplitude

wavelength -

distance

Freeze the waves in time and look at the shape…

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Harmonic wavesA math fact: Any finite function can be

represented as the sum of sine and cosine functions.

So all we need to really understand is sine, or harmonic waves

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Harmonic travelling waves

represents a wave of amplitude ymax travelling in the + x direction.

max max

2 2 2( ) sin siny t y x t y x t

T T

wave speed:

phasev fT

max

2( ) sinso: y t y x vt

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sin sinm m

k f v fk

y y kx t y k x vt

Let and then

Then:

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A travelling sine wave

max 0

2( ) siny t y x vt

A point with a specific phase moves to the right atspeed v.

phase

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iClicker Check 21.1

) ( , ) 2sin(4 - 2 )

) ( , ) sin( -1.5 )

) ( , ) 2sin(3 - 3 )

Here are the equations of three waves:

Which wave travels fastest?

a y x t x t

b y x t x t

c y x t x t

Write the phase in terms of (x-vt) to get the speed.

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iClicker Check 21.2

) ( , ) 2sin(4 - 2 )

) ( , ) sin( -1.5 )

) ( , ) 2sin(3 - 3 )

Here are the equations of three waves:

Which wave has the longest wavelength?

a y x t x t

b y x t x t

c y x t x t

The term that multiplies x is inversely proportional to wavelength.

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iClicker Check 21.3

) ( , ) 2sin(4 - 2 )

) ( , ) sin( -1.5 )

) ( , ) 2sin(3 - 3 )

Here are the equations of three waves:

Which wave oscillates fastest?

a y x t x t

b y x t x t

c y x t x t

The term that multiplies t is proportional tofrequency.

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Phase and Phase Difference

The argument of the sine fn: 2x vt

is called the phase of the wave.

When we talk about phase difference , we are talking about how the argument differs for different times or places in the wave.

For two times (at the same point):

For two points (at the same time):

2v t t

2x k x

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Phase and phase difference

1 0

2x vt

The phase of the wave

at this point is

1x

2 0

2x vt

The phase of the wave

at this point is

2x

2 1

2x x

The phase difference is:

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8t

v

Comparing two waves at different times

t=0

8x

2

1 1

2x vt

A point of constant phase moves distance vt in time t.The phase at a point in space changes by:

2 2

8 4v t v

v

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iClicker Check 21.4

Can you uniquely determine the velocity of a wave from snapshots of amplitude vs position at two different times?

a) Yes

b) No

Because the wave repeats, the change in position can be altered by an integer times the wavelength and you won’t know the difference.

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Wave on a string 1

.Suppose we have a string

with tension

What are the forces on it?

The tension is the only force.

It acts in different directions at each end of

the piece of string and pulls the

string up or down, depending on the

curvature.

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String wave derivation

2

2 2 2

Choose a small section of a string pulse. If this section is curved, then the

tension at the two ends of the section will be along the string at that end.

sin sin tan

ta

y y

l

d yF T T T ma dx

dt

dT

2 2

2 2

2 2 2 2

2 2 2 2

n tan and using tan ;

so so that

d y dy d d y

dx dx dxdt dx

d y d y d y d yT

Tdx dt dx dt

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Waves on a string 2

2 2

2 2

2 2

2 2 2

.

1

y y

x t

y y

x v tv

The resulting force-acceleration relation gives

the differential equation

This is known as the wave equation and is usually seen

in the following form:

where is the spe

.v

ed of the wave

By comparison we see that

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In-class exercise

2 2

2 2

Show that any (doubly differentiable) function of the

form ( - ) is a solution to the wave equation

as long as and are properly related.

What is the relation between and ?

y f x vt

y yb b v

dx dtb v

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Waves on a string 3

( ) sin( )

( ) cos( )

m

transverse m

y t y kx t

dyv t y kx t

dt

Note that the speed of the wave

is but that no piece of the

string travels with the wave.

Each piece of string moves transversely.

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iClicker question 21.4

When the tension in a string is increased, the wave speed

A)Remains the same.B)IncreasesC)Decreases

v

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iClicker question 21.5

When the amplitude of motion of a wave on a string is increased (keeping tension, mass per length, amplitude the same), the wave speed

A)Remains the same.B)IncreasesC)Decreases

v

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iClicker question 21.6

When the amplitude of motion of a wave on a string is increased (keeping tension, mass per length, amplitude the same), the transverse speed

A)Remains the same.B)IncreasesC)Decreases

( ) cos( )transverse m

dyv t y kx t

dt