Chapter 02 Sound and Waves PDF

by  CHAPTER 2 SOUND AND WAVES Properties Of Sound Mechanical Waves Standing Waves Resonance Sound & Pressure Decibels Speed Of Sound & Temperature Human Ear Constructive & Destructive Interference Sounds Produced By...

by  CHAPTER 2 SOUND AND WAVES Properties Of Sound Mechanical Waves Standing Waves Resonance Sound & Pressure Decibels Speed Of Sound & Temperature Human Ear Constructive & Destructive Interference Sounds Produced By Living Creatures Doppler Effect Infrasound & Ultrasound Shock Waves Properties of Sound Sound is a mechanical wave produced by vibrating bodies. Like all waves, sound exhibits the phenomena of reflection, refraction, interference, and diffraction. A vibrating sound source results in compressions and rarefactions in the adjacent molecules of the medium. The surrounding molecules of the medium vibrate and in turn transfer their motion to the adjacent molecules. A material medium is needed between the source and the receiver to propagate sound. Properties of Sound The speed of the sound wave depends on the medium and temperature. In air at 20◦C, the speed of sound is about 3.3×104 cm/sec, and in water, it is about 1.4×105 cm/sec. Human ear can hear sounds in the range of 20-20kHz, lower than 20 Hz, the sound waves are called infrasound, while higher than 20kHz, the waves are called ultrasound. Most natural sounds are a mixture of different frequencies. Some definitions Periodic motion: It is the motion that is regular and repeating. Wavelength λ: is the distance from a point on a wave to the same corresponding point on the next wave (distance between two consecutive crests). Frequency f: is the number of waves that pass a point in one second (measured in Hz). Higher the frequency, higher the pitch of the sound. Period Τ (periodic time): is the time needed to complete one vibration and is the reciprocal of frequency. Some definitions Amplitude A: is the maximum displacement and is related to the energy carried by the wave. Higher the amplitude, louder the sound or the sound intensity. Wave velocity v depends on the type of wave and the medium. In general, the speed of sound wave is given by 𝒗 = 𝝀𝒇 Useful Notes The two important characteristics of any sound are: Intensity: determined by the magnitude of compression and rarefaction. Frequency: determined by how often the compressions and rarefactions take place. Pitch is the perception of frequency. loudness is the perception of amplitude. The speed of a sound wave decreases with decreasing the density of the material (i.e., in vacuum there is no sound). So sound speed is generally highest in solids, medium in liquids and slowest in gases. Mechanical Waves There are two basic types of wave motion for mechanical waves: Transverse wave Elements of the disturbed medium move perpendicular to the direction of propagation Longitudinal wave Elements of the disturbed medium move parallel to the direction of propagation Mechanical Waves Consider a pulse traveling to the right on a long string as shown in Figure. The shape and position of the pulse can be represented by some mathematical function that we will write as 𝒚(𝒙, 𝒕). Mechanical Waves This function describes the transverse position y of the element of the string located at each value of x at time t. 𝒚 = 𝑨 𝒔𝒊𝒏 𝒌𝒙 − 𝝎𝒕 where k is angular wave number and ω is the angular frequency 2𝜋 2𝜋 𝑘= and 𝜔 = 2𝜋𝑓 = 𝜆 Τ where λ is the wavelength of the wave, T is the period (periodic time) and f is the frequency. The speed of the wave (the magnitude of velocity) 𝝎 𝝊 = 𝝀𝒇 = 𝒌 Quiz Example A sinusoidal wave traveling in the positive x-direction has an amplitude of 15 cm, a wavelength of 40 cm, and a frequency of 8 Hz. Find the wavenumber k, period T, angular frequency ω, and speed υ of the wave Answer the wavenumber the period of the wave the angular frequency the wave speed Sound & Pressure A vibrating string produces a sound wave as the string oscillates back and forth, it transfers energy to the air, mostly as thermal energy. A small part of the string’s energy goes into compressing and expanding the surrounding air, creating slightly higher and lower local pressures. These compressions (high pressure regions) and rarefactions (low pressure regions) move out as longitudinal pressure waves having the same frequency as the string—they are the disturbance that is a sound wave. Sound & Pressure The pressure varies due to the sound vibrations, so for a sinusoidal sound wave, the pressure after a time t is 𝑷 = 𝑷𝒂 + ∆𝑷𝒎𝒂𝒙 𝒔𝒊𝒏 𝒌𝒙 − 𝝎𝒕 where Pa is the ambient air pressure and ∆𝑃𝑚𝑎𝑥 is the maximum pressure change due to the sound wave. A Standing Wave A standing wave is a wave that results of two similar waves, but traveling in opposite directions, interfering with each other. This interference causes the wave to appear to remain in one place, with points along the wave called nodes remaining at rest while other points, called antinodes, oscillate with maximum amplitude. Nodes: These are points along the wave where the amplitude is always zero. Antinodes: These are points along the wave where the amplitude is maximum. Wavelength: In a standing wave, it’s twice the distance between two consecutive nodes or antinodes. Resonance Natural frequency: Every object has a natural frequency at which it vibrates most easily. Resonance in sound occurs when a sound wave's frequency matches the natural vibration frequency of an object, causing that object to vibrate with increased amplitude. This results in a louder and more intense sound. Resonance occurs at specific frequencies when the reflected waves interfere constructively, resulting in a high amplitude standing wave. Resonance in Air Columns Resonance in air columns occurs when a sound wave with a specific frequency causes the air molecules within the column to vibrate at their natural frequency. The amplitude of the vibrations is significantly amplified at the resonant frequencies, leading to a noticeable increase in sound intensity. Flutes, clarinets, and oboes rely on air columns to produce their characteristic sounds. The player changes the length of the air column by opening and closing different holes, altering the resonant frequencies and producing different pitches. Resonance in Air Columns Fundamental Frequency (First Harmonic) For an open tube (a tube open at both 𝐿 = 𝜆/2 ends), resonance occurs when the 𝜆 = 2𝐿 wavelength of the wave (λ) satisfies the condition: 2nd Harmonic 𝑳 = 𝒏𝝀/𝟐, 𝒏 = 𝟏, 𝟐, 𝟑, 𝟒, … 𝐿 = 𝜆 These wavelengths allow a standing 𝜆 = 𝐿 wave pattern such that a pressure antinode occurs naturally at the open 3rd Harmonic ends of the tube. 𝐿 = 3𝜆/2 𝜆 = 2𝐿/3 where L = tube length. Resonance in Air Columns Fundamental Frequency (First Harmonic) For a closed tube (by convention, a 𝐿 = 𝜆/4 closed tube is open at one end and 𝜆 = 4𝐿 closed at the other), resonance occurs when the wavelength of the wave (λ) 3rd Harmonic satisfies the condition: 𝐿 = 3𝜆/4 𝑳 = 𝒏𝝀/𝟒, 𝒏 = 𝟏, 𝟑, 𝟓, 𝟕, 𝟗, … 𝜆 = 4𝐿/3 These wavelengths allow a standing wave pattern such that a pressure node 5th Harmonic occurs naturally at the closed end of the tube and a pressure antinode 𝐿 = 5𝜆/4 occurs naturally at the open end of the 𝜆 = 4𝐿/5 tube. where L = tube length. Quizzes In an open tube, the fundamental frequency occurs when the length of the tube is equal to: (a) Half the wavelength of the sound wave (b) One-fourth the wavelength of the sound wave (c) One-eighth the wavelength of the sound wave (d) Twice the wavelength of the sound wave In a closed tube, the fundamental frequency occurs when the length of the tube is equal to: (a) Half the wavelength of the sound wave (b) One-fourth the wavelength of the sound wave (c) One-eighth the wavelength of the sound wave (d) Twice the wavelength of the sound wave Which of the following harmonics are present in an open tube? (a) Only odd harmonics (b) Only even harmonics (c) Both odd and even harmonics (d) None of the above Quizzes Which of the following harmonics are present in a closed tube? (a) Only odd harmonics (b) Only even harmonics (c) Both odd and even harmonics (d) None of the above A tuning fork of frequency 512 Hz produces resonance in a closed tube. The length of the air column for the fundamental mode is: (a) 16.7 cm (b) 33.4 cm (c) 66.8 cm (d) 133.6 cm A tuning fork of frequency 256 Hz produces resonance in an open tube. The length of the air column for the first overtone is: (a) 33.4 cm (b) 66.8 cm (c) 100.2 cm (d) 133.6 cm Quizzes If the length of a closed tube is doubled, the fundamental frequency: (a) Doubles (b) Halves (c) Remains the same (d) Quadruples If the length of an open tube is doubled, the fundamental frequency: (a) Doubles (b) Halves (c) Remains the same (d) Quadruples Decibels Because of the wide range of intensities that the human ear can detect, it is convenient to use a logarithmic scale, where the sound level β is defined by 𝑰 𝜷 = 𝟏𝟎 𝒍𝒐𝒈 𝑰𝟎 where the constant I0 is the reference intensity, taken to be at the threshold of hearing (10-12 W/m2), and I is the intensity (in W/m2) in which the sound level β corresponds, where β is measured in decibels (dB). Decibels The threshold of hearing (TOH), which is the minimum sound intensity of a pure tone a human can hear, is as small as 𝐼0 (𝑇𝑂𝐻) = 10−12 𝑊/𝑚2 The following table shows some typical intensity values given in W/m2 as well as in decibels for some sound sources and dynamics indicators. Quiz Example What in the sound intensity of a truck (80 dB) in W/m2? Answer 𝑰 𝜷 = 𝟏𝟎 𝐥𝐨𝐠 𝑰𝟎 Then 𝐼 80 = 10 log 10−12 𝐼 8 = log 10−12 log 𝐼 𝐼 108 = 10 10−12= −12 10 𝐼 = 108 × 10−12 = 10−4 𝑊/𝑚2 Speed of Sound & Temperature The speed of sound is affected by the temperature of the medium. For air at sea level, the speed of sound is given by 𝑻 𝝊 = 𝟑𝟑𝟏 𝟐𝟕𝟑 where the temperature T is in kelvin. The speed of sound in air and other gases depend on the square room of temperature. Quiz Example Calculate the wavelengths of sounds at the extremes of the audible range, 20 and 20000 Hz, in 30 °C air. Answer The room temperature in Kelvin is 𝑇 = 30 + 273 = 303 Then 𝑇 303 𝜐 = 331 = 331 = 348.7𝑚/𝑠 273 273 but, 𝑣 𝜆= 𝑓 348.7 348.7 𝜆𝑚𝑎𝑥 = = 17 𝑚 and 𝜆𝑚𝑖𝑛 = = 0.017 𝑚 20 20000 Human Ear Part Function Remarks Special structure to detect Pinna sound direction Resonance around 3000 Ear Canal Outer Ear Hz Pressure transfer and Eardrum amplification of vibration Gets the pressure Hammer information from the eardrum Anvil Respond to hummer Middle Ear (Ossicles) Smaller area than Stirrup eardrum, pressure amplification Contains a lot of fibers that resonate and stimulate sensitive cells to Cochlea Inner Ear transfer the sound pulse to the auditory nerve  The hearing mechanism The outer ear (ear canal) carries sound to the eardrum. The lever system of the middle ear takes the force exerted on the eardrum by sound pressure variations and amplifies it. The middle ear transmits this force to the inner ear via the oval window, creating pressure waves in the cochlea approximately 40 times greater than those impinging on the eardrum. Two muscles in the middle ear protect the inner ear from very intense sounds. They react to intense sound in a few milliseconds and reduce the force transmitted to the cochlea. This protective reaction can also be triggered by your own voice, so that humming while shooting a gun, for example, can reduce noise damage. Cochlear Implants Deafness can occur when the hair-like sensors (cilia) in the cochlea break off over a lifetime or sometimes because of prolonged exposure to loud sounds. The cochlear implant electrically stimulates the nerves in the ear to restore hearing loss that is due to damaged or absent cilia. Sound waves are first captured by a microphone, which is worn behind the ear. The microphone converts the sound waves into electrical signals, which are then processed by a speech processor. The speech processor converts the sound signals into a series of electrical pulses that are sent to the implant. The implant itself consists of two main components: The receiver-stimulator which is implanted under the skin behind the ear and is connected to the electrode array. It receives the electrical signals from the speech processor and converts them into electrical pulses that are delivered to the electrodes in the cochlea. The electrode array is a series of tiny electrodes that are inserted into the cochlea, which is a spiral-shaped cavity in the inner ear. The electrodes are arranged along the length of the cochlea, When the electrical pulses are delivered to the electrode array, they stimulate the auditory nerve fibers.  Constructive & Destructive Interference If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves. If the displacements caused by the two pulses are in the same direction, we refer to their superposition as constructive interference. While, if the displacements caused by the two pulses are in opposite directions, we refer to their superposition as destructive interference. Noise Cancelation Many headphones nowadays are designed to cancel noise with destructive interference. The mic in the headphone listen to the outer sound wave and create a sound wave exactly opposite to the incoming sound. These headphones can be more effective than the simple passive attenuation used in most ear protection. Doppler Effect Simply, it's the change in frequency of a wave due to relative motion between source and observer. The apparent or observed frequency f ' depends on the speed of observer vO, the speed of the source vS and the source frequency f as ′ 𝒗 ± 𝒗𝟎 𝒇 =𝒇 𝒗 ∓ 𝒗𝒔 where v is the speed of the sound in this medium. Doppler Effect Note: The signs for the values substituted for vO and vS depend on the direction of the velocity. A positive value is used for the f' f motion of the observer toward the other (associated with an increase in observed frequency). A negative value is used for motion of f f' one away from the other (associated with a decrease in observed frequency). Signs are inverted for vS. Quiz Suppose you’re on a hot air balloon ride, carrying a buzzer that emits a sound of frequency f. If you accidentally drop the buzzer over the side while the balloon is rising at constant speed, what can you conclude about the sound you hear as the buzzer falls toward the ground? (a) The frequency and intensity increase. (b) The frequency decreases and the intensity increases. (c) The frequency decreases and the intensity decreases. (d) The frequency remains the same, but the intensity decreases. Quiz Example An ambulance travels down a highway at a speed of 75 mi/h, its siren emitting sound at a frequency of 4×102 Hz. What frequency is heard by a passenger in a car traveling at 55 mi/h in the opposite direction as the car and the ambulance (a) approach each other and (b) pass and move away from each other (hint: the speed of sound in air is 345 m/s) Answer The corresponding speed of the ambulance vs and the car vo in the units of m/s are: 𝑚 𝑚 𝑣𝑠 = 75 × 0.447 = 33.5 and 𝑣𝑜 = 55 × 0.447 = 24.6 𝑠 𝑠 (a) For the ambulance and car approach each other 𝑣𝑜 = + and 𝑣𝑠 = − ′ 𝒗 + 𝒗𝟎 𝟐 𝟑𝟒𝟓 + 𝟐𝟒. 𝟔 𝒇 =𝒇 = 𝟒 × 𝟏𝟎 = 𝟒𝟕𝟓 𝑯𝒛 𝒗 − 𝒗𝒔 𝟑𝟒𝟓 − 𝟑𝟑. 𝟓 (b) For the ambulance and car move away each other 𝑣𝑜 = − and 𝑣𝑠 = + 𝒗 − 𝒗𝟎 𝟑𝟒𝟓 − 𝟐𝟒. 𝟔 𝒇′ = 𝒇 = 𝟒 × 𝟏𝟎𝟐 = 𝟑𝟑𝟗 𝑯𝒛 𝒗 + 𝒗𝒔 𝟑𝟒𝟓 + 𝟑𝟑. 𝟓 Note: 1 mi/h = 1 × 0.447 m/s Infrasound Infrasound, describes sound waves with a frequency below the lower limit of human audibility (20 Hz down to 0.1 Hz). Infrasound can result from both natural and man-made sources: Natural events: severe weather, earthquakes, volcanoes, waterfalls, calving of icebergs, aurorae, meteors, lightning. Animal communication: whales, elephants, hippopotamuses, rhinoceroses, giraffes, and alligators use infrasound to communicate over distances. Migrating birds use infrasound, from sources such as turbulent airflow over mountain ranges, as a navigational aid. Man-Made sources: sonic booms and explosions, or by machinery such as diesel engines, wind turbines. People use this frequency range for monitoring earthquakes and volcanoes, charting rock and petroleum formations below the earth, and in ballistocardiography (BCG) to study the mechanics of the heart. Ballistocardiography BCG The BCG is a measure of ballistic forces generated by the heart. BCG differs than the electrocardiography (ECG) which is the record of electrical signals from the heart. BCG is aa non-invasive method technique measured from the surface of the body by producing a graphical representation of repetitive motions of the human body arising from the sudden ejection of blood into the great vessels with each heartbeat. BCG measurement and correlation with a synchronized recorded ECG. (A) BCG sensor is positioned under the patient’s chest. (B,C) Mechanical cardiac activity induces a charge shift in the BCG sensor. (D) A BCG signal is recorded and compared to a simultaneously recorded ECG. Ballistocardiography BCG BCG can be used to detect heart problems, such as coronary artery disease, heart failure, and valvular heart disease. How it works: A person lies on a special table or platform that can detect very small movements. With each heartbeat, the heart ejects blood into the aorta, creating a small recoil force that moves the body. These movements are detected by sensors on the platform and recorded as a ballistocardiogram. Advantages of BCG: Non-invasive: It does not require any needles or injections. Safe: It has no known side effects. Sensitive: It can detect subtle changes in heart function. Ultrasound Ultrasound is sound waves with frequencies higher than the upper audible limit of human hearing (20 kHz up to several gigahertz). Ultrasonic devices are used to detect objects, measure distances and fluid flow. Ultrasound is used to detect invisible flaws in the testing of products and structures. Ultrasound is used for cleaning, mixing, and accelerating chemical processes. How Does an Ultrasonic Sensor Work? Ultrasonic sensors work by emitting sound waves at a frequency too high for humans to hear. They then wait for the sound to be reflected, calculating distance based on the time 𝑻 required. The measured distance can be calculated: 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 = ½ 𝑻 𝒙 𝒗 The speed of sound 𝒗 is 343 meters/second, but this varies depending on temperature, humidity and medium On the other hand, for a moving sensor (or object) the echo frequency differs due to Doppler effect. Ultrasound Imaging Ultrasound imaging, also known as sonography or ultrasonography, is a non-invasive imaging technique that uses high-frequency sound waves to create real-time images of internal organs and other soft tissues. How it works: Sound waves: A transducer, which is a small handheld device, is placed on the skin and emits high-frequency sound waves. Echoes: These sound waves travel through the body's tissues and bounce back as echoes when they encounter different tissues. Image creation: The transducer receives these echoes and converts them into electrical signals. These signals are then processed by a computer to create image. Types of Ultrasound Imaging 2D ultrasound: This is the most common type of ultrasound imaging. It produces a two- dimensional image of the internal structures. 3D ultrasound: This type of ultrasound provides a three-dimensional view of the internal structures. It is often used during pregnancy to visualize the fetus in more detail. 4D ultrasound: This is a type of 3D ultrasound that also shows the movement of the internal structures in real-time. It is often used to observe the fetus moving and interacting within the womb. Common Uses of Ultrasound Imaging Pregnancy: Ultrasound is used to monitor the growth and development of the fetus, check for abnormalities, and determine the due date. Abdominal imaging: Ultrasound can be used to visualize the liver, gallbladder, pancreas, spleen, kidneys, and other organs in the abdomen. Heart imaging: Echocardiography is a type of ultrasound that is used to evaluate the structure and function of the heart. Musculoskeletal imaging: Ultrasound can be used to evaluate muscles, tendons, ligaments, and joints. Vascular imaging: Ultrasound can be used to assess blood flow in the blood vessels, such as the