L-5 Sound 1.16.1 Introduction and Classification: Sound waves are mechanical, compression waves which are in general longitudinal in nature-meaning that the particles vibrate parallel to the direction of the wave’s velocity. Sound waves are divided into three categories that cover different frequency ranges. (1) Audible waves They within the range of sensitivity of the human ear. The range of human hearing stretches between 20-20000 Hertz. They can be generated in a variety of ways, such as by musical instruments human voices, or loud speakers (2) Infrasonic waves These waves have frequencies below the audible range, that is less than 20 Hertz. Elephants can use infrasonic waves to communicate with each other, even when separated by many kilometers. (3) Ultrasonic waves They have frequencies above the audible range, that is greater than 20000 Hertz. Some animals can emit these sounds. Bats, for example, emit and hear ultrasound waves, which they use for locating prey and for navigating. 1.16.2 Speed of sound waves The speed of sound waves in a medium depends on the compressibility and density of the medium. The speed of all mechanical waves follows an expression of the general form v elastic property inertial property The speed of sound also depends on the temperature of the medium. For sound traveling through air, the relationship between wave speed and medium temperature is v (331m / s) 1 Tc 273C where 331m/s is the speed of sound in air at 0°C and the T c is the air temperature in degree Celsius. 1.16.3 Decibel (dB) scale The range of sound powers and sound pressures is very wide. In order to cover this wide range while maintaining accuracy, the logarithmic decibel (dB) scale was selected. The intensity Properties of Matter and Sound 1.2 of the faintest sound that the normal person can hear is about 0.0000000000001 watts/m2, while the intensity of the sound produced by a Saturn rocket at liftoff is greater than 100,000,000 watts/ m2. This is a range of 100,000,000,000,000,000,000. Given this extremely large range in values, there needed to be a better way to express or represent these numbers. By using logarithms of these numbers, as compared to a reference value, we can form a new measurement scale in which an increase of 1.0 represents a tenfold increase in the ratio (also called a 1.0 bel increase). The application of logarithms is evolved to the use of 10 subdivisions of a log value.Decibels is abbreviated to the term dB. The lower case “d” represents deci, or 1/10th of a bel. The capital “B” stands for bel, named after Alexander Graham Bell, inventor of the telephone. Decibel isa dimensionless unit related to the logarithm of the ratio of a measured quantity to a reference quantity. Sound power level is the acoustical power radiated by a source with respect to the standard reference of 10-12 watts. Lw = 10 Log (W/Wre) The international reference for power is 10-12 watts. Now because we are converting a sound power into a Level, or dB, the formula is as shown above. The term Lw is used to represent the Sound Power Level. The “w” subscript identifies the fact this equation deals with power in units of watts. 1.16.4 Doppler Effect The Doppler effect is a phenomenon observed whenever the source of waves is moving with respect to an observer. The Doppler effect can be defined as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for the observer and the source are approaching and an apparent downward shift in frequency when the observer and the source are receding. Although the Doppler effect is most typically experienced with sound waves, it is a phenomenon that is common to all waves. For example, the relative motion of source and observer produces a frequency shift in light waves. The Doppler effect is used in police radar systems to measure the speeds of motor vehicles. Likewise, astronomers use the effect to determine the speeds of stars, galaxies, and other celestial objects relative to the earth. 1.17 Shockwaves Definition Shockwave is a wave formed of a zone of extremely high pressure within a fluid, especially the atmosphere, that propagates through the fluid at a speed in excess of the speed of sound. Types Properties of Matter and Sound 1.3 Shockwaves in supersonic flow may be classified as normal or oblique according to whether the orientation of the surface of the abrupt change is perpendicular or at an angle to the direction of flow Description Now consider what happens when the speed vs of a source exceeds the wave speed v. This situation is depicted graphically in Fig.1.18 Fig.1.18 Shock waves The circles represent spherical wave fronts emitted by the source at various times during its motion. At t = 0, the