Chapter 17. Waves-II Sound Waves

Transcription

1 Chapter 17 Waves-II 17.2 Sound Waves Wavefronts are surfaces over which the oscillations due to the sound wave have the same value; such surfaces are represented by whole or partial circles in a twodimensional drawing for a point source. Rays are directed lines perpendicular to the wavefronts that indicate the direction of travel of the wavefronts.

2 17.3 Speed of Sound As sound wave passes through air, potential energy is associated with periodic compressions and expansions of small volume elements of the air. Bulk Modulus, B, determines the extent to which an element of a medium changes in volume when the pressure on it changes. B is defined as: Here ΔV/V is the fractional change in volume produced by a change in pressure Δp. But, and if B replaces τ and ρ replaces μ, This is the speed of sound in a medium with bulk modulus B and density ρ Speed of Sound; Derivation of result We have: Also, And Therefore, But Finally,

3 17.4 Traveling Sound Waves 17.4 Traveling Sound Waves }

4 Example, Pressure and Displacement Amplitudes 17.5: Interference

5 17.5: Interference Phase difference φ can be related to path length difference ΔL, by noting that a phase difference of 2π rad corresponds to one wavelength. Therefore, Fully constructive interference occurs when φ is zero, 2π, or any integer multiple of 2π. Fully destructive interference occurs when φ is an odd multiple of π: Example, Interference:

6 Example, Interference: 17.6: Intensity and Sound Level The intensity I of a sound wave at a surface is the average rate per unit area at which energy is transferred by the wave through or onto the surface. Therefore, I =P/A where P is the time rate of energy transfer (the power) of the sound wave and A is the area of the surface intercepting the sound. The intensity I is related to the displacement amplitude s m of the sound wave by

7 17.6: Intensity and Sound Level Consider a thin slice of air of thickness dx, area A, and mass dm, oscillating back and forth as the sound wave passes through it. The kinetic energy dk of the slice of air is But, Therefore, And, Then the average rate at which kinetic energy is transported is If the potential energy is carried along with the wave at this same average rate, then the wave intensity I, the average rate per unit area at which energy of both kinds is transmitted by the wave, is 17.6: Intensity and Sound Level: Variation with Distance

8 17.6: Intensity and Sound Level: The Decibel Scale Here db is the abbreviation for decibel, the unit of sound level. I 0 is a standard reference intensity ( W/m 2 ), chosen near the lower limit of the human range of hearing. For I =I 0, gives β =10 log 1 = 0, (our standard reference level corresponds to zero decibels). Example, Cylindrical Sound Wave:

9 Example, Decibel, Sound Level, Change in Intensity: Many veteran rockers suffer from acute hearing damage because of the high sound levels they endured for years while playing music near loudspeakers or listening to music on headphones. Recently, many rockers, began wearing special earplugs to protect their hearing during performances. If an earplug decreases the sound level of the sound waves by 20 db, what is the ratio of the final intensity I f of the waves to their initial intensity I i? 17.7: Sources of Musical Sound Musical sounds can be set up by oscillating strings (guitar, piano, violin), membranes (kettledrum, snare drum), air columns (flute, oboe, pipe organ, and the digeridoo of Fig.17-12), wooden blocks or steel bars (marimba, xylophone), and many other oscillating bodies. Most common instruments involve more than a single oscillating part.

10 17.7: Sources of Musical Sound A. Pipe open at both ends B. Pipe open at one end only Example, Double Open and Single Open Pipes:

11 17.8: Beats When two sound waves whose frequencies are close, but not the same, are superimposed, a striking variation in the intensity of the resultant sound wave is heard. This is the beat phenomenon. The wavering of intensity occurs at a frequency which is the difference between the two combining frequencies. Example, Beat Frequencies:

12 17.9: Doppler Effect When the motion of detector or source is toward the other, the sign on its speed must give an upward shift in frequency. When the motion of detector or source is away from the other, the sign on its speed must give a downward shift in frequency. Here the emitted frequency is f, the detected frequency f, v is the speed of sound through the air, v D is the detector s speed relative to the air, and v S is the source s speed relative to the air. 17.9: Doppler Effect In time t, the wavefronts move to the right a distance vt. The number of wavelengths in that distance vt is the number of wavelengths intercepted by D in time t, and that number is vt/λ. The rate at which D intercepts wavelengths, which is the frequency f detected by D, is In this situation, with D stationary, there is no Doppler effect the frequency detected by D is the frequency emitted by S. Fig The planar wavefronts (a) reach and (b) pass a stationary detector D; they move a distance vt to the right in time t.

13 17.9: Doppler Effect; Detector Moving, Source Stationary If D moves in the direction opposite the wavefront velocity, in time t, the wavefronts move to the right a distance vt, but now D moves to the left a distance v D t. Thus, in this time t, the distance moved by the wavefronts relative to D is vt +v D t. The number of wavelengths in this relative distance vt +v D t is (vt +v D t)/λ. The rate at which D intercepts wavelengths in this situation is the frequency f, given by Similarly, we can find the frequency detected by D if D moves away from the source. In this situation, the wavefronts move a distance vt -v D t relative to D in time t, and f is given by 17.9: Doppler Effect; Source Moving, Detector Stationary Detector D is stationary with respect to the body of air, and source S move toward D at speed v S. If T ( =1/f ) is the time between the emission of any pair of successive wavefronts W 1 and W 2, during T, wavefront W 1 moves a distance vt and the source moves a distance v S T. At the end of T, wavefront W 2 is emitted. In the direction in which S moves, the distance between W 1 and W 2, which is the wavelength λ of the waves moving in that direction, is (vt v S T). If D detects those waves, it detects frequency f given by In the direction opposite that taken by S, the wavelength λ of the waves is again the distance between successive waves but now that distance is (vt v S T). D detects frequency f given by

14 Example, Doppler Shift: 17.9: Doppler Effect

15 17.9: Supersonic Speeds, Shock Waves