Ch16 Waves and sound.docx

WAVES AND SOUND A wave consists of a displacement from the equilibrium position that travels from one place to another. As the wave travels, the displacement is the quantity that actually moves from one place to another. The medium that the wave travels through does not experience any permanent displacement. This displacement manifests itself as a temporary change in energy as the disturbance passes through a location. This energy can be converted to another form at some distance from the origin of the disturbance so we say that waves transfer energy. Since there is no transfer of the medium from one place to another, this is a different mechanism from energy transfer by particle. In the case of energy transfer by particle, energy associated with a particle is transferred as the matter composing the particle moves from place to place. The two types of waves we will study are the transverse wave and the longitudinal wave. In transverse waves, the displacement from equilibrium occurs at right angles to the direction of travel of the wave. In longitudinal waves, the displacement from equilibrium occurs in a direction parallel to the direction of travel of the wave. Some waves have some components of both types of motion and particles of the medium can move in circular paths. An example is surface waves in water. The waves we are discussing are called periodic waves since the motion is repeated over and over due to the disturbance being repeated over and over by the source. These waves are represented by the sine or cosine function and can be represented as displacements in terms of distance from the source or distance of a point from the equilibrium position as time changes. The frequency and the period are related by the equation: f = 1/T The period is in seconds and the frequency is in hertz(Hz) which is one cycle per second. The speed of a wave may be expressed as one wavelength divided by one period. v = ?/T v = f? Since f = 1/T, the equation is generally written with f instead of T. Example The speed of a transverse wave on a string is 450 m/s and the wavelength is 0.18 m. If the amplitude is 2.0 mm, find the time required for a particle on the string to move through a distance of 1.0 km. The speed of a wave through a medium depends on certain characteristics of the medium and the type of medium. Whether the medium is a solid, liquid, gas or vacuum has an effect on the speed of a wave. In fact, mechanical waves cannot travel through a vacuum at all since there must be matter present to transmit the wave. A wave travels through matter as one particle exerts a force on the particle next to it to make it move. This second particle exerts a force on the particle next in line and the motion continues. The two factors that affect the transmission rate of this motion are the strength of the binding forces between adjacent particles and the mass of these particles. In the case of a string, increasing the tension on the string increases the restoring force acting on individual parts of the string causing a faster acceleration and return to equilibrium and resulting in increased velocity of the wave along the string. An increase in the mass per unit length will cause a decrease in the speed of the wave since the same restoring force must now accelerate a larger mass. The formula relating these quantities is: _______ v = ?F/(m/L) Example An astronaut hangs a 0.055 kg ball at the end of a 0.95 m wire with a linear density of 1.2 x 10-4 kg/m. The time required for a transverse pulse to travel the length of the wire is 0.016 s. Find the acceleration due to gravity. So far we have discussed the speed of the wave as it moves through a medium. The individual parts of the medium have a speed associated with their motion, too. This motion, however, is controlled by the simple harmonic oscillator equation for speed of a particle and is determined by characteristics of the motion of the source of the wave. vparticle = A?sin?t where amplitude, angular velocity, and position according to time determine the speed of the individual particle. Mathematical Description of a Wave A wave causes particle motion in many different places at the same time. The leading edge of the wave initiates the motion and successive parts of the wave continue that pattern of motion. The shape of the wave can be described mathematically as a sine wave with particle displacement depending on both time and location along the X axis. The equation for a wave is: y = Asin(?t + 2?x/?) where y represents displacement of the particle from the equilibrium position, A is the maximum displacement(amplitude), ?t represents the angular displacement in terms of time, and x/? represents the fraction of a wavelength between the particle and the origin. The equation is generally rewritten as: y = Asin(2?ft + 2?x/?) If we factor out 2?f, we get: y = Asin(2?f(t + x/f?)) y = Asin(2?f(t + x/v)) The time required for the wave to travel a distance x is x/v, so this term is the amount of time that simple harmonic motion at position x lags behind simple harmonic motion at the origin. The figure above illustrates how a wave moving in the positive x direction affects the motion of different locations in the medium through which it is traveling. Each of these graphs represents the displacement of the medium from equilibrium as time goes by. When you calculate the value of y from these equations, your calculator must be set in the radian mode. Remember the 2? in the equations comes from the fact that 2? radians is one complete cycle. Example A wave causes a displacement of the medium given by the equation y = 0.45sin(8.0?t + ?x). where t and x are expressed in seconds and meters. Find (a) the amplitude, (b) frequency, (c) wavelength, and (d) speed of the wave. Is the wave traveling to the right or to the left? Use y = Asin(2?ft - 2?x/?) for a wave traveling to the right. A sound wave is a longitudinal wave consisting of condensations and rarefactions. In terms of air pressure, a condensation is a region of high pressure and a rarefaction is a region of low pressure. The repeated condensations and rarefactions follow each other through the air to form the sound wave. The frequency of a sound wave is defined as the number of cycles per second that passes by a given location. The normal range of frequencies that humans hear is 20 to 20,000 Hz. Animals routinely use frequencies below 20 Hz and above 20,000 Hz for communication