Vibrations Rule the Universe

Vibrations

A repeating back-and-forth motion about an equilibrium position is a vibration (a.k.a. oscillations)

Period

Amplitude

Vibrations

The time required for a full oscillation (one round trip)

Period:

Frequency:

Number of oscillations per unit time

\mathrm{Frequency} = \frac{1}{\mathrm{Period}}

\mathrm{Frequency} = \frac{1}{\mathrm{Period}}

Unit of Frequency: 1 Hertz = [1/second] (1 oscillation per second)

For example, for a period of 2 seconds per oscillation, the frequency is ½ oscillations per second or ½ Hertz

The Pendulum

An ideal pendulum (no froction) would oscillate forever with the same period and amplitude

The Pendulum

For small amplitudes a pendulum's period only depends on its length because the acceleration due to gravity is independent of the object's mass

T= 2 \pi \sqrt(\frac{L}{g})

T= 2 \pi \sqrt(\frac{L}{g})

The Pendulum

Pendulums swing back and forth with such regularity that they have long been used to control the motion of clocks

Checkpoint 1

A pendulum oscillates with a period of 2 seconds. How long would it take to do a round trip if we increase the length of the rope by 4 times?

T= 2 \pi \sqrt(\frac{L}{g})

T= 2 \pi \sqrt(\frac{L}{g})

2 \ \mathrm{seconds} \times \sqrt{4} = 4 \ \mathrm{seconds}

2 \ \mathrm{seconds} \times \sqrt{4} = 4 \ \mathrm{seconds}

Waves

Vibrations create waves. A sine curve is a pictorial representation of a wave

Waves

Vibrations are the source of all waves

For example electrons in the antenna of an AM radio station at 960 kHz vibrate 960,000 times each second, producing 960 kHz radio waves

Transverse Waves

For transverse waves the wave’s amplitude is perpendicular to the wave’s motion

Longitudinal Waves

For longitudinal waves the wave’s amplitude is parallel to the wave’s motion

Waves transport energy without a net displacement of the medium they travel on

Transverse Waves

Waves transport energy without a net displacement of the medium they travel on

Longitudinal Waves

The speed at which waves travel through a medium is related to the frequency and wavelength

Wave Speed

\mathrm{Wave \ Speed} = \frac{\mathrm{Wave \ Length}}{\mathrm{Period}} = \mathrm{Frequency} \times \mathrm{Wave \ Length}

\mathrm{Wave \ Speed} = \frac{\mathrm{Wave \ Length}}{\mathrm{Period}} = \mathrm{Frequency} \times \mathrm{Wave \ Length}

Checkpoint 2

What is the Frequency and Period of a wave set traveling at 6 m/s with a wave length of 3 m ?

\mathrm{Frequency} = \frac{\mathrm{Wave \ Speed}}{\mathrm{Wavelength}} = \frac{6 \ m/s}{3 \ m} = 2 \ \mathrm{per} \ s

\mathrm{Frequency} = \frac{\mathrm{Wave \ Speed}}{\mathrm{Wavelength}} = \frac{6 \ m/s}{3 \ m} = 2 \ \mathrm{per} \ s

\mathrm{Wave \ Speed} = \mathrm{Frequency} \times \mathrm{Wave \ Length}

\mathrm{Wave \ Speed} = \mathrm{Frequency} \times \mathrm{Wave \ Length}

= \frac{\mathrm{Wave \ Length}}{\mathrm{Period}}

= \frac{\mathrm{Wave \ Length}}{\mathrm{Period}}

\mathrm{Period} = \frac{\mathrm{Wavelength}}{\mathrm{Wave \ Speed}} = \frac{3 \ m}{6 \ m/s} = \frac{1}{2} s

\mathrm{Period} = \frac{\mathrm{Wavelength}}{\mathrm{Wave \ Speed}} = \frac{3 \ m}{6 \ m/s} = \frac{1}{2} s

Wave Interference

Multiple waves can share the same space at the same time, this is called interference

Constructive Interference

Destructive Interference

Waves in phase

Waves out of phase

Wave Interference

Noise-canceling headphones use a microphone that listens for noise and a speaker that produces the same noise but out of phase (destructive interference)

Wave Interference

Standing Waves

\lambda = \mathrm{Wavelenght}

\lambda = \mathrm{Wavelenght}

Wave Interference in 2D

Standing Waves in 2D

Doppler Effect

When the source of waves moves the frequency increases in front of the source and decreases behind the source

At point A the frequency is lower than at point B

Source not moving

Source Moving

At point A the frequency is the same than at point B

Doppler Effect

When the source of waves moves the frequency increases in front of the source and decreases behind the source

If moving towards you, wavelength shorter and frequency higher.

