Everything Maths and Science You are here: Home › Grade 12 Physical Science › Doppler effect › The Doppler effect with sound The Doppler effect with sound You, the person hearing the sounds, are called the observer or listener and the thing emitting the sound is called the source. As mentioned in the introduction, there are two situations which lead to the Doppler effect: When the source moves relative to a stationary observer. When the observer moves relative to a stationary source. In points 1 and 2 above there is relative motion between the source and the observer. Both the source and the observer can be moving at the same time but we won't deal with that case in this chapter. Definition 1: Doppler effect The Doppler effect is the change in the observed frequency of a wave when the source or the detector moves relative to the transmitting medium. The Doppler effect occurs when a source of waves and/or observer move relative to each other, resulting in the observer measuring a different frequency of the waves than the frequency that the source is emitting. The medium that the waves are travelling through, the transmitting medium, is also stationary in the cases we will study. The question that probably comes to mind is: "How does the Doppler effect come about?".

We can understand what is happening by thinking through the situation in detail. Case 1: Moving source, stationary observer Let us consider a source of sound waves with a constant frequency and amplitude. The sound waves can be represented as concentric circles where each circle represents a crest or peak as the wavefronts radiate away from the source. This is because the waves travel away from the source in all directions and the distance between consecutive crests or consecutive troughs in a wave is constant (the wavelength as we learnt in Grade 10). In this figure the crests are represented by by the black lines and the troughs by the orange lines. Figure 1: Stationary sound source as more wavefronts are emmitted. The sound source is the police car in the middle and is stationary. For the Doppler effect to take place (manifest), the source must be moving relative to the observer. Let's consider the following situation: The source (represented by the black dot) emits one wave (the black circles represent the crests of the sound wave) that moves away from the source at the same rate in all directions. The distance between the crests represents the wavelength (\(\lambda\)) of the sound.

The closer together the crests, the higher the frequency (or pitch) of the sound according to \(f=\frac{v}{\lambda}\), where \(v\) (speed of sound) is constant. As this crest moves away, the source also moves and then emits more crests. Now the two circles are not concentric any more, but on the one side they are closer together and on the other side they are further apart. This is shown in the next diagram. If the source continues moving at the same speed in the same direction, then the distance between crests on the right of the source is constant. The distance between crests on the left is also constant. The distance between successive crests on the left is constant but larger than the distance between successive crests on the right. When a car approaches you, the sound waves that reach you have a shorter wavelength and a higher frequency. You hear a sound with a higher pitch. When the car moves away from you, the sound waves that reach you have a longer wavelength and lower frequency. You hear a sound with a lower pitch. Figure 2: Moving sound source as more wavefronts are emmitted.

Case 2: Moving observer, stationary source Just as we did before, let us consider a source (a police car) of sound waves with a constant frequency and amplitude. There are two observers, one on the left that will move away from the source and one on the right that will move towards the source. We have three diagrams: shows the overall situation with the siren starting at time \(t_1\); shows the situation at time \(t_2\) when the observers are moving; and shows the situation at \(t_3\) after the observers have been moving for a time interval, \(\Delta t=t_3-t_2\). The crests and troughs are numbered so you can see how they move further away and so that we can track which ones an observer has measured. The observers can hear the sound waves emitted by the police car and they start to move (we ignore the time it takes them to accelerate). The frequency of the wave that an observer measures is the number of complete waves cycles per unit time. By numbering the crests and troughs we can see which complete wave cycles have been measured by each of the observers in time, \(\Delta t\).

To find the frequency we divide the number of wave cycles by \(\Delta t\). In the time interval that passed, the observer moving towards the police car observed the crests and troughs numbered 1 through 5 (the portion of the wave is highlighted below). The observer moving away encountered a smaller portion of the wavefront, crest 3 and trough 4. The time interval for each of them is the same. To the observers this will mean that the frequency they measured is different. The motion of the observer will alter the frequency of the measured sound from a stationary source: An observer moving towards the source measures a higher frequency. An observer moving away from the source measures a lower frequency. It is important to note that we have only looked at the cases where the source and observer are moving directly towards or away from each other and these are the only cases we will consider. Note: We didn't actually need to analyse both cases. We could have used either explanation because of relative motion. The case of a stationary source with moving observer is the same as the case of the stationary observer and the moving source because the relative motion is the same. Do you agree? Discuss with your friends and try to convince yourselves that this is the case.

Being able to explain work to each other will help you understand it better. If you don't understand it, you won't be able to explain it convincingly. For a real conceptual test, discuss what you think will happen if the source and the observer are both moving, in the same direction and at the same speed. The formula that provides the relationship between the frequency emitted by the source and the frequency measured by the listener is: \[\boxed{{f}_{L}=\left(\frac{v\pm{v}_{L}}{v\pm{v}_{S}}\right){f}_{S}}\] where \({f}_{L}\) is the frequency perceived by the observer (listener), \({f}_{S}\) is the frequency of the source, \(v\) is the speed of the waves, \({v}_{L}\) the speed of the listener and \({v}_{S}\) the speed of the source.

Note: The signs show whether or not the relative motion of the source and observer is towards each other or away from each other: Table 1 Sound TIMBRE Complex waveforms like the one shown above are constructed by combining a number of simple waveforms (like the one in the first drawing) of different amplitudes and frequencies. This is why we perceive both high and low pitched sounds at once when we hear most natural sounds. The characteristic sound of a waveform (be it produced by a grand piano or a violin) is called its timbre. Timbre, also referred to as tone color, is said to be rich or full when there are many different frequencies in a sound. A sound from a sine wave is considered dull by most people since it has only one frequency. The different frequencies in a sound, combined with the varying amplitudes of each frequency, make up the spectral content of a waveform. The spectral content, which you might say is the more scientific term for timbre, usually varies over time. Otherwise, the sound remains static and again sounds dull. The spectral characteristic of a waveform over time is the signature of a tone that allows you to describe it as string-like or horn-like. Sound, physical phenomenon that stimulates the sense of hearing.

In humans, hearing takes place whenever vibrations of frequencies between about 15 and 20,000 hertz reach the inner ear. The hertz, or Hz, is a unit of frequency equaling one cycle per second. Such vibrations reach the inner ear when they are transmitted through air, and the term sound is something restricted to such airborne vibrational waves. Modern physicists, however, usually extend the term to include similar vibrations in liquid or solid media. Sounds of frequencies higher than about 20,000 Hz are called ultrasonic. Characteristic Sound Waves Each instrument produces a characteristic vibration. The vibrations travel through the air in sound waves that reach our ears, allowing us to identify the instrument being played even when we cannot see it. The four sound waves shown here demonstrate signature waveforms of some common instruments. A tuning fork makes a pure sound, vibrating regularly in a curving waveform. A violin generates a bright sound and a jagged waveform. The flute produces a mellow, true sound and a relatively curved waveform. The tuning fork, violin, and flute were all playing the same note, so the distance between the peaks (the high points of the wave) is the same for each waveform.

A gong does not vibrate in a regular pattern as do the first three instruments. Its waveform is jagged and random, and its pitch is generally unrecognizable. In general, waves can be propagated transversely or longitudinally. In both cases, only the energy of wave motion is propagated through the medium; no portion of the medium itself actually moves very far. As a simple example, a rope may be tied securely to a post at one end, and the other end pulled almost taut and then shaken once. A wave will travel down the rope to the post, and at that point it will be reflected and returned to the hand. No part of the rope actually moves longitudinally toward the post, but each successive portion of the rope moves transversely. This type of wave motion is called a transverse wave. Similarly, if a rock is thrown into a pool of water, a series of transverse waves moves out from the point of impact. A cork floating near the point of impact will bob up and down, that is, move transversely with respect to the direction of wave motion, but will show little if any outward, or longitudinal, motion. A sound wave, on the other hand, is a longitudinal wave.

As the energy of wave motion is propagated outward from the center of disturbance, the individual air molecules that carry the sound move back and forth, parallel to the direction of wave motion. Thus, a sound wave is a series of alternate compressions and rarefactions of the air. Each individual molecule passes the energy on to neighboring molecules, but after the sound wave has passed, each molecule remains in about the same location. Physical Characteristics Any simple sound, such as a musical note, may be completely described by specifying three perceptual characteristics: pitch, loudness (or intensity), and quality (or timbre). These characteristics correspond exactly to three physical characteristics: frequency, amplitude, and harmonic constitution, or waveform, respectively. Noise is a complex sound, a mixture of many different frequencies or notes not harmonically related. Frequency Sounds can be produced at a desired frequency by different methods. For example, a sound of 440 Hz can be created by actuating a loudspeaker with an oscillator tuned to this frequency. An air blast can be interrupted by a toothed wheel with 44 teeth, rotating at 10 revolutions/sec; this method is used in operating an ordinary siren.

The sound of the speaker and that of the siren at the same frequency are very different in quality, but will correspond closely in pitch, equivalent to the A above middle C on a piano. The next higher A on the piano, the note one octave above, has a frequency of 880 Hz. Similarly, notes one or two octaves below have frequencies of 220 or 110 Hz, respectively. Thus, by definition, an octave is the interval between any two notes the frequencies of which are in a two-to-one ratio. A fundamental law of harmony states that two notes an octave apart, when sounded together, produce a euphonious combination. A fifth and a major third produce successively less euphonious combinations. Physically, an interval of a fifth consists of two notes, the frequencies of which bear the arithmetical ratio three to two, and a major third, the ratio five to four. Fundamentally, then, the law of harmony states that two or more notes sound euphonious when played together if their frequencies bear the ratio of small, whole numbers; if the frequencies do not bear such ratios, a dissonance is produced.

On a fixed-pitch instrument, such as a piano, it is not possible to arrange the notes so that all of these ratios hold exactly, and some compromise is necessary in tuning, called the meantone system, or tempered scale. Amplitude The amplitude of a sound wave is the degree of motion of air molecules within the wave, which corresponds to the extent of rarefaction and compression that accompanies the wave. The greater the amplitude of the wave, the harder the molecules strike the ear drum and the louder the sound that is perceived. The amplitude of a sound wave can be expressed in terms of absolute units by measuring the actual distance of displacement of the air molecules, or the pressure differential in the compression and rarefaction, or the energy involved. Ordinary speech, for example, produces sound energy at the rate of about one hundred-thousandth of a watt. All of these measurements are extremely difficult to make, however, and the intensity of sounds is generally expressed by comparing them to a standard sound, measured in decibels (see Sensations of Tone below). Intensity The distance at which a sound can be heard depends on its intensity, which is the average rate of flow of energy per unit area perpendicular to the direction of propagation.

In the case of spherical waves spreading from a point source, the intensity varies inversely as the square of the distance, provided that no loss of energy is due to viscosity, heat conduction, or other absorption effects. Thus, in a perfectly homogeneous medium, a sound will be nine times as intense at a distance of 1 unit from its origin as at a distance of 3 units; that is, intensity varies inversely as the square of the distance. In the actual propagation of sound through the atmosphere, changes in the physical properties of the air, such as temperature, pressure, and humidity, produce damping and scattering of the directed sound waves, so that the inverse-square law generally is not applicable in direct measurements of the intensity of sound. Quality If A above middle C is played on a violin, a piano, and a tuning fork, all at the same volume, the tones are identical in frequency and amplitude, but very different in quality. Of these three sources, the simplest tone is produced by the tuning fork, the sound in this case consisting almost entirely of vibrations having frequencies of 440 Hz.

Because of the acoustical properties of the ear and the resonance properties of the ear's vibrating membrane, however, it is doubtful whether a pure tone reaches the inner hearing mechanism in an unmodified form. The principal component of the note produced by the piano or violin also has a frequency of 440 Hz, but these notes also contain components with frequencies that are exact multiples of 440, called overtones, such as 880, 1320, and 1760. The exact intensity of these other components, which are called harmonics, determines the quality of the note. Velocity of Sound The frequency of a sound wave is a measure of the number of waves passing a given point in 1 second. The distance between two successive crests of the wave is called the wavelength. The product of the wavelength and the frequency must equal the speed of propagation of the wave, and is the same for sounds of all frequencies (if the sound is propagated through the same medium at the same temperature).

Thus, the wavelength of A above middle C is about 78.2 cm (about 2.6 ft), and the wavelength of A below middle C is about 156.4 cm (about 5.1 ft). The speed of propagation of sound in dry air at a temperature of 0ï¿½ C (32ï¿½ F) is 331.6 m/sec (1088 ft/sec). If the temperature is increased, the speed of sound increases; thus, at 20ï¿½ C (68ï¿½ F), the velocity of sound is 344 m/sec (1129 ft/sec). Changes in pressure at controlled density have virtually no effect on the speed of s ound. The velocity of sound in many other gases depends only on their density. If the molecules are heavy, they move less readily, and sound progresses through such a medium more slowly. Thus, sound travels slightly faster in moist air than in dry air, because moist air contains a greater number of lighter molecules. The velocity of sound in most gases depends also on one other factor, the specific heat, which affects the propagation of sound waves.

Sou nd generally moves much faster in liquids and solids than in gases. In both liquids and solids, density has the same effect as in gases; that is, velocity varies inversely as the square root of the density. The velocity also varies directly as the square root of the elasticity. The speed of sound in water, for example, is slightly less than 1525 m/sec (5000 ft/sec) at ordinary temperatures but increases greatly with an increase in temperature. The speed of sound in copper is about 3353 m/sec (about 11,000 ft/sec) at ordinary temperatures and decreases as the temperature is increased (due to decreasing elasticity); in steel, which is more elastic, sound moves at a speed of about 4877 m/sec (about 16,000 ft/sec). Sound is propagated very efficiently in steel. Refraction, Reflection, and Interference Sound moves forward in a straight line when traveling through a medium having uniform density. Like light, however, sound is subject to refraction, which bends sound waves from their original path.

In polar regions, for example, where air close to the ground is colder than air that is somewhat higher, a rising sound wave entering the warmer region, in which sound moves with greater speed, is bent downward by refraction. The excellent reception of sound downwind and the poor reception upwind are also due to refraction. The velocity of wind is generally greater at an altitude of many meters than near the ground; a rising sound wave moving downwind is bent back toward the ground, whereas a similar sound wave moving upwind is bent upward over the head of the hearer. Sound is also governed by reflection, obeying the fundamental law that the angle of incidence equals the angle of reflection. An echo is the result of reflection of sound. Sonar depends on the reflection of sounds propagated in water. A megaphone is a funnel-like tube that forms a beam of sound waves by reflecting some of the diverging rays from the sides of the tube. A similar tube can gather sound waves if the large end is pointed at the source of the sound; an ear trumpet is such a device. Sound is also subject to diffraction and interference.

If sound from a single source reaches a listener by two different pathsï¿½one direct and the other reflectedï¿½the two sounds may reinforce one another; but if they are out of phase they may interfere, so that the resultant sound is actually less intense than the direct sound without reflection. Interference paths are different for sounds of different frequencies, so that interference produces distortion in complex sounds. Two sounds of different frequencies may combine to produce a third sound, the frequency of which is equal to the sum or difference of the original two frequencies. Sensations of Tone If the ear of an average young person is tested by an audiometer, it will be found to be sensitive to all sounds from 15 to 20 Hz to 15,000 or 20,000 Hz. The hearing of older persons is less acute, particularly to the higher frequencies. The ear is most sensitive in the range from A above middle C up to A four octaves higher; in this range a sound can be perceived hundreds of times fainter than a sound an octave higher or two octaves lower.

The degree to which a sensitive ear can distinguish between two pure notes of slightly different loudness or slightly different frequency varies in different ranges of loudness and frequency of the tones. A difference in loudness of about 20 percent (1 decibel, dB), and a difference in frequency of 1/3 percent (about 1/20 of a note) can be distinguished in sounds of moderate intensity at the frequencies to which the ear is most sensitive (about 1000 to 2000 Hz). In this same range, the difference between the softest sound that can be heard and the loudest sound that can be distinguished as sound (louder sounds are "felt," or perceived, as painful stimuli) is about 120 dB (about 1 trillion times as loud). All of these sensitivity tests refer to pure tones, such as those produced by an electronic oscillator. Even for such pure tones the ear is imperfect. Notes of identical frequency but differing greatly in intensity may seem to differ slightly in pitch. More important is the difference in apparent relative intensities with different frequencies.

At high intensities the ear is approximately equally sensitive to most frequencies, but at low intensities the ear is much more sensitive to the middle high frequencies than to the lowest and highest. Thus, sound-reproducing equipment that is functioning perfectly will seem to fail to reproduce the lowest and highest notes if the volume is decreased. Three Important Types of Ordinary Sound In speech, music, and noise, pure tones are seldom heard. A musical note contains, in addition to a fundamental frequency, higher tones that are harmonics of the fundamental frequency. Speech contains a complex mixture of sounds, some (but not all) of which are in harmonic relation to one another. Noise consists of a mixture of many different frequencies within a certain range; it is thus comparable to white light, which consists of a mixture of light of all different colors. Different noises are distinguished by different distributions of energy in the various frequency ranges. When a musical tone containing some harmonics of a fundamental tone, but missing other harmonics or the fundamental itself, is transmitted to the ear, the ear forms various beats in the form of sum and difference frequencies, thus producing the missing harmonics or the fundamental not present in the original sound. These notes are also harmonics of the original fundamental note.

This incorrect response of the ear may be valuable. Sound-reproducing equipment without a large speaker, for example, cannot generally produce sounds of pitch lower than two octaves below middle C; nonetheless, a human ear listening to such equipment can resupply the fundamental note by resolving beat frequencies from its harmonics. Another imperfection of the ear in the presence of ordinary sounds is the inability to hear high-frequency notes when low-frequency sound of considerable intensity is present. This phenomenon is called masking. In general, speech is understandable and musical themes can be satisfactorily understood if only the frequencies between 250 and 3000 Hz, the frequency range of ordinary telephones, are reproduced, although a few speech sounds, such as th, require frequencies as high as 6000 Hz. For naturalness, however, the range of about 100 to 10,000 Hz must be reproduced. Sounds produced by a few musical instruments can be reproduced naturally only at somewhat lower frequencies, and a few noises can be reproduced at somewhat higher frequencies.

