L 23a – Vibrations and Waves [4]L 23a – Vibrations and Waves [4] resonance resonance clocks – pendulum clocks – pendulum springs springs harmonic motion harmonic motion mechanical waves mechanical waves sound waves sound waves golden rule for waves golden rule for waves musical instruments musical instruments wave interferencewave interference The Doppler effect The Doppler effect

Doppler radarDoppler radar radar gunsradar guns

shock wavesshock waves

ReviewReview

A wave is a disturbance that travels through A wave is a disturbance that travels through something – solids, liquids or gasessomething – solids, liquids or gases

The disturbance moves because of the The disturbance moves because of the elastic nature of the materialelastic nature of the material

As the disturbance moves, the parts of the As the disturbance moves, the parts of the material (segment of string, air molecules) material (segment of string, air molecules) execute harmonic motion (move up and execute harmonic motion (move up and down or back and forth)down or back and forth)

transverse wave on a string

• jiggle the end of the string to create a disturbance• the disturbance moves down the string• as it passes, the string moves up and then down• the string motion in vertical but the wave moves in the

horizontal (perpendicular) direction transverse wave• this is a single pulse wave (non-repetitive)• the “wave” in the football stadium is a transverse wave

Harmonic waves – keep jiggling the end of the string up and down

Slinky waves

• you can create a longitudinal wave on a slinky

• instead of jiggling the slinky up and down, you jiggle it in and out

• the coils of the slinky move along the same direction (horizontal) as the wave

SOUND WAVES

• longitudinal pressure disturbances in a gas

• the air molecules jiggle back and forth in the same direction as the wave

the diaphragm of thespeaker moves in and

out

Sound – a longitudinal wave

The golden rule for waves

• This last result was not a coincidence• The wavelength = wave speed / frequency

= v / f or v = f Wave speed = wavelength frequency• This applies to all waves water waves,

waves on strings, sound, radio, light . . • This rule is important for understanding

how musical instruments work

Vibration modes of a string

L

L

Fundamental modeWavelength = 2 LFrequency = fo

First harmonic modeWavelength = LFrequency = 2 fo

N N

N N N

A

AA

N = nodes, A = antinodes

Standing waves

• At the NODE positions, the string does not move

• At the ANTINODES the string moves up and down harmonically

• Only certain wavelengths can fit into the distance L

• The frequency is determined by the velocity and mode number (wavelength)

Vibration frequencies

• In general, f = v / , where v is the propagation speed of the string

• The propagation speed depends on the diameter and tension

• Modes– Fundamental: fo = v / 2L

– First harmonic: f1 = v / L = 2 fo

• The effective length can be changed by the musician “fingering” the strings

Bowed instruments

• In violins, violas, cellos and basses, a bow made of horse hair is used to excite the strings into vibration

• Each of these instruments are successively bigger (longer and heavier strings).

• The shorter strings make the high frequencies and the long strings make the low frequencies

• Bowing excites many vibration modes simultaneously mixture of tones (richness)

Organ pipes• The air pressure inside the

pipe can vibrate, in some places it is high and in other places low

• Depending on the length of the pipe, various resonant modes are excited, just like blowing across a pop bottle

• The long pipes make the low notes, the short pipes make the high notes

Beats – wave interference• Waves show a special property called

interference

• When two waves are combined together, the waves can add or subtract

• We call this constructive and destructive interference

• When a wave is launched on a string it can reflect back from the far end. The reflected wave can combine with the original wave to make a standing wave

Constructive interference

Waves add to double amplitude

Destructive interference

waves add to give 0 amplitude

Combining 2 waves of the same frequency

Red + Blue

Combining 2 waves of slightly different frequencies

BeatsRed + Blue

Room Acoustics

• Destructive interference accounts for bad room acoustics

• Sound that bounces off a wall can interfere destructively (cancel out) sound from the speakers resulting in dead spots

Wave interference can be used to eliminate noise

Take one wave, turn it upside down (invertits phase) then add it to the original wave

The Doppler Effect• If a source of sound is moving toward you,

you hear a higher frequency than when it is at rest

• If a source of sound is moving away from you, you hear a lower frequency than when it is at rest

• You can hear this effect with sirens on fire engines of train whistles

• A similar effect occurs with light waves and radar waves

A science teacherdemonstrating the

Doppler effect

Doppler effect Radar guns

http://auto.howstuffworks.com/radar-detector1.htm

When radar waves bounce off a moving object(echo )the frequency of the reflected radar changesby an amount that depends on how fast the objectis moving. The detector senses the frequencyshift and translates this into a speed.

Once you see the cop, he’s got you!

Bow waves

• When the speed of the source is as great as the speed of the wave (Mach 1) something interesting happens

The waves pile up in front of the source-this is what is meant by “breaking thesound barrier”.

Shock waves – sonic boom• V is the speed of the jet, Us is the speed of

sound

V less than UsV equals Us V exceeds Us

shock wave

When the conical shell of compressed air (shock) sweeps over you onthe ground, the listener nears a sharp crack – SONIC BOOM