Waves

Terms and concepts:

•  Wave
•  Universal wave equation
•  Frequency
•  Wavelength
•  Impulsive wave
•  Transverse wave
•  Longitudinal wave
•  Tension
•  Crests
•  Troughs
•  Condensations
•  Rarefactions
•  Compression Wave
•  Sine wave

Vibrations in musical bodies create sound.  Audible sound is a wave.  Physicists call a sound a “periodic disturbance” because its impulses happen repeatedly between 20 and 20,000 times a second.  Below 20 times a second (20 hertz), we feel, but do not hear the vibrations.  At the other end, most people cannot hear impulses that are faster than 20,000 times a second, but dogs and some other animals can do so.

Airborne sound waves work like this:  When a surface such as a drum skin vibrates in air, it makes nearby air molecules collectively push together or spread apart in time with the vibrating skin.  The adjoining molecules pick up this kinetic message, so they also begin to push together and pull apart in the same timing.  This collective motion passes through the air as a rapid series of pressure and density changes that match the drum’s vibrations.  The air passes these vibrating pressure changes to our ears, which the brain, as a later lesson will show, interprets to be sound.

Other instruments, for example, a guitar, work the same way.  The waves move along the strings, then spread into the guitar’s body.  The air inside and outside the guitar pick up the body’s vibrations and radiates them outward as musical sound.

A wave’s fundamental quality is that it relays energy through a medium without significantly moving its molecules from one place to another.  Thus, while the drum and guitar vibrations do not propel individual air molecules through the countryside or across a room, the force travels among its molecules, creating audible energy that we hear as drumbeats and guitar notes.  The air responds the same way to all musical instruments.  The point is that we do not hear the instruments themselves.  They make the air vibrate and our brains interpret these rapid pressure-changes to be sound.  Thanks to the air, a room or even the hills are alive with the sound of music.

Perhaps, this, too, is similar to traditional knowledge.  Sound moves as travelling vibrations through objects, musical instruments and the air.  But, sound waves essentially leave the molecules where they are.  Similarly, traditional knowledge, like a wave, connects humanity both through place and time.  Each generation and every person remains in place, yet the essence and energy of the people – that which defines the culture – passes one person, community or nation to the next.

Tunkasila Sydney Keith described the four colours of humanity as quadrants in a medicine wheel.  Black, white, yellow and red, he said, each has its place in the circle.  If, however, one colour thinks it is better than another and leaves, it breaks the circle.  The quadrants must remain connected so that the power can move like a wave throughout the circle.  In that way, the circle will be powerful and its waves will continue on and on and on and on.

The Universal Wave Equation

Simple waves have special qualities, including the wave’s frequency, which is the number of times it cycles per second and is its wavelength, which is the distance from any point on the wave to its corresponding point in the next cycle.  The graph below illustrates both:

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Fig. 1: Wavelength

Wavelength (ë)  =    If the wave requires one second to complete the distance, then the frequency is (f) = 1 cycle per second (1 Hz).  The wavelength can be microscopic or huge.  Scientists usually measure the length metrically, for example, in  millimetres, meters or kilometres.
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This provides everything we need to use the UNIVERSAL WAVE EQUATION, a simple formula that we can apply to all waves. The equation is as follows:

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THE UNIVERSAL WAVE EQUATION:
V = f .λ

V = velocity (speed) of the wave
f = the frequency of the wave *
λ = the wavelength

*One way to measure frequency is cycles per second, or hertz.
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Transverse and Longitudinal Waves

The most basic form of a wave is an impulsive wave, a pulse that is simply one brief disturbance that travels through the medium.  Music requires sound waves that vary and last much longer than a single pulse.  These waves take two major forms.

A transverse wave occurs when the medium vibrates perpendicular (at right angles) to the direction of the wave.  Think of what happens when a person flips a long rope into a wave. The rope, the medium in this example, moves up and down as the wave moves forward.  The rope itself remains where it is.  Only the wave travels across the room.

A mouth harp, guitar or violin string is like a tiny very dense rope stretched tightly between two fixed points.  When a musician plucks the string, it vibrates as the sound wave moves back and forth along the string’s length.  The string stays where it is; only the wave moves.  That is a transverse wave.

The string’s tension affects whether the string vibrates quickly or slowly.  The tighter the musician stretches the string before he plucks it, the shorter is its wavelength, which makes the musical note sound higher.  The string vibrates from side to side (or up and down).  Scientists call a simple wave that looks like this a sine wave.

The accompanying diagram shows this action.  The x-axis represents the distance the wave travels (wavelength) and the y-axis shows displacements from the equilibrium position.  This action creates a series of high points, or crests (displacements above the equilibrium position) and low points, or troughs (displacements below the equilibrium position).  These crests and troughs progress sideways at individual points on the string.

The second type, longitudinal waves, include the sound we hear.  These occur in air, gases, water and other fluids when molecules rapidly push together and pull apart in the same direction the wave is moving.

A key difference between fluids and solid objects is this:  Solids can transmit both transverse and longitudinal waves.  Fluids (air and water) can transmit only longitudinal waves.