Historical Development The elementary phenomena of sound were the subject of much speculation among the ancient peoples; however, with the exception of a few lucky guesses, little was known about the science of sound until about AD 1600. Starting at that time, the knowledge of sound increased more rapidly than knowledge of the corresponding phenomena of light, because the latter are more difficult to observe and measure. The ancient Greeks cared little for the scientific study of sound, but they had a great interest in music, and considered music to represent "applied number," in contrast to "pure number," the science of arithmetic. The Greek philosopher Pythagoras discovered that an octave represents a two-to-one frequency ratio and enunciated the law connecting consonance with numerical ratios; on this law, however, he built a fantastic and unscientific edifice of mystical speculation. Aristotle, in brief remarks on sound, made a fairly accurate guess concerning the nature of the generation and transmission of sound, but no scientifically valid experimental studies were made until about 1600, when Galileo made a scientific study of sound and enunciated many of its fundamental laws.

Galileo stated the relationship between pitch and frequency and the laws of musical harmony and dissonance, essentially as stated above in this article. He also explained theoretically how the natural frequency of vibration of a stretched string, and hence the frequency of sound produced by a string instrument, depends on the length, weight, and tension of the string. The 16th, 17th, and 18th Centuries Quantitative measurement of sound was made by the French mathematician Marin Mersenne, who measured the time of return of an echo, and arrived at a figure that was in error by less than 10 percent. Mersenne also made the first crude determination of the actual frequency of a note of a given pitch. He measured the frequency of vibration of a long, heavy wire that moved so slowly that its motion could be followed by the eye; then, from theoretical considerations, he calculated the frequency of a short, light wire that produced an audible sound.

In 1660, the dependence of sound on a gaseous, liquid, or solid medium for transmission was demonstrated by the Anglo-Irish scientist Robert Boyle, who suspended a bell in a vacuum by means of a string and showed that, although the clapper could be seen to strike the bell, no sound was heard. The mathematical treatment of the theory of sound was begun by the English mathematician and physicist Sir Isaac Newton in his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy, 1687). The propagation of sound through any fluid was shown to depend only on measurable physical properties of the fluid, such as elasticity and density, and Newton calculated from theoretical considerations the velocity of sound in air. The 18th century was primarily a period of theoretical development. The calculus provided a powerful new tool to scientists in many fields, and such mathematicians as the French Jean le Rond d'Alembert and Joseph Louis Lagrange, the Dutch mathematician Johann Bernoulli, and the Swiss mathematician Leonhard Euler contributed to the knowledge of such subjects as the pitch and quality of sound produced by a particular musical instrument and the speed and nature of transmission of sound in various media.

The complete mathematical treatment of sound, however, depends on harmonic analysis, which was discovered by the French mathematician Baron Jean Baptiste Joseph Fourier in 1822 and applied to sound by the German physicist Georg Simon Ohm. Variations in sound, called beats, inherent in sound waves, were discovered about 1740 by the Italian violinist Giuseppe Tartini and the German organist Georg Sorge. The German physicist Ernst Chladni made numerous discoveries in acoustics at the close of the 18th century, notably concerning the vibration of strings and rods. The 19th and 20th Centuries The 19th century was primarily a period of experimental development. The first accurate measurements of the speed of sound in water were made in 1826 by the French mathematician Jacques Sturm, and throughout the century numerous experiments were made determining the speed of sound of various frequencies in various media with extreme accuracy. The fundamental law that the speed is the same for sounds of different frequencies and depends on the density and elasticity of the medium was determined in these experiments.

The stroboscope, the stethoscope, and the siren were all used in the study of sound during the 19th century. The standardization of pitch occupied much attention in the 19th century. The first suggestion for a standard had been made about 1700 by the French physicist Joseph Sauveur, who proposed C equals 256, a convenient standard for mathematical purposes. The German physicist Johann Heinrich Scheibler made the first accurate determination of pitch corresponding to frequency and proposed the standard A equals 440 in 1834. In 1859 the French government decreed that the standard should be A equals 435, based on the research of the French physicist Jules Antoine Lissajous. This standard was accepted in many parts of the world, including the United States, until well into the 20th century. During the 19th century the telephone, the microphone, and the phonograph, all of which were useful for further study of sound, were invented. In the 20th century, physicists for the first time had instruments that made possible simple, accurate, quantitative study of sound.

