Acoustics

Acoustics is the study of the creation, transmission, and reception of sound.  The word acoustics also refers to the quality of sound as heard or transmitted in a room or concert hall.  Conic sections are often used in acoustics to amplify sound.

Vibrating objects produce sound.  As the object moves outward, it compresses the surrounding medium; the region of compression is called a condensation.  As the object then moves inward, the medium expands into the space previously occupied by the object.  This region is called a rarefaction.  As the object continues to move outward and inward, a series of condensations and rarefactions leave the object.  These series are sound waves.  Sound waves move outward from the object in all directions.

When sound waves in one medium strike a large object of another medium, some of the sound usually reflects off the new medium’s surface, and some refracts into the new medium.  The speed of sound in the new medium helps determine the amount of reflection off new medium’s surface.  When sound refracts, its speed and direction change.  If sound waves travel slower in the second medium, the waves refract towards the “normal,” an imaginary line perpendicular to the boundary between mediums.  If sound travels faster in the second medium, the waves will refract away from the normal.

Sound waves bend toward a region of slower speed.  For example, sound carries farther at night than during a sunny day because sound favors cooler temperatures.  During the day, air near the ground is warmer than the air above, so sound refracts away from it.  At night, air near the ground is cooler than the air above, so the sound bends toward it and travels across it for a greater distance.

Acoustics is most often used in the reflection and refraction of sound.  Acoustics is used for ultrasound and monitoring earthquakes.  Federally supported research on long-range underwater communication in World War II led to sonar and ultrasound techniques.  With sonar, ships “see” other vessels, map the ocean floor, identify mines, and isolate spots for oil drilling with acoustics.  Ultrasound is used to monitor pregnancies, image the inside of the body, stop internal bleeding, and destroy kidney stones.

Architectural acoustics involve making rooms and buildings quiet and providing good conditions for listening to speech and music.  It is an important part of the planning and construction of auditoriums, churches, halls, libraries, music rooms, and so on.  Various factors affect the acoustical quality of a room, including the size and shape of the room, the ability of the ceiling, walls and floor to keep out unwanted sound, and furnishings made of sound-absorbing materials, like carpets, drapes, and upholstered furniture.

Another factor is the room’s reflection of sound.  Sounds made by a speaker or musical instrument bounce back and forth against the surfaces of the room.  These reflections of sound make up its reverberation.  The reverberation time of a room is the time it takes for a sound to fade to one-millionth of its original energy.  Reverberations should last about one second in a speech auditorium and two seconds in a music hall.  If a strong reflection reaches a listener’s ear more than 1/20 of a second after the arrival of the direct sound, or else the listener will hear an echo.  Echoes should be avoided.

Acoustics have many other fields.  In sonochemistry, sound waves create high-energy bubbles in liquids.  When these bubbles heat and cool, they speed up chemical reactions for the creation of pharmaceuticals.  Defects in airplanes, bridges, and pipelines can be identified by sound waves.  Since sound waves can change pressure and temperature in gases, thermo-acoustic refrigerators can cool objects while being good to the environment.

Other areas of acoustics are environmental acoustics (controlling noise pollution), physiological acoustics (hearing sound), psychological acoustics (interpreting sound), musical acoustics (instruments and voices producing sound), and speech communication (producing and hearing speech).  Acoustics also encompasses sound waves we cannot hear, such as infrasound (too low) and ultrasound (too high.)  Sound waves within the earth and underwater are also outside human hearing and merit study.

Conic sections are often used in acoustics to amplify sound.  Orchestra shells, for example, have a parabolic shape.  A parabola is the set of all points equidistant from a fixed point and fixed line; it is shaped like a wave.  When a wave hits a parabola, it reflects towards the focus.  The orchestra’s sound reflects off the shell and back onto the audience.  This amplifies the orchestra’s sound.  The same principle is applied to enhance musical instruments.  String instruments are built with parabolas in their shape; when sound is made, it bounces off these parabolas and back into the focus, the sound post.

An ellipse, the set of all points equidistant from two fixed points (foci), is also key in acoustics.  When a line is drawn from the focus of an ellipse, it bounces off the side of the ellipse and touches the other focus.  This property is used in the lithiopter, a device that destroys kidney stones.  With this device, scientists fire shockwaves from one focus of an ellipse to destroy kidney stones at the other.

This property is also used in elliptical ceilings.  When sound is made at one focus of an ellipse, it bounces off the ceiling and reaches the other focus.  Thus, a person at one focus can even hear a whisper from the person at the other focus.  The most famous whisper gallery may be St. Paul’s Cathedral, built by the mathematician Sir Christopher Wren.

Some believe that Benjamin Franklin discovered this property at the Rotunda of the Capital Building, and he would stand at one focus to easily hear sound from the other focus, even in a crowded room.  Others credit John Quincy Adams.  While he was a member of the U. S. House of Representatives, he positioned his desk at one focus of the elliptical ceiling at the Capitol’s Statutory Hall and eavesdropped on private conversations of house members at the other focus.

The uses of acoustics are many, and conic sections are invariably important to them.  Research continues on how acoustics can be used to benefit humanity, and, as in the lithiopter, conic sections will be very important to the cause.  This “mathematical magic” can improve our lives for years to come.

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