Strings and violin bodies vibrate transversely, back and forth on both sides of an equilibrium point.  The musical instrument’s transverse waves become longitudinal air waves like this:  Imagine a vibrating violin string.  It moves from side to side and transfers this same motion to the violin’s body.  The vibrating string and body compress nearby air molecules closer together and stretch them apart.  The distances are tiny, and the vibrations very rapid – between 20 and 20,000 times per second.  The air molecules crash into other air molecules which, in turn, crash into other air molecules.  This creates waves that move sound energy through the countryside or across a room.

Those are the vibrations we hear as sound.  Faster vibrations create higher sounds.  Slower vibrations create lower ones.  The compressed areas are condensations, while the expanded areas are rarefactions.  Because the waves compress the air, another term for a longitudinal wave is a compression wave.

The compressed areas are condensations, while the expanded areas are rarefactions.  Because the waves compress the air, another term for a longitudinal wave is a compression wave.

Flutes use vibrating air, namely, compression waves.  Suppose we seal both ends of a tube.  This traps a long column of air inside so the vibrations can’t spread out in all directions.  You might think that the air column would vibrate like a string, but it doesn’t.  In one way a string and an air column  are similar.  If something disturbs an air column at one end, its individual molecules do pass their vibrations back and forth along the column’s length.  But, that is where the similarity stops.  A string vibrates sideways in a series of crests and troughs.  The air vibrates rapidly through compressing and thinning out.

To visualize what happens in the air column of a wind instrument imagine water sloshing along a rain gutter.  Instead of regular crest and troughs, the water’s vertical motion can have several possible nodes. These points of large horizontal motion correspond to specific nodes in the air column.  Air vibrations in the tube still squeeze molecules together and spread them apart.  If we open the tube’s ends – as in a flute – these compression waves travel through the air to our ears as musical notes.  In other words, the wave moves through the tube as rapidly oscillating pulses of higher pressure and lower pressure.

Tunkasila Sydney Keith said the smallest action touches the whole world.  This truth affects physics, too.  Suppose someone is playing a violin.  The violin vibrates transversely.  But, then the vibrating violin excites the air molecules, too, so they begin rapidly compressing together and pulling apart.  Thus, the violin’s transverse wave becomes the air’s longitudinal wave – and, then, the music – in other ways – moves us, too.  As Grandfather Sydney and my elders have said, what anyone does always affects everyone else – each in their own way.

Interference

Sound waves are complex and interact in interesting ways.  The superposition of oscillations describes what happens when two waves of different amplitudes pass through each other while keeping their respective shapes.  When these two waves meet, they produce an audible effect, interference, which is the outcome of combining their amplitudes and waveforms.

Interference is important in music.  So are overtones, the fact vibrating bodies produce harmonious additional sounds that are multiples of their lowest (fundamental) frequencies.  While overtones are any higher frequency modes, exact multiples produce the harmonics without which rich and complex music would be impossible.  Without these interactions simple single-note sine-wave tones would be all we would hear.  For example, if they existed, a single-note sine-wave flute would sound exactly like a single-note sine-wave mouth harp.  In other words, wave actions and interactions equal individuality.  These effects combine to create music’s variety and character.

Let’s look more closely at interference.  As we will see in the next section, interference affects loudness.

The example below shows what happens when we blend two equally-loud waves that have the same frequency and are in step (in phase) with each other.  The crests and troughs of the first wave will coincide with the crests and troughs of the second wave.  This makes the combined wave’s amplitude double and the sound becomes louder.  We call this complete constructive interference.

But, doubling the amplitude does not double the loudness because our ears are logarithmic sound detectors, the math of which is beyond this lesson’s scope.  It would, however, make the sound noticeably louder, about six decibels or so.

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Complete Constructive Interference

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If, however, two waves are moving in the same direction and the crests of the first wave coincides with the troughs of the second, the two waves will cancel each other out at the point of interference; this is called complete destructive interference.

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Fig. 5 – Complete Destructive Interference

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As we can see, interference’s two forms have opposite effects:  Constructive interference increases a sound’s volume, while destructive interference decreases volume or even silences it completely.

Resonance

This brings us back to the picture of a parent pushing a child higher and higher on a swing.  When identical waves are moving in the same direction at precisely the same time, they are in phase and resonate with each other.  If we use physics’ jargon, we would say:  If a body that is already vibrating at its natural frequency is again acted upon from the outside by a periodic disturbance that matches its natural frequency or frequencies, then this can produce vibrations of ever increasing amplitude.  Now, we are ready to explore standing waves, a key phenomenon that produces musical sounds in all their richness and variety.

Standing Waves

We’ve investigated force, the physics of kinetic and potential energy, periodic motion, transverse waves, longitudinal waves and other phenomena that make music possible.  This section discusses standing waves, a topic that combines much of what we have learned so far.

Standing waves result from constructive and destructive interferences that take place when waves combine as they move along the same medium from opposite directions.  This is very important to the physics of musical sound.  Standing waves are among the basic forms that interference patterns take

All musical instruments are wave catchers, that is, they are designed to catch very specific sound waves.

