Wave - III

Sound Resonances Consider a pipe of length L, open at one end, closed at the other end. At resonance, a displacement antinode at the open end, and a displacement node at the closed end. The longest wavelength to satisfy this condition is Fundamental resonant frequency Harmonics:

Pipe open at both ends: displacement antinodes at both ends Pipe open at both ends: displacement antinodes at both ends. open end closed at the other end. Pipe closed at both ends: displacement nodes at both ends. In both cases: The same expression as in string with both ends fixed.

Beats Two sound waves with different but close frequencies give rise to BEATS Consider Very small ≈w1≈w2

On top of the almost same frequency, the amplitude takes maximum twice in a cycle: cosw’t = 1 and -1: Beats Beat frequency fbeat:

The Doppler Effect The Doppler Effect: the frequency change related to the motions of the source or/and detector In the following, the speed is measured with respect to the air, through which the sound wave travels

Detector Moving, Source Stationary The detector stationary: Divided by l to get the number of periods in time t Distance the sound travels in time t Periods in unit time: frequency The detector moving toward the source: more periods reaches detector. Equivalently:

The detector moving toward the source: vD is the SPEED, always positive In general: + : toward S -: away from S

Source Moving, Detector Stationary The source stationary: Distance between two wavefronts period T apart The source moving toward the detector : waves are squeezed. Equivalently:

The source moving toward the detector : vS is the SPEED, always positive The source moving toward the detector : In general: -: toward D +: away fromD

In General + : toward S -: away from S +: away from D -: toward D All speeds are measured with respect to the medium of propagation: the air

At Low Speed Relative speed: + : toward each other -: away from each other Relative speed:

Supersonic Speed When vS>v, the equation no longer applicable: Supersonic speed The wavefronts form a Mach Cone A Shock Wave is generated: abrupt change of air pressure The source moving toward the detector :

HRW 51E (5th ed. ). The water level in a vertical glass tube 1 HRW 51E (5th ed.). The water level in a vertical glass tube 1.00 m long can be adjusted to any position in the tube. A tuning fork vibrating at 686 Hz is held just over the open top end of the tube. At what positions of the water level will there be a resonance? Let L be the length of the air column. Then the condition for resonance is:

Mass increases  f1 decreases HRW 61E (5th ed.). A tuning fork of unknown frequency makes three beats per second with a standard fork of frequency 384 Hz. The beat frequency decreases when a small piece of wax in put on a prong of the first fork. What is the frequency of this fork? fbeat = 3 Hz  f1 = 381 or 387 Hz Resonant frequency Mass increases  f1 decreases fbeat decreases  f1 becomes closer to 384 Hz Therefore, f1 = 387 Hz

HRW 68E (5th ed.). The 16,000 Hz whine of the turbines in the jet engines of an aircraft moving with speed 200 m/s is heard at what frequency by the pilot of a second aircraft trying to overtake the first at a speed of 250 m/s? The detector moves toward the source: take the plus sign for vD. The source moves away from the detector : take the plus sign for vS.

(a) The source moving toward the detector : HRW 80P (5th ed.). A person on a railroad car blows a trumpet note at 440 Hz. The car is moving toward a wall at 20.0 m/s. Calculate (a) the frequency of the sound as received at the wall and (b) the frequency of the reflected sound arriving back at the source. (a) The source moving toward the detector : (b) The person (detector) moves toward the source at the wall with f’ = 467 Hz: