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Unit 8: Mechanical Waves and SoundFor this unit you must:Use a visual representation to construct an explanation of the distinction between transverse and longitudinal waves by focusing on the vibration that generates the waveDescribe representations of transverse and longitudinal wavesDescribe sound in terms of transfer of energy and momentum in a medium and relate the concepts to everyday examplesUse graphical representation of a periodic mechanical wave to determine the amplitude of the waveExplain and/or predict qualitatively how the energy carried by a sound wave relates to the amplitude of the wave, and/or apply this concept to a real-world exampleUse a graphical representation of a periodic mechanical wave (position versus time) to determine the period and frequency of the wave and describe how a change in the frequency would modify features of the representationUse a visual representation of a periodic mechanical wave to determine wavelength of the waveDesign an experiment to determine the relationship between periodic wave speed, wavelength, and frequency and relate these concepts to everyday examplesCreate or use a wave front diagram to demonstrate or interpret qualitatively the observed frequency of a wave, dependent upon relative motions of source and observerUse representations of individual pulses and construct representations to model the interaction of two wave pulses to analyze the superposition of two pulsesDesign a suitable experiment and analyze data illustrating the superposition of mechanical waves (only for wave pulses or standing waves)Design a plan for collecting data to quantify the amplitude variations when two or more traveling waves or wave pulses interact in a given mediumAnalyze data or observations or evaluate evidence of the interaction of two or more traveling waves in one or two dimensions (i.e., circular wave fronts) to evaluate the variations in resultant amplitudesRefine a scientific question related to standing waves and design a detailed plan for the experiment that can be conducted to examine the phenomenon qualitatively or quantitativelyPredict properties of standing waves that result from the addition of incident and reflected waves that are confined to a region and have nodes and antinodesPlan data collection strategies, predict the outcome based on the relationship under test, perform data analysis, evaluate evidence compared to the prediction, explain any discrepancy and, if necessary, revise the relationship among variables responsible for establishing standing waves on a string or in a column of airDescribe representations and models of situations in which standing waves result from the addition of incident and reflected waves confined to a regionChallenge with evidence the claim that the wavelengths of standing waves are determined by the frequency of the source regardless of the size of the regionCalculate wavelengths and frequencies (if given wave speed) of standing waves based on boundary conditions and length of region within which the wave is confined, and calculate numerical values of wavelengths and frequencies. Examples include musical instrumentsUse a visual representation to explain how waves of slightly different frequency give rise to phenomenon of beatsChapter 14: Waves and SoundSection 14-1 Types of WavesWaves can propagate via different oscillation modes such as transverse and longitudinal.Mechanical waves can be either transverse or longitudinal. Examples include waves on a stretched string and sound wavesThis includes, as part of the mechanism of “propagation,” the idea that the speed of a wave depends only on properties of the mediumThe propagation of sound waves included in this EK includes the idea that the traveling disturbance consists of pressure variations coupled to displacement variations (studied later)This applies to both periodic waves and to wave pulsesNote: AP Physics 1 only deals with mechanical waves. The superposition of no more than two wave pulses and properties of standing waves is included.For propagation, mechanical waves require a medium, while electromagnetic waves do not require a physical medium. Examples include light traveling through a vacuum and sound not traveling through a vacuum.Distinguish between Transverse and Longitudinal Waves.Diagram of a waveThe amplitude is the maximum displacement of a wave from its equilibrium valueFor a periodic wave, the period is the repeat time of the wave: The frequency is the number of repetitions of the wave per unit time:For a periodic wave, the wavelength is the repeat distance of the wave.For a periodic wave, wavelength is the ratio of speed over frequency:Section 14-2 Waves on a StringWave speed is a property of the medium:Note: changing the frequency or wavelength does not change the speed of the wave.Example 1: All spiders are very sensitive to vibrations. An orb spider will sit at the center of its large, circular web and monitor radial threads for vibrations created when an insect lands. Assume that these threads are made of silk with a linear density of 1.0 105 kg/m under a tension of 0.40 N, both typical numbers. If an insect lands in the web 30 cm from the spider, how long will it take for the spider to find out?Example 2: A 12 m rope is pulled tight with a tension of 49 N. When one end is given a quick “flick,” a wave is generated that takes 0.54 s to travel to the other end of the rope. What is the mass of the rope?Example 3: A particular species of spider spins a web with silk threads of density 1300 kg/m3 and diameter 3.0 μm. A typical tension in the radial threads of such a web is 7.0 mN. If a fly lands in this web, which will reach the spider first, the sound or the wave on the web silk? Assume the speed of sound in air is 343 m/s.Example 4: A rope of length L and mass M hangs from a ceiling. If the bottom of the rope is given a gentle wiggle, a wave will travel to the top of the rope. As the wave travels upward, does its speed increase, decrease, or stay the same? Justify your response.Section 14-3 Harmonic Wave FunctionsFrom SHM, displacement of a wave as a function of time is given byWe can also write the displacement of a wave as a function of distance travelled at time t:DiagramExample 5: Suppose a boat is at rest in the open ocean. The wind has created a steady wave with wavelength 190 m traveling at 14 m/s. The top of the crests of the waves is 2.0 m above the bottom of the troughs. What is the maximum vertical speed of the boat as it bobs up and down on the passing wave? What is the maximum vertical acceleration? Section 14-4 Sound WavesThe amplitude is the maximum displacement from equilibrium of the wave. A sound wave may be represented by either the pressure or the displacement of atoms or molecules. This covers both periodic waves and wave pulses.The pressure amplitude of a sound wave is the maximum difference between local pressure and atmospheric pressureClassically, the energy carried by a wave depends upon and increases with amplitude. Examples include sound waves.Higher amplitude refers to both greater pressure variations and greater displacement variationsExamples include both periodic waves and wave pulsesIn a periodic sound wave, pressure variations and displacement variations are both present and with the same frequencyThe speed of sound in air is the same for all frequencies and follows the relationship:Sound produced with different frequencies have different wavelengths, but we hear each frequency at the same time.Example 6: You drop a stone into a well that is 7.35 m deep. How long does it take before you hear the splash if the temperature outside is 20.0°C (that is, the speed of sound is 343 m/s)?Section 14-5 Sound IntensityA travelling wave transfers energy from one point to another. How energy is related to sound is evident in the definition of sound intensity. The power of a wave is the rate at which the wave transfers energy.Definition of IntensityIntensity from a Point SourceThe loudness of a sound is determined by its intensity, that is, by the amount of energy that passes through a given area in a given time.Example 7: Sounds from the street below come through an open window measuring 0.75 m by 0.88 m. If the power of the sounds is 1.2 x 10-6 W, what is its intensity?Example 8: Two people relaxing on a deck listen to a songbird sing. One person, only 1.00 m from the bird, hears a sound with an intensity of 2.80 x 10-6 W/m2. What intensity is heard by the second person, who is 4.25 m from the bird? Assume that no reflected sound is heard by either person.What is the power output of the bird’s song?Section 14-6 The Doppler EffectThe observed frequency of a wave depends on the relative motion of source and observer. The Doppler Effect is the change in the apparent frequency of a source due to the relative motion between the source and the observer. For sound waves, the sound wave’s speed and frequency both appear to be higher when an observer moves towards a source for instance. This is a qualitative treatment only.General equation for the Doppler EffectwhereSourceObserverObserved FrequencyDiagram of a Moving Sound WaveSection 14-7 Superposition and InterferenceTwo or more wave pulses can interact in such a way as to produce amplitude variations in the resultant wave. When two pulses cross, they travel through each other; they do not bounce off each other. Where the pulses overlap, the resulting displacement can be determined by adding the displacements of the two pulses. This is called superposition.Two or more traveling waves can interact in such a was as to produce amplitude variations in the resultant wave.SuperpositionConstructive InterferenceDestructive InterferenceConstructive Interference of Wave PulsesDestructive Interference of Wave PulsesInterference of Circular WavesSection 14-8 Standing WavesStanding waves are the result of the addition of incident and reflected waves that are confined to a region and have nodes and antinodes. Examples include waves on a fixed length of string and sound waves in both closed and open tubes.Reflection of waves and wave pulses, even if a standing wave is not created, is covered in AP Physics 1For standing sound waves, pressure nodes correspond to displacement antinodes, and vice versa. For example, the open end of a tube is a pressure node because the pressure equalizes with the surrounding air pressure and therefore does not oscillate. The closed end of a tube is a displacement node because the air adjacent to the closed end is blocked from oscillating.The possible wavelengths of a standing wave are determined by the size of the region to which it is confined.A standing wave with zero amplitude at both ends can only have certain wavelengths. Examples include fundamental frequencies and harmonicsOther boundary conditions or other region sizes will result in different sets of possible wavelengthsThe