If a source is moving toward a stationary detector at a speed equal to the speed of sound—that is, if νS = ν—Eqs. 17-47 and 17-55 predict that the detected frequency f′ will be infinitely great. This means that the source is moving so fast that it keeps pace with its own spherical wavefronts, as Fig. 17-23a suggests. What happens when the speed of the source exceeds the speed of sound?
For such supersonic speeds, Eqs. 17-47 and 17-55 no longer apply. Figure 17-23b depicts the spherical wavefronts that originated at various positions of the source. The radius of any wavefront in this figure is νt, where ν is the speed of sound and t is the time that has elapsed since the source emitted that wavefront. Note that all the wavefronts bunch along a V-shaped envelope in the two-dimensional drawing of Fig. 17-23b. The wavefronts actually extend in three dimensions, and the bunching actually forms a cone called the Mach cone. A shock wave is said to exist along the surface of this cone, because the bunching of wavefronts causes an abrupt rise and fall of air pressure as the surface passes through any point. From Fig. 17-23b, we see that the half-angle θ of the cone, called the Mach cone angle, is given by
The ratio νS/ν is called the Mach number. When you hear that a particular plane has flown at Mach 2.3, it means that its speed was 2.3 times the speed of sound in the air through which the plane was flying. The shock wave generated by a supersonic aircraft (Fig. 17-24) or projectile produces a burst of sound, called a sonic boom, in which the air pressure first suddenly increases and then suddenly decreases below normal before returning to normal. Part of the sound that is heard when a rifle is fired is the sonic boom produced by the bullet. A sonic boom can also be heard from a long bullwhip when it is snapped quickly: Near the end of the whip’s motion, its tip is moving faster than sound and produces a small sonic boom—the crack of the whip.
Fig. 17-23 (a) A source of sound S moves at speed vS equal to the speed of sound and thus as fast as the wavefronts it generates. (b) A source S moves at speed vS faster than the speed of sound and thus faster than the wavefronts. When the source was at position S1 it generated wavefront W1, and at position S6 it generated W6. All the spherical wavefronts expand at the speed of sound v and bunch along the surface of a cone called the Mach cone, forming a shock wave. The surface of the cone has halfangle θ and is tangent to all the wavefronts.
The photograph that opens Lesson 11 shows the jet-powered car Thrust SSC as it was traveling faster than the speed of sound in the Black Rock Desert of Nevada. The shock wave generated by the car sent a sonic boom across the desert as a sure signal that the car was supersonic. You can see evidence for the shock wave just above the car in the photograph. Our view of the background in that direction is distorted because light from the background encountered significant changes in air pressure and density as it passed into and out of the shock wave to reach the camera. These changes caused slight deflections in the light, distorting the background in the photograph taken by the camera.
Fig. 17-24 Shock waves produced by the wings of a Navy FA 18 jet. The shock waves are visible because the sudden decrease in air pressure in them caused water molecules in the air to condense, forming a fog.
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