and detection of objects. Frequencies below 20 Hz are called infrasonic. Frequencies above 20,000 Hz are called ultrasonic. The brain's interpretation of frequency is called pitch. High frequency is interpreted as high pitch. A sound wave can be represented on a graph as a variation in air pressure. The amplitude of the wave depends on the amount of variation in the pressure and is graphed on the Y axis. Remember the equilibrium position for air pressure is not zero Loudness of a sound depends on the size of the variation in air pressure(conversation = .03 Pa) and the sensitivity of the listener. While changes in pressure can be measured, loudness depends on perception. Sound travels at different speeds in different substances. Overall, sound travels fastest in solids and slowest in gasses. In dry air at 0°C the speed of sound is 331m/s. The speed increases about .6 m/s for each Celsius degree increase in temperature. The sound from thunder travels about 1 mile every 5 seconds. To determine the distance to a lightning bolt, count the seconds between the flash of light and the sound of the thunder and divide by 5. Example Two microphones are located 1.5 m apart on the y axis. A source of sound is located on the positive x axis. If the speed of sound is 343 m/s and it takes 1.46 ms longer for the sound to travel to the most distant microphone, find the distance to each microphone. Sound intensity is the sound power per unit of area in watts per square meter. The equation is: I = P/A When a sound leaves its source it spreads out in a pattern shaped like the surface of a sphere. The total amount of power close to the source is the same as the total amount farther away. It is, however spread out over a larger area. The surface area of a sphere is calculated with the formula A = 4?r2. The intensity equation becomes: I = P/4?r2 If we consider the ratio of two intensities at different radii, we get: I2/I1 = (P/4?r22 )/(P/4?r12 ) Since P/4? divides out, I2/I1 = (1/ r22 )/(1/ r12 ) I2/I1 = r12/ r22 This is called the inverse square law and shows us that the intensity decreases as the square of the distance. If the distance doubles, the intensity is reduced to one fourth, and if the distance triples, the intensity reduces to one ninth. Example A loudspeaker has a circular opening with a radius of 0.0950 m. The power operating the speaker is 25 watts. If the average sound intensity at the opening is 17.5 w/m2, find the percent of the input energy that is converted into sound. Find the intensity at (a) 1.0 m and (b) 3.0 m from the speaker. Sound intensity is often expressed in decibels(dB) named for Alexander Graham Bell. This unit is a relative intensity unit which compares an intensity to that of the threshold of hearing(quietest sound that can be heard by humans). The average threshold of hearing intensity is 1 x 10-12w/m2 and the maximum intensity(live rock concert) before the threshold of pain is about 1 w/m2. This gives us a relative range of sound intensities of 1012 which is inconvenient to graph on the standard XY plane. In order to make the numbers more manageable, a logarithmic scale is used. Each increase of 1 Bel corresponds to multiplying the initial intensity by a factor of 10 which results in a larger than desirable increment.An increase of one decibel corresponds to multiplying the original intensity by a factor of 1.259 which is acceptable. The equation for sound intensity using decibels is: ? = 10(dB)log(I/Io) The decibel is a dimensionless unit since it is the log of a ratio for which the units cancel. Remember that a relative intensity of zero decibels means that the observed intensity is equal to the threshold intensity since the log of 1 = 0. Doubling the absolute intensity produces a 3 dB increase in the relative intensity. This does not sound twice as loud. A 10 dB increase will produce a sound that is percieved as twice as loud. If we use the equation for relative intensities to find the change in absolute intensity that corresponds to a 10 dB difference, we find that we must multiply the original intensity by a factor of 10 in order to double the percieved loudness. 200 watts sounds twice as loud as 20. Example A speaker in a room produces a sound level of 75.0 dB at a point in a room. Another speaker aimed at the same point produces a sound level of 72 dB at that same point. When both speakers produce sound at the same time, what is the sound level in dB at that point? The Doppler effect is a change in wavelength of a sound caused by relative motion between the source and the observer. The wavelengths are shortened in the direction of motion and lengthened in the direction opposite to the motion. The pitch will be higher in front of the fire truck and lower behind it. It is also possible for the observer to move and the source remain stationary. In this case the actual wavelength does not change but the moving observer detects the wavelengths per second of a stationary observer plus an additional number depending on his rate of motion. In this case, the observer detects an increased frequency. If the observer is moving away from the source, the frequency will be decreased. It is important to understand that the wavelength changes in the case of the moving source but the frequency changes in the case of the moving observer. It is also possible for both source and observer to be in motion relative to the air. This gives us the general equation for the Doppler effect which covers all of these possibilities. fo = fs(1±vo/v) (1±vs/v) fo is the observed frequency, fs is the frequency emitted by the source, vo is the speed of the observer, vs is the speed of the source, and v is the speed of sound at the temperature of the medium. In the numerator, the + sign is used when the observer moves towards the source and the negative sign when he moves away. In the denominator, the minus sign is used when the source moves towards the observer and the plus sign when it moves away. Example A car alarm sounds at a frequency of 960 hz when the speed of sound is 343 m/s. As you drive towards it, pass it and drive away, you notice a 95 hz drop in the frequency you hear. How fast are you going? Applications of sound in medicine Ultrasound is used to monitor fetal development, produce images of internal organs for evaluation, and detect malignancies in soft tissue. Echocardiography is used to evaluate heart function. Doppler flow meters can be used to detect areas of resricted blood flow.