Vibrating objects can produce sound.  The vibrating object can be a string, an air column, a membrane, etc.  When the objects moves outward, it pushes the air molecules, creating a region of high pressure.  When it moves inward it creates a region of low pressure.  As the vibrating object alternately compresses and expands, the surrounding air, the disturbance travels outward from the source as a longitudinal wave.

The speed of sound at standard temperature and pressure (25^oC, 760mm of mercury) is 343m/s (767 miles/h).  It is determined by how often the air molecules collide.  The speed of sound increases by about 6m/s if the temperature increases by 10^oC.  Sound travels faster in liquids and solids than in gases, since the particles in liquids and solids are closer together and can respond more quickly to the motion of their neighbors.  The speed of sound in water is 1500m/s, in iron it is 5000m/s.

The chart below shows the speed of sound in various media.

The speed of a sound wave through a medium does not depend on the frequency of the wave.

Problem:

Our ears are most sensitive to sounds in the frequency range of 3000Hz - 4000Hz.  Almost nobody hears sounds outside the range of 20Hz - 20000Hz.  Sound waves whose frequencies are less than 20Hz are called infrasonic waves and sound waves whose frequencies are higher than 20000Hz are called ultrasonic waves.

The energy carried by a sound wave is proportional to the square of its amplitude.  The energy passing a unit area per unit time is called the intensity of the wave.  The higher the intensity, the louder is the sound.  Our ears, however, do not respond linearly to the intensity.  A wave that carries twice the energy does not sound twice as loud.

Sound levels b are measured using a logarithmic scale.

b = 10 log₁₀(I/I₀)

The units of b are decibels (dB).  I₀ = 10^-12W/m² is the reference intensity.

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Like all waves, two or more sound waves traveling trough the same medium will interfere.  We can have constructive and destructive interference.  If a person stands equidistant from two speakers which are playing the same sound in phase, i.e. which are moving in and out together, then the two waves arrive in phase after traveling the same distance.  Crest meets crest and trough meets trough at the location of the person.  The amplitudes of the two waves add and the sound is loudest here.  If the two speakers play the same sound but are out of phase, i.e. one is moving out while the other is moving in, then the sound has a low volume at the location of the person equidistant from the two speakers.  This can easily be demonstrated by switching the wires on one of the speakers.  (This is why you need to pay attention to the color of the wires when setting up your stereo).  Dead spots in an auditorium are sometimes produced by destructive interference

Beats occur when two sounds have nearly, but not exactly, the same frequency.  They are also the result of interference.  Even though a crest may meet a trough at one instant in time at some point in space (destructive interference), at some later time at the same point a crest will meet a crest and the amplitudes will add (constructive interference).  The frequency of the beats is equal to the difference in the frequencies of the two sound waves.

The two waves in the figure above have wavelengths of 20 and 21 units respectively.  Their frequencies are v/20 and v/21, where v is the speed of sound.  The frequency of the intensity variations is v/20 – v/21 = v/420.

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Exercise (You can earn up to 5 points extra credit by completing this exercise.)

When a sound wave hits a wall, it is partially absorbed and partially reflected.  A person far enough from the wall will hear the sound twice.  This is an echo.  In a small room the sound is also heard more than once, but the time differences are so small that the sound just seems to loom.  This is known as reverberation.

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Music is the Sound that is produced by instruments or voices.  To play most musical instruments you have to create standing waves on a string or in a tube or pipe.

The pitch of the sound is equal to the frequency of the wave.  The higher the frequency, the higher is the pitch.

Wind instruments produce sounds by means of vibrating air columns.  To play a wind instrument you push the air in a tube with your mouth or a reed.  The air in the tube starts to vibrate with the same frequency as your lips or the reed.  Resonance increases the amplitude of the vibrations, which can form standing waves in the tube.  The length of the air column determines the resonant frequencies. The mouth or the reed produces a mixture if different frequencies, but the resonating air column amplifies only the natural frequencies.  The shorter the tube the higher is the pitch.  Many instruments have holes, whose opening and closing controls the effective pitch.

We can create a standing wave in a tube, which is open on both ends, and in a tube, which is open on one end and closed on the other end.  Open and closed ends reflect waves differently.  The closed end of a tube is an antinode in the pressure (or a node in the longitudinal displacement).  The open end of a tube is approximately a node in the pressure (or an antinode in the longitudinal displacement).

The longest standing wave in a tube of length L with two open ends has displacement antinodes (pressure nodes) at both ends. It is called the fundamental.

The next longest standing wave in a tube of length L with two open ends is the second harmonic.  It also has displacement antinodes at each end.

An integer number of half wavelength have to fit into the tube of length L.
L = nl/2,  l = 2L/n,  f = v/l  = nv/(2L).
For a tube with two open ends all frequencies f_n = nv/(2L) = nf₁, with n equal to an integer, are natural frequencies.

The longest standing wave in a tube of length L with one open end and one closed end has a displacement antinode at the open end and a displacement node at the closed end.  This is the fundamental.

The next longest standing wave in a tube of length in a tube of length L with one open end and one closed end is the third harmonic.  It also has a displacement antinode at one end and a node at the other.

The next longest standing wave in a tube of length L with one open end and one closed end is the fifth harmonic. 

An odd-integer number of quarter wavelength have to fit into the tube of length L.
L = nl/4,  l = 4L/n,  f = v/l  = nv/(4L),  n = odd.
For a tube with one open end and one closed end all frequencies f_n = nv/(4L) = nf₁, with n equal to an odd integer are natural frequencies, i.e. only odd harmonics of the fundamental are natural frequencies.

The quality of a sound depends on the relative intensities of the waves with the natural frequencies.  It depends on the spectrum of the sound.  The sound quality of a musical instrument is called its timbre.

