PH 105

Dr. Cecilia VogelLecture 6

OUTLINE

Natural or Normal Modes Driving force Resonance

Helmholtz resonator Standing Waves

Strings and tubes Longitudinal vs transverse waves

SuperpositionWhen two disturbances (or waves)are at the same place at the same time,total disturbance is the sum of the two.

watch impulsive waves

InterferenceBecause of superposition,Waves, when they meet

can add or interfere constructively so the total is

periodic waves, when they meetcan cancel or interfere destructively so the total is

BeatsTwo waves with slightly different frequency (period) go in and out of phase

InterferenceWaves from two source,will have places where they interfere constructively

what does it sound like with sound?what does it look like with light?

and other places where they interfere destructively

what does it sound like with sound?what does it look like with light?

video

Normal Modes

A normal or natural mode isa way the system behaves when left to move naturally.

How does a pendulum behave naturally?How does mass on a spring behave naturally? How does string vibrate naturally?

Some systems have multiple normal modes

Driving ForceYou can apply a periodic driving forcea force that pushes the system periodically

Period of driving force =

Example: pushing a swing

Sympathetic VibrationsA driving force will often cause the driven system to vibrate

with the same period as the driving force.If the driving vibrator is vibrating naturally, these vibrations are called sympathetic vibrations.

Listen to the tuning fork;listen again when on box

box driven by tuning fork.both emit sound

ResonanceWhen the frequency of the driving force matches a natural frequency,

the driving force hasthe vibrator is resonating

Why push a swing each time it swings?Observe spring on and off resonance.

Helmholtz ResonatorA bottle with a neck is analogous to a mass on a spring.

the air in the neck is the mass which oscillates the volume of air in the bottle acts as a spring

Called a Helmholtz resonator f=resonant frequencyV = volume of bottlebigger bottle, _____ r freq (pitch)a = neck area, l= neck length long, skinny neck, _____ freq

2

v af

Vl

Closed Tube Resonances If tube is closed at both ends

the pressure has no there is a pressure antinode at ends

Observe slinky “pressure” hi & lo at fixed end observe that a pressure antinode is a

displacement (motion) node!

Closed Tube Resonances How can there be antinodes at both

ends?

If etc

L = L = L =

Resonant Frequencies of Closed Tube L = n/2

n = 1, 2, 3, 4, 5, ….

Since f = v

2

vf n

L

n shows there are many resonant frequencies

Resonances of Open Tube

If tube is open at both ends, it has a pressure node at both ends displacement __________ analysis is similar

2

vf n

L 1f nf

Tube with One Closed End

If tube is closed at one end there is a pressure _________ at that

end _______ at the other end

Closed Tube Resonances How?

If etc

L = L = L =

Resonant Frequencies L = n/4

n = 1, 3, 5, 7, 9…. (only odd!)

Since f = v

4

vf n

L

n odd

1f nf

Standing Wave in String String is generally fixed at both

ends node at analysis like

2

vf n

L 1f nf

L = n/2 n = 1, 2, 3, 4, 5, ….

Were measured resonantfrequencies integer times f1?

Standing Wave in String Combine

Can change resonant freq’s by changing

Tv

2

vf n

L

2

n Tf

L

Impedance and Resonance A reflection can occur any time there is a

change in impedance. Acoustic Impedance means difficulty of air

flow observe wave machines

There can be resonance in each part of a complex tube:

L1

L2

L3

SummaryInterference is the addition of waves at point where they meet

constructive interference destructive interference

Normal modes are natural behaviorsometimes multiple natural frequencies

At resonancedriving frequency matches natural freqdriving force has a huge effect

Resonance of Helmholtz resonator, open and closed tubes, strings