Sound-Science and Math for Technology

  • Chapter 9- Sound

    SCIENCE and MATH FOR TECHNOLOGY by Thomas Barrowman, Copyright 2001


    Author: Thomas Barrowman   Pages: 241   Binding: paperback   ISBN: 0-9713542-0-0

    Copyright: 2004, 2001  Publisher: Thomas R. Barrowman and Ruth K. Barrowman

    Price:  $21.95 at

    Techno Terms:

    • Wave motion
    • Frequency
    • Sound Waves
    • Sound Quality
    • Stereo Speakers
    • Observing Sound
    • Frequencies Humans Can Hear
    • Loudness and Intensity
    • Echo
    • Doppler Effect and Sonic Boom
    • Sound Transmission Through Walls

    Wave Motion

    Energy can be transmitted from one point to another through wave motion. The most visible is a water wave. Water in the wave appears to be traveling in the direction of the wave. If you observe a floating object on the crest of the wave, it actually bobs up and down in a circular pattern rather than moving forward. What does move forward is the energy which makes the water wave move. This wave motion does not produce sound until the energy is slowed near land. Breaking waves disturb the air and produce sound.

    Another type of wave can be demonstrated by fastening a length of rope to a stationary object. Grasp the other end and give it a sudden up and down motion. Each part of the rope will move up and down at right angles or perpendicular to the direction the wave is moving along the rope. This type of wave is called a transverse wave. Light waves and radio waves are examples of transverse waves.


    A pattern of motion which repeats itself during equal itervals of time is called periodic motion. Periodic motion takes place when materials are bent, stretched, or compressed from their normal shape and then released. A guitar, bells, and drums are examples of instruments that produce a periodic wave.


    The time that it takes for a point on a vibrating object to return to its original position is called the period of motion. The number of times that the point returns to its original position in one second is called the frequency of motion. The standard unit of frequency is the hertz [Hz], named after a German physicist Heinrich Hertz. The amplitude is the maximum distance the vibrating material moves away from its normal position.

    Every material has a natural frequency at which it will vibrate. If an object has a natural frequency of 300 hertz and an external frequency of 200 hertz is applied, the object begins a forced vibration. Because it is a forced vibration, the distance the object moves will be small. If an external frequency of 300 hertz is applied to the object with a natural frequency of 300 hertz, the object will begin vibrating and will move or oscillate a greater distance. This is called the resonant frequency of the material.

    Bridges can be destroyed by wind-generated resonance. If the frequency of the wind force is the same as the natural frequency of the bridge, a steady increasing movement of the bridge will occur. Soldiers are taught to march out of time when crossing bridges or their timed footsteps could set up resonant vibrations which might destroy the bridge.

    Sound Quality

    Sounds are produced by a material that is vibrating. In order for a sound to be transmitted and heard, three conditions have to be met. First, there has to be a source of wave energy caused by a vibrating body. Second, there has to be a medium for the sound to travel in such as air, liquid or a solid. Third, there has to be a device for receiving the sound, such as the ear.

    Pitch, tone, and frequency of sound are all closely related and are used to describe sound. Pitch is the term used to describe a specific frequency usually produced by a musical instrument. A musical note begins at approximately 20 hertz. Increasing the frequency to 200 hertz produces a higher pitch.

    Tone is the sound a musical instrument produces when a certain pitch is played. As the pitch is being produced, other overtones or harmonic frequencies are also being produced by the musical instrument. For example a 60 hertz note will produce harmonic overtones at 120, 180, 240, 300, and so on. The added frequencies produce the pleasing tones and timbre that make a musical instrument desireable. 04/19/2010


    Stereo Speakers

    Steroe speaker cones move back and forth when producing sound. This back-and-forth movement, which must occur at the same time in both speakers for quality sound to be produced, is called condensation and rarefaction. Speakers must be synchronized so the cones move in unison. Condensation and rarefaction then occur at the same time.

    The size of a speaker is an indication of the frequency range that the speaker can produce. Low frequencies require more power tobe heard by the ear than do high frequencies. Low frequencies require a large volume of air movement. Thus you will find that a low-frequency speaker, called a woofer, is the largest speaker in a stereo system. A midrange speaker is just that- midsize. A tweeter, which produces high frequencies, is the smallest speaker. Speaker cabinets should not vibrate when producing music. If they vibrate, they produce disturbing harmonics and change the music that you hear. 04-20-2010


    Observing Sound

    One of the best examples of the quality or timbre of a musical instrument is the Stradavarius violin. Scientists have studied this violin for years in an attempt to copy the quality of the music that this violin produces. In 1822, Joseph Fourier discovered a mathematical way of breaking down the sound of a musical instrument such as a violin and displaying the results as a series of curved waves. Today, we use an oscilloscope, which displays these curved waves as the sound produced from a single instrument or an entire band. 04-20-2010


    The fundamental note being played is the largest wave seen on an oscilloscope. This wave determines the pitch of the note. The higher-frequency waves are the overtones that give the characteristic quality. These overtones are seen as smaller curves or dips on the larger fundamental tone. Frequency sound filters are used to display the overtones as single waves so they can be identified and studied.