If moving away, wavelength longer and frequency lower.

Doppler Effect

When the source of waves moves the frequency increases in front of the source and decreases behind the source

Police use the Doppler effect of radar waves to measure the speeds of cars on the highway.

Doppler Effect

When the source of waves moves the frequency increases in front of the source and decreases behind the source

Astronomers use the Doppler effect to measure the rotation speed of galaxies.

Sound and Music

Origin of Sound

Sound is a wave that is produced by the vibrations of material objects

Nature of Sound

Sound in air is a longitudinal wave created by compressions and rarefactions

Sound Needs a Material Medium to Travel

Sound cannot travel in vacuum

Media That Transmit Sound

Sound travels faster in liquids and solids than in gasses (air)

The closer the molecules are and the more elastic the material is, the better sound travels

Media That Transmit Sound

Sound travels faster in liquids and solids than in gasses (air)

The closer the molecules are and the more elastic the material is, the better sound travels

Speed of Sound

Air 20°C

Water 20°C

Metal 20°C

The speed of sound depends on temperature, the faster the molecules move (the higher the temperature) the faster sound travels

1484 m/s

5120 m/s

343 m/s

Light is a million times faster than sound on air

Speed of Sound

Light Speed: ~3x10^8 m/s

Sound Speed: ~3.5x10^2 m/s

Checkpoint 1

If the speed of sound on air is about 340 m/s, and you hear a thunder 3 s after you see it, how far away is the storm?

About 1000 m = 1 km

Loudness and Amplitude

Loudness depends on amplitude of pressure and density variations in sound waves

Loudness Scale

Loudness is measured in Decibels (dB), which is a logarithmic scale

Decibels Scale

B_1 - B_2 = 10 \log_{10}(\frac{I_1}{I_2})

B_1 - B_2 = 10 \log_{10}(\frac{I_1}{I_2})

=> 10^\frac{B_1 - B_2}{10} = \frac{I_1}{I_2}

=> 10^\frac{B_1 - B_2}{10} = \frac{I_1}{I_2}

A subway train is 50 db louder than a conversation. Thus it is

10^\frac{50}{10} = 10^5 = 100000

10^\frac{50}{10} = 10^5 = 100000

more intense. The amplitude/energy of the sound of a subway is 100000 greater than that of a conversation

Checkpoint 2

What is the intensity/amplitude difference between sound waves from normal speech at 60 dB, and sound waves from a loud concert at 120 dB?

Difference in dB scale = 120 dB - 60 dB = 60 dB

Difference in intensity = 10^(60/10) = 10^6 = 1000000

10^\frac{B_1 - B_2}{10} = \frac{I_1}{I_2}

10^\frac{B_1 - B_2}{10} = \frac{I_1}{I_2}

Sound Reflection

Soft surfaces absorb sound

Is quiet after a fresh snowfall because soft snow absorbs sound

Symphony halls use hard plates to reflect sound and produce reverberation

Sound reflects strongly from rigid surfaces

Natural Frequency

All objects have a natural frequency at which they vibrate if they are struck

The natural frequency of an object depends on shape and composition

Music

Musical Notes

Musical notes correspond to specific frequencies

Middle C has the same fundamental frequency of 262Hz when played in different instruments

Musical Notes

Pythagoras discovered that different musical notes were related by mathematical ratios, such as the ratios of lengths or sizes in musical instruments or even in simple objects

The note produced by two strings, one half the length of the other, sounded similar

In Western music these two notes are said to be an octave apart

Notes and Frequencies

An octave has 12 steps and going up an octave doubles the frequency

C

524 Hz

After 12 steps you get back to C: Same not but double the frequency (one octave apart)

Piano Keyboard (Upper Half)

Woodwind Instruments

Resonant standing waves also produced in a pipe, but the pipe length varied by air holes (finger-holes, keys, or pads)

The End

Beats

When the two frequencies are almost the same, the sum is about the same frequency but periodically varying amplitude. This is called a beat

Beats

When the two frequencies are almost the same, the sum is about the same frequency but periodically varying amplitude. This is called a beat

Same effect seen with two combs with different tooth spacing.

Sound Refraction

Sound speed can vary by material or conditions.

This causes the sound to bend in direction, in the same way that light bends when it passes through a glass lens

Wave Refraction

Resonance

Resonance occurs when forced/induced vibrations match an object’s natural frequency

Resonance dramatically increases the amplitude

Resonance

Resonance occurs whenever successive impulses are applied to a vibrating object in rhythm with its natural frequency

Resonance dramatically increases the amplitude

The Fundamental is the lowest frequency standing wave. The Overtones are twice, three times, etc., the frequency of the Fundamental

Oscillations, Waves and Sound