carotid arteries and veins in the legs, named as Doppler imaging. Doppler Ultrasound Imaging Ultrasound can measure the blood flow by recoding the reflected echo frequencies. Ultrasound Imaging Advantages and Limitations Advantages: Non-invasive and painless No radiation exposure Real-time imaging Portable and relatively inexpensive Limitations: Limited depth of penetration Cannot visualize structures through bone or air-filled structures like the lungs Operator-dependent Safety: Ultrasound imaging is generally considered safe for adults and children. It does not involve ionizing radiation. Ultrasound Therapy Ultrasound therapy is a non-invasive treatment that uses high-frequency sound waves to penetrate soft tissues. It works by increasing blood flow to the targeted area, which can help reduce pain, inflammation, and promote healing. How it works: A device emits high-frequency sound waves that are applied to the skin using a transducer. These sound waves penetrate the tissue and cause vibrations, which generate heat. The heat helps increase blood flow to the area, which can reduce pain and inflammation. The vibrations can also help break up scar tissue and promote tissue repair. Ultrasound Therapy Common Uses: Pain relief: Ultrasound therapy can help relieve pain associated with conditions such as arthritis, bursitis, tendonitis, and muscle strains. Soft tissue injuries: It can help accelerate the healing process of soft tissue injuries, such as sprains and strains. Increased range of motion: By reducing pain and inflammation, ultrasound therapy can improve joint mobility and flexibility. Potential Benefits: Non-invasive and generally painless May help reduce the need for medication Sounds Produced By Living Creatures  Sound production is often associated with the respiratory system.  The frequency of the sounds is determined by the tension on the vocal cords and the size of the larynx.  The infant has a small larynx with high tensioned vocal cords, which produces a high-frequency voice.  By getting older, the size of the larynx is getting bigger for men and tender vocal cords, which produces lower sound pitch.  One can change his voice tone by changing the tension in his vocal cords.  Some insects produce sounds by rubbing their wings.  Birds’ chirps are like whistling in humans, the airflows vibrate the beak of the birds.  Some reptiles, like a rattlesnake, produce its characteristic sound (hissing) by shaking its tail. Ultrasound & Animals Many insects have good ultrasonic hearing, and most of these are nocturnal insects. Dogs and cats' hearing range extends into the ultrasound; the top end of a dog's hearing range is about 45 kHz, while a cat's is 64 kHz. A dog whistle is a whistle that emits ultrasound, used for training and calling dogs. Several types of fish can detect ultrasound. Contrary to popular belief, birds cannot hear ultrasonic sound. Ultrasound generator/speaker systems are sold as electronic pest control devices, which are claimed to frighten away rodents and insects. Bats Animals such as bats and dolphins use ultrasound for locating prey and obstacles. The bat emits waves with speed of sound 𝒗 at a frequency 𝒇. If the bat is moving towards the reflector with a speed 𝒗𝑩𝒂𝒕 and the reflector (insect) is moving towards the bat with a speed 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 then these waves reach the reflector at a frequency f’. ′ 𝒗 + 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 𝒇 =𝒇 𝒗 − 𝒗𝑩𝒂𝒕 Bats After that, the reflector (insect) acts as an emitter and the bat acts as a receiver and these waves with frequency f‘ are reflected and received by the bat at a frequency f''. ′′ ′ 𝒗 + 𝒗𝑩𝒂𝒕 𝒇 =𝒇 𝒗 − 𝒗𝑹𝒇𝒆𝒍𝒆𝒄𝒕𝒐𝒓 Hence, the bat hears the echoes of his sound with a frequency for a bat and reflector moving toward each other. ′′ 𝒗 + 𝒗𝑩𝒂𝒕 𝒗 + 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 𝒇 =𝒇 𝒗 − 𝒗𝑩𝒂𝒕 𝒗 − 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 Quiz A bat is chasing a pray at constant speed 7 m/s. The insect is moving away from the bat at constant speed. If the returning echo has a pitch 2% lower than the original sound. Can the bat catch its pray with this speed? find the speed of the insect. Hint: speed of sound 340 m/s Answer ′′ 𝟐 𝒇 =𝒇 𝟏− = 𝟎. 