By means of electronic oscillators, waves of any type may be produced electronically, then converted into sounds by electromagnetic or piezoelectric means . Conversely, sounds may be converted into electrical currents by means of a microphone, amplified electronically without distortion, and then analyzed by means of a cathode-ray oscilloscope. Modern techniques permit extremely high-fidelity recording and reproduction of sound. Military necessity in World War I led to the first use of sound for underwater detection of vessels; sound is now also used for studies of ocean currents and layers, and for sea-bottom mapping. In addition, ultrahigh-frequency sound waves are now used in a wide range of technical and medical applications. Regents Physics Wave Characteristics Regents Physics - Wave Characteristics Defining & Describing Waves A pulse is a single disturbance which carries energy through a medium or through space. Imagine you and your friend holding opposite ends of a slinky. If you quickly move your arm up and down, a single pulse will travel down the slinky toward your friend. If, instead, you generate several pulses at regular time intervals, you now have a wave carrying energy down the slinky. A wave, therefore is a repeated disturbance which carries energy.

The mass of the slinky doesn't move from end of the slinky to the other, but the energy it carries does. When a pulse or wave reaches a hard boundary, it reflects off the boundary, and is inverted. If a pulse or wave reaches a soft, or flexible, boundary, it still reflects off the boundary, but does not invert. Courtesy Penn State Schuylkill Waves at Boundaries Waves can be classified in several different waves. One type of wave, known as a mechanical wave, requires a medium (or material) through which to travel. Examples of mechanical waves include water waves, sound waves, slinky waves, and even seismic waves. Electromagnetic waves, on the other hand, do not require a medium in order to travel. Electromagnetic waves (or EM waves) are considered part of the Electromagnetic Spectrum. Examples of EM waves include light, radio waves, microwaves, and even X-rays. Further, waves can be classified based upon their direction of vibration. Waves in which the "particles" of the wave vibrate in the same direction as the wave direction are known as longitudinal, or compressional, waves. Examples of longitudinal waves include sound waves and seismic P waves.

Waves in which the particles of the wave vibrate perpendicular to the wave's direction of motion are known as transverse waves. Examples of transverse waves include seismic S waves, electromagnetic waves, and even stadium waves (the "human" waves you see at a baseball or football game!). Courtesy Penn State Schuylkill Transverse vs. Longitudinal It is important to note that in a transverse wave, the motion of the medium is perpendicular to the direction of the wave's velocity. This is illustrated clearly in the animation below. Wave Anatomy Waves have a number of characterisics which define their behavior. Looking at a transverse wave, we can identify specific locations on the wave. The highest points on the wave are known as crests. The lowest points on the wave are known as troughs. The amplitude of the wave, corresponding to the energy of the wave, is the distance from the baseline to a crest or the baseline to a trough. The length of the wave, or wavelength, noted with the Greek letter lambda ( ), is the distance between corresponding points on consecutive waves (i.e. crest to crest or trough to trough).

Points on the same wave with the same displacement from equilibrium moving in the same direction (such as a crest to a crest or a trough to a trough) are said to be in phase (phase difference is 0° or 360°). Points with opposite displacements from equilibrium (such as a crest to a trough) are said to be 180° out of phase. Question: The diagram represents a periodic wave. Which point on the wave is in phase with point P? Answer: Point C is in phase with point P, since point C represents a point with the same displacement from equilibrium moving in the same direction. Points that are in phase are located one or more whole wavelengths apart on a wave. Question: Two waves having the same frequency and amplitude are traveling in the same medium. Maximum constructive interference occurs at points where the phase difference between the two superposed waves is: Answer: 0°. In similar fashion, longitudinal, or compressional, waves also have amplitude and wavelength.

In the case of longitudinal waves, however, instead of crests and troughs, the longitudinal waves have areas of high density (compressions) and areas of low density (rarefactions), as shown in the representation of the particles of a sound wave. The wavelength, then, of a compressional wave is the distance between compressions, or the distance between rarefactions. Once again, the amplitude corresponds to the energy of the wave. Question: A longitudinal wave moves to the right through a uniform medium, as shown below. Points A, B, C, D, and E represent the positions of particles of the medium. What is the direction of the motion of the particles at position C as the wave moves to the right? Answer: The particles move to the left and right at position C, as the particles in a longitudinal wave vibrate parallel to the wave velocity: Question: Between which two points on the wave could you measure a complete wavelength? Answer: You could measure a complete wavelength between points A and C, since A and C represent the same point on successive waves. The Wave Equation The frequency (f) of a wave describes the number of waves that pass a given point in a time period of one second.

The higher the frequency, the more waves that pass. Frequency is measured in number of waves per second (1/s), also known as a Hertz (Hz). If 60 waves pass a given point in a second, the frequency of the wave would be 60 Hz. Closely related to frequency, the period (T) of a wave describes how long it takes for a single wave to pass a given point, and can be found as the reciprocal of the frequency. Period is a measurement of time, and therefore is measured in seconds. Both frequency and period were introduced earlier in our discussion of circular motion . Question: What is the period of a 60-hertz electromagnetic wave traveling at 3.0×108 meters per second? Answer: Because waves move through space, they must have a velocity. The velocity of a wave is a function of the type of wave, and the medium it travels through. Electromagnetic waves moving through a vacuum, for instance, travel at roughly 3*108 m/s. This value is so famous and common in physics it is given its own symbol, c. When an electromagnetic wave enters a different medium, such as glass, it slows down.

