A wave pulse is a disturbance that moves through a medium.

A periodic wave is a periodic disturbance that moves through a medium.  The medium itself goes nowhere.  The individual atoms and molecules in the medium oscillate about their equilibrium position, but their average position does not change.  As they interact with their neighbors, they transfer some of their energy to them.  The neighboring atoms in turn transfer this energy to their neighbors down the line.  In this way the energy is transported throughout the medium, without the transport of any matter.

The animation above portrays a medium as a series of particles connected by springs.  As one individual particle is disturbed, it transmits the disturbance to the next interconnected particle.  This disturbance continues to be passed on to the next particle.  The result is that energy is transported from one end of the medium to the other end of the medium without the actual transport of matter.  The individual atoms and molecules act like coupled harmonic oscillators.

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Periodic waves are characterized by a frequency, a wavelength, and by their speed.  The wave frequency f is the oscillation frequency of the individual atoms or molecules.  The period T = 1/f is the time it takes any particular atom or molecule to go through one complete cycle of its motion.  The wavelength is the distance along the direction of propagation between two atoms which oscillate in phase.

In a periodic wave a pulse travels a distance of one wavelength l in a time equal to one period T.  The speed v of the wave can be expressed in terms of these quantities.

v = l/T = lf

This relationship holds true for any periodic wave.

If the individual atoms and molecules in the medium move with simple harmonic motion, the resulting periodic wave has a sinusoidal form.  We call it a harmonic wave or a sinusoidal wave.

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Transverse and longitudinal waves

If the displacement of the individual atoms or molecules is perpendicular to the direction the wave is traveling, the wave is called a transverse wave.

If the displacement is parallel to the direction of travel the wave is called a longitudinal wave or a compression wave.

Transverse waves can occur only in solids, whereas longitudinal waves can travel in solids, fluids, and gases.  Transverse motion requires that each particle drag with it adjacent particles to which it is tightly bound.  In a fluid this is impossible, because adjacent particles can easily slide past each other.  Longitudinal motion only requires that each particle push on its neighbors, which can easily happen in a fluid or gas.  The fact that longitudinal waves originating in an earthquake pass through the center of the earth while transverse waves do not is one of the reasons the earth is believed to have a liquid core.

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Consider a transverse harmonic wave traveling in the positive x-direction.  The displacement y of a particle in the medium is given as a function of x and t by

y(x,t) = Asin(kx - wt + f)

Here k is the wavenumber, k = 2p/l, and w = 2p/T = 2pf is the angular frequency of the wave.  f is called the phase constant.  We may write

y(x,t) = Asin((2p/l)x - (2pf)t + f)=sin((2p/l)(x - lft) + f)

or, using lf = v,

y(x,t) = Asin((2p/l)(x - vt) + f).

For a transverse harmonic wave traveling in the negative x-direction we have

y(x,t) = Asin(kx + wt + f)

or

y(x,t) = Asin((2p/l)(x + vt) + f).

To visualize the traveling wave, download this Excel spreadsheet.  Macros must be enabled.  (On Excel's menu bar click Tools, Macro, Security, and  choose Security Level, Medium.)

The amplitude A of a wave is the maximum displacement of the individual particles from their equilibrium position.  The energy carried by a wave is proportional to the square of its amplitude.

E µ A²

The power delivered by the wave if it is absorbed is proportional to the square of its amplitude times its speed.

P µ A²v

Interference

Two or more waves traveling in the same medium travel independently and can pass through each other.  In regions where they overlap we only observe a single disturbance.  We observe interference.  When two or more waves interfere, the resulting displacement is equal to the vector sum of the individual displacements.  If two waves with equal amplitudes overlap in phase, i.e. if crest meets crest and trough meets trough, then we observe a resultant wave with twice the amplitude.  We have constructive interference.  If the two overlapping waves, however, are completely out of phase, i.e. if crest meets trough, then the two waves cancel each other out completely.  We have destructive interference.

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When a wave in a medium reaches the end of the medium, it often is reflected and travels back in the opposite direction.  If a periodic wave is reflected, the reflected wave interferes with the incoming wave.  The resulting pattern can become very complex and confusing.

Standing waves

Two or more waves traveling in the same medium travel independently and can pass through each other.  In regions where they overlap, the disturbances add like vectors.

The two wave pulses above interfere constructively when they meet.

These two wave pulses interfere destructively.

When the medium through which a wave travels abruptly changes, the wave may be partially or totally reflected.  When a wave pulse traveling along a rope reaches the end of the rope, it is totally reflected.  The details of the reflection depend on if the end of the rope is tied down and fixed, or if it is allowed to swing loose.

This wave pulse is totally reflected from a rope with a fixed end.  Upon reflection, it is inverted.

This wave pulse is totally reflected from a rope with a loose end.  It is not inverted upon reflection.

When a periodic wave is totally reflected, then the incident wave and the reflected wave travel in the same medium in opposite directions and interfere.

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When the medium is of finite extend, then the waves reflect on both ends.  Consider a string fixed on both ends.  When the wavelength of the waves is just right, so that an integral number of half wavelengths fit into the length of the string, a standing wave forms.

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A standing wave is a pattern which results from the interference of two or more waves traveling in the same medium.  All standing waves are characterized by positions along the medium which are standing still.  Such positions are referred to as nodes.  Nodes are the result of the meeting of a crest with a trough.  This leads to a point of no displacement.  Standing waves are also characterized by antinodes.  These are positions along the medium where the particles oscillate about their equilibrium position with maximum amplitude.  Antinodes are the result of a crest meeting a crest and a trough meeting a trough.  Standing wave patterns are always characterized by an alternating pattern of nodes and antinodes.

Standing waves of many different wavelengths can be produced on a string with two fixed ends, as long as an integral number of half wavelength fits into the length of the string.  Each wavelength corresponds to a particular frequency and is known as a harmonic.  Wavelength and frequency are related through lf = v, where v is the speed of waves along the string.  The shorter the wavelength, the higher is the frequency.  The lowest possible frequency of a standing wave is known as the fundamental frequency or the first harmonic.  Only half a wavelength fits into the length of the string

The second lowest frequency at which a string could vibrate is known as the second harmonic, the third lowest frequency is known as the third harmonic, and so on.

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The frequency associated with each harmonic depends on the speed with which waves move through the medium and the wavelength of the wave.  For a string the speed of the waves is a function of the mass per unit length m = m/L of the string and the tension F in the string

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If a guitar string is simply plucked, the fundamental frequency dominates.  The first harmonic can be produced by touching the string lightly in the middle when plucking it.  Touching the string lightly one-third the length of the string from one end will produce the second harmonic.

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Standing wave patterns can be set up in almost any structure.  In two and three dimensions, the patterns can become quite complex.  When a structure is vibrating with its fundamental frequency, then all the particles oscillate in phase with the same frequency.  A harmonic driving force with the same frequency can very efficiently pump energy into this mode.  At resonance, the amplitude of the standing wave increases without limit, until the structure is damaged.

A famous example of a structure driven into resonance in a windstorm and collapsing is the Tacoma Narrows bridge failure.  The original, 5,939-foot-long Tacoma Narrows Bridge, was opened to traffic on July 1, 1940 after two years of construction, linking Tacoma and Gig Harbor.  It collapsed 4 months and 7 days after it opened.  The collapse occurred during a 42-mile-per-hour wind storm on November 7, 1940, around 11:00 am.

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