•WAVE MOTIONS AND SOUND

Vibrations are common in many

elastic materials, and you can see and hear the results of many

in your surroundings. Other vibrations in your surroundings, such as those involved in heat, electricity, and light, are invisible to

the senses.

• Forces and Elastic Materials.

• Forces and Vibrations.– A vibration is a repeating motion that moves back and

forth.

A mass on a frictionless surface is at rest at an equilibrium position (A) when undisturbed. When the

spring is stretched (B) or compressed (D), then released (C), the mass vibrates back and forth because

restoring forces pull opposite to and proportional to the displacement.

• Describing Vibrations.– A vibrating mass is described by measuring several

variables.• The extent of displacement from the equilibrium

position.• A cycle is the movement from some point, to another

point and back again.• A period (T) is the time required for one complete

cycle.• Frequency (f) is the number of cycles per second.

– Frequency is measured in Hertz (Hz)– The period is the time for one cycle and the

frequency is the cycles per second, the relationship is: T=1/f

» F = 1/T

A vibrating mass attached to a spring is displaced from rest, or equilibrium, position, and then released. The maximum displacement is called the amplitude of the vibration. A cycle is one complete vibration. The period is the time required for one complete cycle. The frequency is a count of how many cycles it completes in 1s.

A graph of simple harmonic motion is described by a sinusoidal curve.

• Waves.

• Kinds of Waves.– Longitudinal Wave

• A wave that travels in a back and forth movement

– Transverse Wave• A wave that disturbs particles in a perpendicular motion in the

direction of the wave.

(A) Longitudinal waves are created in a spring when the free end is moved back and forth parallel to the spring. (B) Transverse waves are created in a spring

when the free end is moved up and down.

• Waves in Air.– Condensation

• When a longitudinal wave forces particles closer together.

• This results in a pulse of increased density and pressure.

– Rarefaction• A zone of reduced density and pressure.

– Sound• A vibrating object produces condensation and rarefactions that

expand from the source.

• The vibrations can be interpreted as sound by the human ear if the frequency of the waves is between 20 and 20,000 Hz.

When you open one door into this room, the other door closes. Why does this happen? The answer is that

the first door creates a pulse of compression that moves through the air like a sound wave. The pulse of

compression pushes on the second door, closing it.

(A) Swinging the door inward produces pulses of increased density and pressure called

condensations. Pulling the door outward produces pulses of

decreased density and pressure called rarefactions. (B) In a condensation, the average

distance between gas molecules is momentarily

decreased as the pulse passes. In a rarefaction, the average

distance is momentarily increased.

A vibrating tuning fork produces a series of condensations and rarefactions that move away from

the tuning fork. The pulses of increased and decreased pressure reach your ear, vibrating the eardrum. The

ear sends nerve signals to the brain about the vibrations, and the brain interprets the signals as

sounds.

Compare the (A) back-and-forth vibrations of a tuning fork with (B) the resulting condensations and

rarefactions that move through the air and (C) the resulting increases and decreases of air pressure on a

surface that intercepts the condensations and rarefactions

• Hearing Waves in Air.– Infrasonic

• Longitudinal waves with frequencies below 20 Hz

– Ultrasonic• Longitudinal waves with frequencies greater that 20,000 Hz

– Since humans can only hear waves in the 20 – 20,000 Hz range, they hear neither infrasonic nor ultrasonic waves.

– Waves move the eardrum in and out with the same frequency as the wave, which the brain interprets as sound.

• Wave Terms.

• Wave Crest– The maximum disturbance a wave will create from the resting

position

• Wave trough– Maximum displacement a wave will create in the opposite direction

from the resting position.

• Amplitude– The magnitude of the displacement to either the crest or the trough.

• Period– The time required for a wave to repeat itself– This is the time that is required to move through one full wave cycle.

Here are some terms associated with periodic waves. The wavelength is the distance from a part of one wave to the same part in the next wave, such as from one crest to the

next. The amplitude is the displacement from the rest position. The period is the time required for a wave to

repeat itself, that is the time for one complete wavelength to move past a given location.

• Wavelength– The distance from one crest of a wave to the crest of the

next wave.– Given the Greek symbol lambda ()

• Wave Equation– The wave equation tells us that the relationship between

the velocity of sound waves and the frequency is:• v=f

• Sound Waves.

• Introduction– The movement of sound waves requires a medium

through which the waves can travel.– The nature of the medium determines the velocity of the

sound through the medium• This is due to the fact that the waves are propagated through

molecular interactions and is determined by:

• Inertia of the molecules

• Strength of the interactions between molecules

(A) Spherical waves move outward from a sounding source much as a rapidly expanding balloon. This two-

dimensional sketch shows the repeating condensation as spherical wave fronts. (B) Some distance from the source,

a spherical wave front is considered a linear, or plane, wave front.

(A) Since sound travels faster in warmer air, a wave front becomes bent, or refracted, toward the earth's surface

when the air is cooler near the surface. (B) When the air is warmer near the surface, a wave front is refracted upward,

away from the surface.

This closed-circuit TV control room is acoustically treated by covering the walls with sound-absorbing

baffles.

• Velocity in Air.