𝟗𝟖𝒇 𝟏𝟎𝟎 ′′ 𝒗 + 𝒗𝑩𝒂𝒕 𝒗 − 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 𝒇 =𝒇 𝒗 − 𝒗𝑩𝒂𝒕 𝒗 + 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 𝟑𝟒𝟎 + 𝟕 𝟑𝟒𝟎 − 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 𝟎. 𝟗𝟖𝒇 = 𝒇 𝟑𝟒𝟎 − 𝟕 𝟑𝟒𝟎 + 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 𝒗𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒐𝒓 = 𝟏𝟎. 𝟒𝟑𝟐 𝒎/𝒔 No, the bat can’t catch its pray with this speed. Shockwaves and Sonic Booms Suppose a jet airplane is coming nearly straight at you (stationary observer 𝒗𝟎 = 𝟎 ), emitting a sound of frequency f. 𝒗 𝒇′ =𝒇 𝒗 − 𝒗𝒔 The greater the plane’s speed vs, the greater the value of 𝒇′ and as 𝒗𝒔 approaches the speed of 𝒗 > 𝒗𝒔 𝒗 = 𝒗𝒔 sound, 𝒇′ approaches infinity, because the denominator approaches zero. At the speed of sound 𝒗 = 𝒗𝒔 , this means that, each successive wave is superimposed on the previous one 𝒗𝒔 increasing to because the source moves forward at the speed of reach a value sound. The observer gets them all at the same instant, larger than 𝒗 and so the frequency is infinite. Shockwaves and Sonic Booms If the source exceeds the speed of sound 𝒗 < 𝒗𝒔 , no sound is received by the observer until the source has passed. These waves travel at the speed of sound and, as the speed of the object increases, the waves are forced together, or compressed, because they cannot get out of each other's way quickly enough and they merge into a single shock wave, which travels at the speed of sound. 𝒗𝒔 The ratio ( ) is called the Mach number. 𝒗 The critical source speed is known as 𝑴𝒂𝒄𝒉 𝒏𝒖𝒎𝒃𝒆𝒓 = 𝟏 and is approximately 1,235 km/h (767 mph) at sea level and 20 °C (68 °F). Shockwaves and Sonic Booms The line tangent to all wave fronts drawn from source lie on the surface of a cone. The angle θ between one of these tangent lines and the direction of travel is given by 𝒗 𝟏 sin 𝜽 = = 𝒗𝒔 𝑴𝒂𝒄𝒉 𝒏𝒖𝒎𝒃𝒆𝒓 The large pressure variation in the shock wave condenses water vapor into droplets and is visible as a fog of water. Shockwaves Therapy Shock waves are high-energy, audible sound waves. They are characterized by a rapid increase in pressure and a short pulse length. In comparison to ultrasound waves, the shockwave peak pressure is approximately 1000 times greater than the peak pressure of an ultrasound wave and utilizes lower frequency waves. How does shockwave therapy work? Shock wave therapy is delivered directly onto the affected area via the use of a ‘generator’ or device, then the waves penetrate the skin and treat this area. Shock wave therapy has two main ‘modes of action’ that can help with persistent pain. Firstly, the shock waves work to ‘desensitize’ nerve endings which can immediately reduce pain in the local area. Secondly, the waves stimulate blood flow in the area, causing a small amount of localized inflammation. In the days immediately following the treatment, the body naturally tries to heal the inflammation and in doing so, encourages the regeneration of cells, repairing damaged tissue and reducing pain. Shock wave therapy can also help with issues relating to scar tissue. Because scar tissue is much denser - and much less elastic - than normal tissue, the sound waves can help break it down, improving mobility and reducing discomfort. Similarly, the waves can be used to break down ‘disorganized’ tissue or any build-up. Types of shockwave therapy Shockwave therapy is a popular, non-invasive alternative to many surgical or steroid- related procedures. Extracorporeal Shockwave Therapy (ESWT) gained popularity as a method for treating foot- and ankle-related injuries, it has also been used in treating kidney stones and gallstones for two decades. Shockwave effects cause hematoma formation and focal cell death, which then stimulate new bone or tissue formation. There are two common types : 1) Radial ESWT 2) Focused ESWT