If the same wave were to then re-emerge from glass back into a vacuum, it would again travel at c, or 3*108 m/s. The speed of a wave can be easily related to its frequency and wavelength. Speed of a wave is determined by the wave type and the medium its traveling through. For a given wave speed, as frequency increases, wavelength must decrease, and vice versa. This can be shown mathematically using the wave equation: . Question: In a vacuum, all electromagnetic waves have the same: speed wavelength Answer: (1) speed. Question: The diagram below represents a periodic wave traveling through a uniform medium. If the frequency of the wave is 2.0 hertz, find the speed of the wave. Answer: The diagram shows a length of 6m for 1.5 wavelengths, therefore the wavelength ( ) must be 4m. Given both wavelength and frequency, we can use the wave equation to find the wave speed: Sound Waves Sound is a mechanical wave which we observe by detecting vibrations in the inner ear. Typically, we think of sound as traveling through air, therefore the particles vibrating are air molecules.

Sound can travel through other media as well, including water, wood, and even steel. The particles of a sound wave vibrate in a direction parallel with the direction of the sound wave's velocity, therefore sound is a longitudinal wave. The speed of sound in air at standard temperature and pressure (STP) is 331 m/s, a value which is supplied on the front of the Regents Physics Reference Table . Question: At an outdoor physics demonstration, a delay of 0.50 second was observed between the time sound waves left a loudspeaker and the time these sound waves reached a student through the air. If the air is at STP, how far was the student from the speaker? Answer: When we observe sound waves through hearing, we pick up the amplitude, or energy, of the wave as loudness. The frequency of the wave is perceived as pitch, with higher frequencies observed as a higher pitch. Typically, humans can hear a frequency range of 20Hz to 20,000 Hz, although young observers can often detect frequencies above 20,000 Hz, an ability which declines with age. Certain devices create strong sound waves at a single specific frequency.

If another object, having the same "natural frequency," is impacted by these sound waves, it may begin to vibrate at this frequency, producing more sound waves. The phenomenon where one object emitting a sound wave with a specific frequency causes another object with the same natural frequency to vibrate is known as resonance. A dramatic demonstration of resonance involves a singer breaking a glass by singing a high pitch note. The singer creates a sound wave with a frequency equal to the natural frequency of the glass, causing the glass to vibrate at its natural, or resonant, frequency so energetically that it shatters. Content on this page requires a newer version of Adobe Flash Player. = wavelength in ft The amplitude of vibration is the maximum displacement of the particle from its equilibrium position. Two particles are in phase when they are vibrating with the same frequency, and continually pass through corresponding points of their paths at the same time. For any other condition the particles are out of phase. The two particles are in phase opposition when they reach their maximum displacement in opposite directions at the same time. The wavelength is the distance measured along the direction of propagation between two corresponding points of equal intensity that are in phase on adjacent waves.

This length can be represented by the distance between the adjacent maximum rarefaction points in the traveling sound wave ( figure 7-47 ). When referring to figure 7-47 , keep in mind that the transverse wave drawn below the compressional wave is merely a device for simplifying the concept and relating it to the type of wave illustration commonly used in discussions of electromagnetic waves. When an advancing wave encounters a medium of different character, some of its energy is reflected back into the initial medium, and some is transmitted into the second medium. Reflection of Sound Waves To understand wave reflection, it is helpful to think of the wave as a ray. A ray is a line which indicates the direction the wave is traveling. In a uniform medium, a ray will travel in a straight line. Only at the boundary of two media or in an area where the medium is changing do the rays change their direction. If a line, called a "normal," is drawn perpendicular to a boundary, the angle between an incoming ray and this normal is called the angle of incidence, "i" as shown in figure 7-48 .

The angle which the reflected ray makes with the normal is called the angle of reflection "r." Any wave being reflected is reflected in such a way that the angle of incidence equals the angle of reflection. Light is often thought of first when reflection is discussed; however, reflection is equally common in other types of waves. As an example, echoes are caused by reflection of sound waves. When a hard surface is situated so that a sound reflection from it is outstanding, it appears as a distinct echo, and is heard an appreciable interval later than the direct sound. If the surface is concave, it may have a focusing effect and concentrate the reflected sound energy at one locality. Such a reflection may be several levels higher in intensity than the direct sound, and its arrival at a later time may have particular significance in such applications as sonar. Speed of Sound In any uniform medium, under given physical conditions, sound travels at a definite speed. In some substances, the velocity of sound is higher than in others. Even in the same medium under different conditions of temperature, pressure, etc., the velocity of sound varies. Density and elasticity of a medium are the two basic physical properties which govern the velocity of sound.