– As the gas molecules that make up the air increase in temperature, the velocity of sound waves increases due to increased kinetic energy (energy of motion).

– This increase is 0.60 m/s for each degree Celsius increase in temperature.

– At sea level, in dry air the velocity of sound is 331.0 m/s.

– The velocity of sound at different temperatures can be calculated from the following equation:

• For feet per second simply substitute in 2.0 ft/s for 0.60 m/s.

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(A) At room temperature, sound travels at 343 m/s. In 0.10 s, sound would travel 34 m. Since the sound must travel to a surface and back in order for you to hear the

echo, the distance to the surface is one-half the total

distance. (B) Sonar measures a depth by

measuring the elapsed time between an ultrasonic

sound pulse and the echo. The depth is one-half the

round trip.

• Refraction and Reflection– Sound waves are reflected or refracted from a boundary,

which means a change in the medium through which they are being transmitted.

• Interference.– Constructive interference

• Reflected waves that are in phase with the incoming waves undergo constructive interference.

– Destructive interference• Waves that are out of phase undergo destructive interference.

(A) Constructive interference occurs when two equal, in-phase waves meet. (B) Destructive interference occurs

when two equal, out-of-phase waves meet. In both cases, the wave displacements are superimposed when they

meet, but they then pass through one another and return to their original amplitudes.

Two waves of equal amplitude but slightly different frequencies interfere destructively and constructively. The

result is an alternation of loudness called a beat.

• Energy and Sound.

• Loudness.– The energy of a sound wave is called the wave intensity

and is measured in Watts per square meter.– The intensity of wound is expressed on the decibel scale,

which relates to changes in loudness as perceived by the human ear.

The intensity, or energy, of a sound wave is the rate of energy transferred to an area perpendicular to the waves. Intensity is measured in watts per square meter, W/m2.

• Resonance.– All elastic objects have natural frequencies of vibration

that are determined by the materials they are made of and their shapes.

– When energy is transferred at the natural frequencies, there is a dramatic increase of amplitude called resonance.

– The natural frequencies are also called resonant frequencies.

When the frequency of an applied force, including the force of a sound wave, matches the natural frequency of

an object, energy is transferred very efficiently. The condition is called resonance.

Different sounds that you hear include (A) noise, (B) pure tones, and (C) musical notes.

• Sources of Sounds.

• Vibrating Strings.– Standing Waves

• When reflected waves interfere with incoming waves

• Created by a patter on nodes and antinodes

– Nodes• Places of destructive interference, which show no disturbance

– Antinodes• Loops of constructive interference which take place where

crests and troughs produce a disturbance that rapidly alternates upward and downward.

– Fundamental Frequency• The longest wave that can make a standing wave on a string has

a wavelength that is twice the length of the string

• This longest wavelength has the lowest frequency and is called the fundamental frequency.

• The fundamental frequency determines the pitch of the basic musical note being sounded and is called the first harmonic.

An incoming wave on a chord with a fixed end (A) meets a reflected

wave (B) with the same amplitude and

frequency, producing a standing wave (C). Note that a standing

wave of one wavelength has three

nodes and two antinodes.

A stretched string of a given length has a number of possible resonant frequencies. The

lowest frequency is the fundamental,

f1; the next higher frequencies, or

overtones, shown are f2 and f3.

A combination of the fundamental and overtone frequencies produces a composite waveform with

characteristic sound quality.

– Overtones• Any whole number of halves of the wavelength will permit a

standing wave to form.

• The frequencies of these wavelengths are called overtones or harmonics.

– The relationship for finding the fundamental frequency and the overtones when the string length and velocity are known is:

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• Vibrating Air Columns.– Closed Tube.

• The air in a closed tube can be made to vibrate by a reed, turbulence, or some other way.

• In a closed tube the longest wave that can make a standing wave has a wavelength that is four times the length of the column.

• The relationship between the fundamental frequency, the overtones, the length of the column and the velocity are given by the equation:

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Standing sine wave patterns of air vibrating in a closed tube. Note the node at the closed end and the antinode at the open end. Only odd multiples of the fundamental are

therefore possible.

– Open Tube.• An open tube can have an antinode at both ends since the air is

free to vibrate at both places.

• The longest wavelength that can fit into the open tube is therefore the distance between the 2 antinodes or ½

• The relationship between frequency, velocity, and length for an open column is give by:

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Standing waves in these open tubes have an antinode at the open end, where air is free to vibrate.

Standing sine wave patterns of air vibrating in an open tube. Note that both ends have anitnodes. Any whole number of multiples of the fundamental are therefore

possible.

• Sounds from Moving Sources.– A moving source of sound or a moving observer

experiences an apparent shift of frequency called the Doppler Effect.

– If the source is moving as fast or faster than the speed of sound, the sound waves pile up into a shock wave called a sonic boom.

– A sonic boom sound very much like the pressure wave from an explosion

(A) Spherical sound waves from a stationary source spread out evenly in all

directions. (B) If the source is moving, an observer at position P will experience

more wave crests per second than an observer at position

P'. The observer at P interprets this as a higher

pitch, and the phenomenon is called the Doppler effect.