Problems:

Water waves

Standing on a beach and watching the waves roll in and break, one might guess that water is moving bodily towards the shore.  But no water is piling up on the beach.  Watching a piece of floating debris beyond the breakers, we can see it move towards the shore on the crest of a wave, and move the same distance backward with the trough of the wave.  The debris moves in a roughly circular path perpendicular to the water’s surface.  Water waves are surface waves, a mixture of longitudinal and transverse waves.  Surface waves in oceanography are deformations of the sea surface.  The deformations propagate with the wave speed, while the water molecules remain at the same positions on average.  Energy, however, moves towards the shore.  Most ocean waves are produced by wind, and the energy from the wind offshore is carried by the waves towards the shore.

We distinguish between deep-water waves and shallow water waves.  The distinction between deep and shallow water waves has nothing to do with absolute water depth.  It is determined by the ratio of the water's depth to the wavelength of the wave.

The water molecules of a deep-water wave move in a circular orbit. The diameter of the orbit decreases with the distance from the surface. The motion is felt down to a distance of approximately one wavelength, where the wave's energy becomes negligible.

The orbits of the molecules of shallow-water waves are more elliptical.

The change from deep to shallow water waves occurs when the depth of the water, d, becomes less than one half of the wavelength of the wave, l.  When d is much greater than l/2 we have a deep-water wave or a short wave.  When d is much less than l /2 we have a shallow-water wave or a long wave.

The speed of deep-water waves depends on the wavelength of the waves.  We say that deep-water waves show dispersion.  A waves with a longer wavelength travels at higher speed.  In contrast, shallow- water waves show no dispersion.  Their speed is independent of their wavelength.  It depends, however, on the depth of the water.  Shallow-water waves move at a speed that is equal to the square root of the product of the acceleration of gravity and the water depth.

Deep-water waves in the ocean are wind-generated waves.  They can be generated by the local winds (sea) or by distant winds (swell).

Ocean waves are produce by a variety of forces.  Meteorological forces (wind, air pressure) produce seas and swells.  Astronomical forces produce the tides.  Earthquakes produce tsunamis.  Tides and tsunamis are shallow-water waves, even in the deep ocean.  The deep ocean is shallow with respect to a wave with a wavelength longer than twice the ocean's depth.

Tsunamis

A tsunami, also called seismic sea wave or tidal wave, is a catastrophic ocean wave, usually caused by a submarine earthquake occurring less than 50 km (30 miles) beneath the seafloor, with a magnitude greater than 6.5 on the Richter scale.  Underwater or coastal landslides or volcanic eruptions also may cause a tsunami.  The term tidal wave is more frequently used for such a wave, but it is a misnomer, for the wave has no connection with the tides.  A tsunami can have a wavelength in excess of 100 km and period on the order of one hour.  Because it has such a long wavelength, a tsunami is a shallow-water wave. Shallow-water waves move with a speed equal to the square root of the product of the acceleration of gravity and the water depth.

Problem:

The rate at which a wave loses its energy is inversely related to its wavelength.  A tsunami not only propagates with a high speed, it also can travel a great, transoceanic distance with only limited energy loss.

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In the deep ocean, the amplitude of a tsunami is only a few feet.  It cannot be felt aboard a ship or seen from the air in the open ocean.  When the tsunami approaches the coastline, its speed decreases and its amplitude increases.  (The power is proportional to the square of the amplitude times the speed. As the speed decreases, the amplitude increases.)  The amplitude can grow to a height exceeding 100 feet.  The tsunami can strike with devastating force.

Earthquakes generate tsunamis when the sea floor abruptly deforms and displaces the water above from its equilibrium position.  Waves are formed as the displaced water under the influence of gravity attempts to regain its equilibrium.  The initial size of a tsunami is determined by the amount of vertical sea floor deformation.

Tides

The earth and the moon orbit each other.  They revolve about their common center of mass.  Gravity provides the centripetal acceleration.  The moon is constantly falling towards the earth and the earth is constantly falling towards the moon.  But while each body moves on a curved trajectory, the distance between it and the other body stays constant.

The gravitational force between two objects is inverse proportional to the square of their distance.  The distance between the earth and the moon is usually taken to be the distance between their centers.  But the earth is an extended object.  The gravitational acceleration due to the moon is larger than average on the side facing the moon, and smaller than average on the side facing away from the moon.  Any loose material on the side of the earth facing the moon would accelerate at a higher than average rate towards the moon and move in a tighter orbit if not bound to the earth by gravity.  Any loose material on the side of the earth facing away from the moon would accelerate at a lower than average rate towards the moon and move in a wider orbit if not bound to the earth by gravity.  The waters of the oceans try to fall into these natural orbits, but the earth's gravity pulls them back.  Water on the side of the earth facing the moon therefore forms a bulge outward from the center of the earth and toward the moon.  Water on the side of the earth facing away from the moon forms a bulge outward from the center of the earth and away from the moon.

There are thus two separate tidal bulges in the earth's oceans, one on the side nearest the moon and one on the side farthest from the moon.  The earth rotates once a day, so these bulges move across the earth's surface.  There are two bulges, so each shore passes through two bulges a day.  At those times, the tide is high.  During the times when the seashore is between bulges, the tide is low.  Because the moon moves as the earth turns, the time interval between high tides is about 12 hours and 26 minutes, not exactly 12 hours.  Since local water must flow to form the bulges as the earth rotates, there are cases where the tides are delayed as the water struggles to move through a channel.  However, even in those cases, the high tides occur every 12 hours and 26 minutes.

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The sun's gravity also contributes to the tides, but its effects are smaller and serve mostly to vary the heights of high and low tide.

Problem:

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