In general, a difference in density between two substances is sufficient to indicate which one will be the faster transmission medium for sound. For example, sound travels faster through water than it does through air at the same temperature. However, there are some surprising exceptions to this rule of thumb. An outstanding example among these exceptions involves comparison of the speed of sound in lead and aluminum at the same temperature. Sound travels at 16,700 fps in aluminum at 20° C, and only 4,030 fps in lead at 20° C, despite the fact that lead is much more dense than aluminum. The reason for such exceptions is found in the fact, mentioned above, that sound velocity depends on elasticity as well as density. Using density as a rough indication of the speed of sound in a given substance, it can be stated as a general rule that sound travels fastest in solid materials, slower in liquids, and slowest in gases. For a fixed temperature, the velocity of sound is constant for any medium and is independent of the period, frequency, or amplitude of the disturbance.

Thus, the velocity of sound in air at 0° C (32° F) is 1,087 fps and increases by 2 fps for each centigrade degree of temperature rise (1.1 fps for each degree Fahrenheit). For practical purposes, the speed of sound in air may be considered 1,100 fps. Mach Number In the study of aircraft that fly at supersonic speeds, it is customary to discuss aircraft speed in relation to the velocity of sound (approximately 750 miles per hour). The term "Mach number" has been given to the ratio of the speed of an aircraft to the speed of sound, in honor of Ernst Mach, an Austrian scientist. Thus, if the speed of sound at sea level is 750 miles per hour, an aircraft flying at a Mach number of 2.2 would be traveling at a speed of 750 mph x 2.2 = 1,650 miles per hour. Frequency of Sound The term "pitch" is used to describe the frequency of a sound. The outstanding recognizable difference between the tones produced by two different keys on a piano is a difference in pitch.

The pitch of a tone is proportional to the number of compressions and rarefactions received per second, which in turn, is determined by the vibration frequency of the sounding source. Frequency, or pitch, is usually measured by comparison with a standard. The standard tone may be produced by a tuning fork of known frequency or by a siren whose frequency is computed for a particular speed of rotation. By regulating the speed, the pitch of the siren is made equal to that of the tone being measured. Loudness When a bell rings, the sound waves spread out in all directions and the sound is heard in all directions. When a bell is struck lightly, the vibrations are of small amplitude and the sound is weak. A stronger blow produces vibrations of greater amplitude in the bell, and the sound is louder. It is evident that the amplitude of the air vibrations is greater when the amplitude of the vibrations of the source is increased. Hence, the loudness of the sound depends on the amplitude of the vibrations of the sound waves. As the distance from the source increases, the energy in each wave spreads out, and the sound becomes weaker. The intensity of sound is the energy per unit area per second.

In a sound wave of simple harmonic motion, the energy is half kinetic and half potential; half is due to the speed of the particles, and half is due to the compression and rarefaction of the medium. These two energies are 90 degrees out of phase at any instant. That is, when the speed of particle motion is at a maximum, the pressure is normal, and when the pressure is at a maximum or a minimum, the speed of the particles is zero. The loudness of sound depends upon both intensity and frequency. The intensity of a sound wave in a given medium is proportional to the following quantities: (1) Square of the frequency of vibration. (2) Square of the amplitude. (3) Density of the medium. (4) Velocity of propagation. At any distance from a source of sound (point), the intensity of the wave varies inversely as the square of the distance from the source. As the sound wave advances, variations in pressure occur at all points in the transmitting medium. The greater the pressure variations, the more intense the sound wave will be. It can be shown that the intensity is proportional to the square of the pressure variation regardless of the frequency.

Thus, by measuring pressure changes, the intensities of sounds having different frequencies can be compared directly. Measurement of Sound Intensity The loudness (intensity) of sound is not measured by the same type of scale used to measure length. The human ear has a nonlinear response pattern, and units of sound measurement are used that vary logarithmically with the amplitude of the sound variations. These units are the "bel" and "decibel," which refer to the difference between sounds of unequal intensity or sound levels. The decibel, which is one-tenth of a bel, is the minimum change of sound level perceptible to the human ear. Hence, the decibel merely describes the ratio of two sound levels. For example, 5 decibels may represent almost any volume of sound, depending on the intensity of the reference level or the sound level on which the ratio is based.

Question: For a constant sound wave, what is the time interval between successive crests per second called?
Answer: Frequency