term first harmonic refers to the standing waves corresponding to the fundamental frequency, i.e., the lowest frequency corresponding to a standing wave. The second harmonic is the standing wave corresponding to the second lowest frequency that generates a standing wave in the given scenarioResonance is another term for standing sound wave.Reflections (Insert with Section 14-2)Recall that the speed of a wave on a string is given by:So strings with a smaller linear density will allow waves to travel fasterWe have already reviewed reflections at fixed and free boundaries, that isAt a fixed boundary, wave pulses reflect on the opposite endAt a free boundary, wave pulses reflect on the same sideBut, what happens when the properties of the medium change?Standing Waves on a String1.2.3.In general, the frequency and wavelength for standing waves on a string are given bywhereExample 9: A 2.50-m-long string vibrates as a 100 Hz standing wave with nodes at 1.00 m and 1.50 m from one end of the string and at no points in between these two. Which harmonic is this? What is the string’s fundamental frequency? And what is the speed of the traveling waves on the string?Example 10: A particular species of spider spins a web with silk threads of density 1300 kg/m3 and diameter 3.0 μm. A passing insect brushes a 12-cm-long strand of the web, which has a tension of 1.0 mN, and excites the lowest frequency standing wave. With what frequency will the strand vibrate?Example 11: A string of length 10 m and mass 300 g is fixed at both ends, and the tension in the string is 40 N. What is the frequency of the standing wave for which the distance between a node and the closest antinode is 1 m?Example 12: The fifth string on a guitar plays the musical note A, at a frequency of 110 Hz. On a typical guitar, this string is stretched between two fixed points 0.640 m apart, and this length of string has a mass of 2.86 g. What is the tension in the string?Example 13: A standing wave pattern is created on a string with linear density 3.1 x 10-4 kg/m. A wave generator with frequency 69 Hz is attached to one end of the string and the other end goes over a pulley connected to a mass. The distance between the generator and the pulley is 0.66 m. Initially the 3rd harmonic is formed. Find the tension in the string and the value of the hanging mass.If the frequency is fixed at 69 Hz, what is the maximum mass that can be used to still create a coherent standing wave pattern?The hanging mass is adjusted to created the 2nd harmonic and the frequency is fixed at 69 Hz. What is the mass hanging on the string?Standing Waves in a Column of Air Closed at One End1.2.3.In general, the frequency and wavelength for standing waves in a column closed at one end are given bywhereStanding Waves in a Column of Air Open at Both Ends1.2.3.In general, the frequency and wavelength for standing waves in a column open at both ends are given bywhereExample 14: Wind instruments have an adjustable joint to change the tube length. Players may need to adjust this joint to stay in tune (to stay at the correct frequency). Suppose a “cold” flute plays A at 440 Hz when the temperature is 20°C.How long is the tube? At 20°C, the speed of sound in air is 343 m/s.As the player blows air through the flute, the air inside the instrument warms up. Once the air temperature inside the flute has risen to 32°C, increasing the speed of sound to 350 m/s, what is the frequency?At the higher temperature, how must the length of the tube be changed to bring the frequency back to 440 Hz?Example 15: An empty soda pop bottle is to be used as a musical instrument in a band. In order to be tuned properly, the fundamental frequency of the bottle must be 440.0 Hz.If the bottle is 26.0 cm tall and the speed of sound is 343 m/s, how high should it be filled with water to produce the desired frequency?What is the frequency of the next higher harmonic for this bottle?Section 14-9 BeatsBeats arise from the addition of waves of slightly different frequencyBecause of the different frequencies, the two waves are sometimes in phase and sometimes out of phase. The resulting regularly spaced amplitude changes are called beats. Examples include the tuning of an instrumentThe beat frequency is the difference in frequency between the two wavesOnly qualitatively understanding is necessaryExample 16: Two musicians are comparing their clarinets. The first clarinet produces a tone that is known to be 441 Hz. When the two clarinets play together they produce eights beats every 2.00 s. If the second clarinet produces a higher pitched tone than the first, what is the second clarinet’s frequency?Example 17: A guitarist tunes one of her strings to be 440 Hz tuning fork. Initially, a beat frequency of 5 Hz is heard when the string and tuning fork are sounded. When the guitarist tightens the string slowly and steadily, the beat frequency decreases to 4 Hz. What was the initial frequency of the string?Example 18: A tuning fork with a frequency of 320.0 Hz and a tuning fork of unknown frequency produce beats with a frequency of 4.5 Hz. If the frequency of the 320.0 Hz tuning fork is lowered slightly by placing a bit of putty on one of its tines, the new beat frequency is 7.5 Hz. Which tuning fork has the lower frequency? Explain.What is the final frequency of the 320.0 Hz tuning fork?What is the frequency of the other tuning fork? ................
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