    An electric guitar would be an example of a musical system without many overtones. The note played is picked up immediately and fed into the amplifier. This eliminates the amplification of any overtones that occur. 04-21-2010


    Frequencies Humans Can Hear

    The range of frequencies that the human ear can detect varies with the individual, but usually has a range from 20 to 20,000 hertz. This range of hearing decreases with age. By examining these charts, you will gain a lot of information about the range that you can hear. 04-21-2010



    Sound equipment is sometimes designed with frequencies around 2,000 Hz. Very little energy is needed for these frequencies to be heard. Higher frequencies can be easily heard at approximately 15 dB. dB stands for decibels and is a measure of loudness. Lower frequencies require much more energy to produce sound that a human can hear. Many components and a high amount of power is needed to produce the bass notes that are often heard in popular music. Most stereos have bass and treble controls which help adjust the power used to produce the loudness of the low and high frequencies.

    Loudness and Intensity of Sound

    The amplitude of the vibrating source and the distance the source is from your ear determines how loud the sound is. As the intensity of the sound increases, the amount of energy needed increases by a much greater amount. If the music seems twice as loud, the electrical power would have to be increased about eight times.

    Alexander Graham Bell, the inventor of the telephone, determined that a bel would be the measurement used to identify the loudness or intensity of sound. He increased the power by units of ten and said the resulting sound would be the standard for one bel. People soon found out that the bel was too large for measuring sound from electrical devices or musical instruments. It was determined that 1/10 of a bel could be heard as a change in loudness. The standard for measuring sound is now a decibel which is 1/10 of a bel.


    The decibel level and the time the sound continues to be heard determines if the noise is a minor irritant, a definite disturbance, or even a threat to your hearing. Federal, state and local agenicies have established standards for just how much noise is acceptable without damage to your ear.



    When a ball is thrown against a wall, it rebounds and bounces back toward the person that threw the ball. Similarly, waves can travel through air, reflect off a surface, and return to the human ear. These reflecting sounds are called echos. Buildings, hallways, canyons, and large rock faces can cause sound to bounce back and form echoes. The human ear can distinguish between the first sound and the echo only if the sounds are more than 1/10 of a second apart. If the sounds are less than 1/10 of a second apart, the two sounds are heard as one. The time required for the echo to return depends on how far away the reflecting surface islocated. 04-21-2010


    Sound travels through the air at approximately 1100 ft per second. In order for an echo to be heard as a separate sound, the distance of the reflecting surface has to be more than 110 feet away. This will allow enough time before the sound returns to the ear for the echo to be heard as a separate sound.

    Reflecting Surface Distance = sec. x 1100 diivded by 2

    Whales, dolphins, and bats use echo location to find their way and find food. They send out sound at very high frequencies. Ships use echoes to find the depth of water. Electronic instruments time the return echo and then display the depth. This system is called SONAR, which stands for 'sound navigation ranging'. 04-21-2010


    Doppler Effect and Sonic Boom

    Imagine a bug bouncing up and down on top of the water. Water ripples would be created and act much like sound waves in the air. 04-21-2010


    If the bug now begins to swim in one direction the ripples in front of the bug get closer together. This is the same situation that happens when a firetruck is approaching you. You hear a higher sound from the siren as the truck approached because the speed of sound is added to the speed of the truck. A lower sound is heard after the truck passes by and moves away from you. This time you have to subtract the speed of the truck from the speed of sound. This is called the Doppler effect. 04-21-2010



    Now suppose the bug is swimming as fast as the ripples that are being made. A large bow or a front wave begins to appear. If the bug can overcome this bow wave and move ahead of the wave the swimming will be smooth. This pattern of waves takes place when a plane travels faster than the speed of sound. A bow wave of air builds up in front of the plane. If the plane travels faster than the air is being compressed, the sound is heard on earth as a sonic boom. This compressed wave continues to follow the plane until the plane slows to less than the speed of sound. 04-21-2010


    Speeds greater than the speed of sound are measured by mach numbers. Mach one is the speed of sound or approximately 1130 ft/sec, or 770 miles per hour. Use the following formula to find the mach speed.

    Mach number = spped of plane divided by 1130 [speed of sound]

    Sound Transmission Through Walls

    Some of the sound produced within a room is absorbed by the surfaces of the walls. Because of this absorption, the sound intensity will be reduced as it moves through the walls. A wall with a transmission loss of 30 decibels will reduce the intensity of loudness level from 65 dB to 35 dB. Sound deadening materials are designed to reduce the decibel level of frequencies between 25o hertz and 2000 hertz. Wall materials have a sound rating which is established by the National Bureau of Standards. is called the Sound Transmission Class [STC] system. The higher the number, the better the sound barrier. 04-21-2010


    Sounds within a room that are present most of the time are called masking sounds. Masking sounds of quiet homes measure approximately 30 dB. Loud speech is approximately 70 dB. Thus, when a sound-deadening wall between a living room and a bedroom is being designed, the sound will have to be reduced by 40 dB to match the everpresent 30 dB masking sound. 40-21-2010