As a sound wave travels along a medium, it eventually comes up against a different surface and reflects back on itself.  Now, if the sound waves are reflected back in such a way that constructive (and not destructive) interference happens and music becomes possible.

If we want to examine standing waves, we need to look at how a musical instrument “catches” them.  That is, we must consider the big picture of wave motion within an instrument.

When two waves with the same frequency and wavelength are moving along the same medium, but in opposite directions, they interfere with one another in such a way as to create the stationary wave pattern we call a standing wave.  These are blended waves that strengthen each other at some points and weaken each other at others.  They constructively and destructively interact in ways that affect both the amplitude and the tonality

Imagine a medium such as a string.  Wave interactions along a string can be very complex, so, let’s keep this example simple.  We’ll pretend that the string is so long that a wave could travel along it forever and never bounce back.

Unlike other interference patterns (see above), standing waves require the original waves to be coming from opposite directions.

A single wave moving from the left to the right looks like this:
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A single wave moving from the right to the left looks like this:
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But, if these identical wave forms pass through each other, you get a unique result: a stationary wave.  The string moves up and down, but the wave doesn’t travel forward or backward along the string, hence the term, “standing” wave.  It looks like this:
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N= Node
A = Antinode
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The solid curve shows the wave interaction when it moves the string up and down.  The dotted curve shows the wave interaction when, half a cycle later, it moves the same parts of the string down and up.  The standing wave repeats itself every full cycle.  The points where the waves intersect don’t move at all.

Please note two things: First, the two travelling waves add their amplitudes together.  In this case, the combined wave is twice as high.  Second, you can see the string going up and down at certain places, but every half wavelength, it stays perfectly still.  The places where the string doesn’t move are nodes (see N above).  The segments that move up and down are loops and their highest displacements are antinodes (see A above).

Standing waves can be very complex.  Remember what we learned earlier about harmonics?  The two nodes at the string’s farthest ends can create a wave at the lowest frequency (the fundamental frequency).  Further in, each additional node, if present, corresponds to the next higher harmonic.  The two nodes at the string’s farthest ends can create a wave at the lowest frequency (the fundamental frequency).

That is one reason why standing waves are so important to music.  Fundamentals and harmonics are among the most basic building blocks of musical sounds.

In our simple example, we imagined a string that has no ends.  We sent two waves toward each other.  But, a guitar string is fixed at both ends, so it creates standing waves differently than two vibration sources along an infinite string.

How do we get harmonious standing waves when we have only one original source?  Let’s reflect on that (pun intended).  When a musician plucks a guitar string, the vibrations travel along the string then bounce off its fixed ends.  When they reflect back, the waves invert.  In other words, the upward curves come back as downward curves and the downward curves come back as upward curves.  When these reflected waves flow through the waves moving in the original direction, they blend into a standing wave.

However, the action gets more complex.  On a guitar string, the transverse waves keep reflecting back and forth, creating many standing wave patterns at once.  These patterns include the fundamental (lowest) tone and its harmonics, those numerous partials of the fundamental.  Also, the instruments’ bodies have much more surface area than strings, so when the bodies pick up these complex vibrations, the sounds become much louder.

Drums and other transversely-vibrating instruments also generate complex wave interactions.  All transversely-vibrating instruments stimulate the air to vibrate longitudinally.  These air waves are what we hear as music.

Flutes and other aerophones involve standing waves, too, but because they make air vibrate directly, an aerophone’s waves are already longitudinal.  For us to hear the music, the air does not have to convert the waves.

Let’s investigate aerophones in more detail:

Air Column Closed At Both Ends

No flute is closed at both ends, but let’s see what happens anyway.  A standing wave in a closed tube’s air column works the same way as in a string.  The longitudinal wave moves down the tube, pushing and pulling molecules until it reaches the closed end, which is a denser medium than the air.

However, unlike the fixed string’s reflected transverse wave, the reflected longitudinal wave will not undergo phase reversal.  When it hits a displacement node (the closed end), the molecules crash into each other and build up greater pressure.  This bounces back the same way – as greater pressure (a condensation).

Conversely, those points that experience the greatest displacement and thus create the least amount of pressure in their areas are displacement antinodes and pressure nodes

Just like the fixed string, some air molecules in the tube will not compress together or expand apart.  These points are displacement nodes.  Since they remain immovable when other molecules act upon them, they create areas of great pressure.  This makes them into pressure antinodes.

Air Column in a tube that is open at one end

Though useful for explaining these actions, closed tubes cannot make music.  We should also consider the standing wave in a tube that is open at one end.  In flutes, when the wave reaches the tube’s end, it is moving from a denser medium (confined air) into a less dense medium (natural air pressure).  The open end is thus a pressure node, which is also a displacement antinode.  Because no displacement node exists as in the form of a closed end, the reflected wave then bounces back into the tube and creates a standing wave.

©2019 This project was made possible with the support of the Department of Canadian Heritage through Canadian Culture Online
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