source is at S0 and at a later time t, the source is at Sn. At the time t, the wave front centered at S0 reaches a radius of vt. In this same time interval, the source travels a distance vst to Sn. At the instant the source is at Sn, waves are just beginning to be generated at this location, and hence the wave front has zero radius at this point. The tangent line drawn from Sn to the wave front centered on S0 is tangent to all other wave fronts generated at intermediate times. Thus, we see that the envelope of these wave fronts is a cone whose apex half-angle (the “Mach angle”) is given by sin vt v vs t vs and the conical wave front produced when vs > v (supersonic speeds) is known as a shock wave. An interesting analogy to shock waves is the V-shaped wave fronts producted by a boat (the bow wave) when the boat’s speed exceeds the speed of the surface-water waves Sonic Boom Properties of Matter and Sound 1.4 Jet airplanes traveling at supersonic speeds produce shock waves, which are responsible for the loud “sonic boom” one hears. The shock wave carries a great deal of energy concentrated on the surface of the cone, with correspondingly great pressure variations. Such shock waves are unpleasant to hear and can cause damage to buildings when aircraft fly supersonically at low altitudes. In fact, an airplane flying at supersonic speeds produces a double boom because two shock waves are formed, one from the nose of the plane and one from the tail. People near the path of the space shuttle as it glides toward its landing point often report hearing what sounds like two very closely spaced cracks of thunder. Applications Shock waves have applications outside of aviation. They are used to break up kidney stones and gallstones without invasive surgery, using a technique with the impressive name extracorporeal shock-wave lithotripsy. A shock wave produced outside the body is focused by a reflector or acoustic lens so that as much of it as possible converges on the stone. When the resulting stresses in the stone exceed its tensile strength, it breaks into small pieces and can be eliminated. 1.18 Mach Number Mach number is a dimensionless measure of relative speed. It is defined as the speed of an object relative to a fluid medium, divided by the speed of sound in that medium. M v vs where M is the Mach number, v is the speed of the object relative to the medium and vs is the speed of sound in the medium. Mach number is named after Austrian physicist and philosopher Ernst Mach. It can be shown that the mach number is also the ratio of inertial forces (also referred to aerodynamic forces). The square of the Mach number is Cauchy number. M2 = C, Cauchy number. High speed flights can be classified in five categories i. Sonic : ii. Subsonic iii. Transonic : 0.8 < M < 1.2 iv. Supersonic : 1.2 < M<5 v. Hypersonic : M>5 : M=1 M<1 Properties of Matter and Sound 1.5 For supersonic and hypersonic flows, small disturbances are transmitted downstream within a cone as shown in Fig.1.19 Properties of Matter and Sound 1.6 Fig.1.19 Shock wave v vs Fig 1.20 Mach angle The wave front is a cone with angle α called the Mach angle as given in Fig. 1.20. sin v vs Mach number M :. sin vs v 1 M The speed of sound depends primarily on the fluid temperature around it and is given as v RT Properties of Matter and Sound 1.7 where T is the temperature (Kelvin), R is the gas constant of fluid and γ is the adiabatic index of the gas (that is the ratio of specific heats of a gas at constant pressure and volume). For most calculations, standard air conditions are assumed and a value of γ = 1.4 and R = 287 J/(kg K) are used. M vs RT The Mach number is commonly used both with objects traveling at high speed in a fluid, and with high speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. At a temperature of 15 degree Celsius and at sea level, Mach 1 is 340 3m/s(1,225 km/h) in the Earth’s atmosphere. The speed represented by Mach 1 is not a constant, it is temperature dependent. Hence in the stratosphere it remains about the same regardless of height, though the air pressure changes with height. Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the fluid temperature around it. Mach number is useful because the fluid behaves in a similar way at the same Mach number. So, an aircraft traveling at a Mach 1 at sea level will experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m when it is traveling at Mach 1 at 11,000 m even though it is traveling at 295 m/s( 1.062 km/h, 86% of its speed at sea level). Critical Mach number A critical mach number is the speed of an aircraft (below Mach 1)when the air flowing over some area of the airfoil has reached the speed of sound. For instance, if the air flowing over a wing reaches Mach 1 when the wing is only moving at Mach 0.8, then the wing’s critical Mach number is 0.8. Mach Tuck For a subsonic aircraft traveling significantly below Mach 1.0, Mach tuck is an aerodynamic effect, whereby the nose of an aircraft tends to pitch downwards as the air flow around the wing reaches supersonic speeds. Mach meter A Mach meter is an aircraft instrument that shows the ratio of the speed of sound to the true airspeed, a dimensionless quantity called Mach number. That is, Mach meter is an aircraft instrument that indicates speed in Mach numbers. Worked Example 1.12 An aircraft is flying at speed 370m/s at an attitude where the speed of sound is 320m/s. Calculate the Mach number Mach number = Aircraft speed 370m / s Speed of sound 320m / s 1.156 Properties of Matter and Sound 1.8 Worked Example 1.13: The Concorde is flying at Mach 1.25 at an altitude where the speed of sound is 325 m/s. Calculate the speed of the Concorde Concorde speed = Mach number × speed of sound = 1.25 × 325m/s = 406.25 m/s Worked Example 1.14: The Concorde is flying at Mach 1.75 at an altitude of 8000 m, where the speed of sound is 320 m/s. How long after the plane passes directly overhead will you hear the sonic boom? The shockwave forms a cone trailing backward from the airplane, so the problem is really asking for how much time elapses from when the Concorde flies overhead to when the shockwave reaches you. vS = Mach 1.75 vS 8000m Shock wave L a vst The angle α of the shock cone is = arcsin 1 34.8 o The speed of the plane is the speed of sound 1.75 multiplied by the Mach number vs = (1.75) (320 m/s) = 560 m/s From figure, we have, tan α = t 8000m (560m / s) tan 34.8 8000 m vs t o 20.5s Properties of Matter and Sound 1.9 Worked Example 1.15: A plane is flying at supersonic speed at an altitude where the speed of sound is 320m/s. The shock angle makes an angle of 33.5° with the direction of the plane. What is the plane’s speed and its Mach number Mach number 1 1 1.81 sin sin 33.5 Aircraft speed = Mach number × speed of sound = (1.81) ( 320 m/s) = 579.2 m/s Worked Example 1.16: A sonic boom is heard 20.5s after the Concorde passes overhead. Assuming the Mach 1.75 and speed of sound is 320 m/s, calculate the distance traveled by the flight at this time. Distance traveled = speed of flight × time = (560 m/s) (20.5s) = 11500 m Worked Example 1.17: Determine the velocity of a bullet fired in the air if the Mach angle is observed to be 30°. Given that the temperature of the air is 22°C Take γ = 1.4 and R = 287.43 J/kg.K T = 273.15 +22 = 295.15 K RT (1.4) (287.4) (295.15 ) Sonic velocity = =344.6 m/s For the Mach cone, Sin α = 1 0.5 M 2.0 M Bullet velocity = (2.0) (344.6 m/s) = 689.2 m/s Worked Example 1.18: A Mach cone of Mach angle π/6 radian is observed for a fighter aircraft at an altitude where the temperature is 280K Calculate the aircraft velocity. Sonic velocity = = RT (11.4) (287) (280) = 336 m/s 1 sin sin ( / 6) 0.5 M Properties of Matter and Sound 1.10 M=2 Aircraft velocity = (2) (336 m/s) = 772 m/s Worked Example 1.19: An observer on the ground hears the sonic boom of a plane 15km above when the plane has gone 20km ahead of him. Estimate the speed of flight of the plane. T 20km Plane 15 km Observer The observer hears the sound after the plane has passed over his head because he must be in the zone of silence before he hears it. As soon as he falls within the conical zone of awareness, he hears it.In that position sin α = 1 M tan α = 15 km 0.75 20 km α = 36.87° sin 36.87° = 0.6 = 1/M M = 1.67 The plane must be flying at a supersonic speed corresponding to a local mach number of 1.67. Exercise Problem1.5: An aircraft is flying horizontally at Mach 1.8 over a flat desert. A sonic boom is heard on the ground 8.1s after the aircraft has passed directly overhead. Assume the speed of sound in the air is 350 m/s. At what altitude is the aircraft flying? Hint : Altitude = vst sinα = 283lm Properties of Matter and Sound 1.11 Exercise Problem 1.6: The speed of the first Indian satellite Aryabhatta at an altitude of 8000m was 1000 m/s. Take R = 287.3J/kg.K, γ = 1.4. Calculate the Mach number for local conditions at T = 240K and the angle of the Mach cone. Hint: Sonic velocity = M RT satellite velocity 3.22 sonic velocity sin α = 1/M α = 18.1° Exercise Problem1.7: A supersonic fighter plane moves with a Mach number of 1.5 in atmosphere at an altitude of 500m above the ground level. What is the time that lapses, by which the acoustic disturbance reaches an observer on the ground after it is directly overhead? Take T = 20°C, γ = 1.4 and R = 287 J/kg.K Hint: Sonic velocity = Time elapsed = 1.09 s RT