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### University Physics: Waves and Electricity

Ch17. Longitudinal Waves

Lecture 5

Dr.-Ing. Erwin Sitompul

http://zitompul.wordpress.com

2014

Homework 4: Two Speakers

Two speakers separated by distance d1 = 2 m are in phase. A listener observes at distance d2 = 3.75 m directly in front of one speaker. Consider the full audible range for normal human hearing, 20 Hz to 20 kHz. Sound velocity is 343 m/s.

What is the lowest frequency fmin,1 that gives minimum signal (destructive interference) at the listener’s ear?

What is the second lowest frequency fmin,2 that gives minimum signal?

What is the lowest frequency fmax,1 that gives maximum signal (constructive interference) at the listener’s ear?

What is the highest frequency fmax,n that gives maximum signal?

Solution of Homework 4: Two Speakers

(a) What is the lowest frequency fmin,1 that gives minimum signal (destructive interference) at the listener’s ear?

- Fully destructive interference

(b) What is the second lowest frequency fmin,2 that gives minimum signal?

Solution of Homework 4: Two Speakers

(c) What is the lowest frequency fmax,1 that gives maximum signal (constructive interference) at the listener’s ear?

- Fully constructive interference

(d) What is the lowest frequency fmax,1 that gives maximum signal (constructive interference) at the listener’s ear?

- Highest constructive frequency that still can be listened by human, < 20 kHz

Beats

- If two sounds whose frequencies are nearly equal reach our ears simulta-neously, what we hear is a sound whose frequency is the average of the two combining frequencies.
- We also hear a striking variation in the intensity of this sound –it increases and decreases in slow, wavering beats that repeat at a frequency equal to the difference between the two combining frequencies.

Beats

- Let the time-dependent variations of the displacements due to two sound waves of equal amplitude sm be

- From superposition principle, the resultant displacement is:

Amplitude modulation, depends on Δk/2 and Δω/2

Oscillating term, a traveling wave, depends on k and ω

Beats

- In 1 amplitude cycle, we will hear 2 beats (maximum amplitude magnitude)
- The "beat" wave oscillates with the frequency average, and its amplitude varies according to the frequency difference

Example: Beats

The A string of a violin is not properly tuned. Beats at 4 per second are heard when the string is sounded together with a tuning fork that is oscillating accurately at concert A (440 Hz).

(a) What are the possible frequencies produced by the string?

(b) If the string is stretched a little bit more, beats at 5 per second are heard. Which of the possible frequencies are the the frequency of the string?

- A string is stretched tighter The frequency will be higher
- The frequency of beats increases The frequency difference increases
- If the string frequency becomes higher and its difference to 440 Hz increases The frequency of the string is 444 Hz.

The Doppler Effect

- The Doppler Effect deals with the relation between motion and frequency.
- The body of air is taken as the reference frame.
- We measure the speeds of a sound sourceS and a sound detectorD relatif to that body of air.
- We shall assume that S and D move either directly toward or directly away from each other, at speeds less than the speed of sound (vsound = 343 m/s).

The Doppler Effect: D Moving S Stationary

- If the detector moves toward the source, the number of wavefronts received by the detector increased.
- The motion increases the detected frequency.
- If the detector moves away from the source, the number of wavefronts received by the detector decreased.
- The motion decreases the detected frequency.

The Doppler Effect: S Moving D Stationary

- If the source moves toward the detector, the wavefronts is compressed. The number of wavefronts received by the detector increased.
- The motion increases the detected frequency.
- If the source moves away from the detector, the distance between wavefronts increases. The number of wavefronts received by the detector decreased
- The motion decreases the detected frequency.

The Doppler Effect

- The emitted frequency f and the detected frequency f’ are related by:

where v is the speed of sound through the air, vD is the detector’s speed relative to the air, and vS is the source’s speed relative to the air.

+ The detector moves toward the source

– The detector moves away from the source

– The source moves toward the detector

+ The source moves away from the detector

Checkpoint

The figure indicates the directions of motion of a sound source and a detector for six situations in stationary air. For each situation, is the detected frequency greater than or less than the emitted frequency, or can’t we tell without more information about the actual speeds?

Source

Detector

Source

Detector

(a)

(d)

zero speed

Greater

Need moreinformation

(e)

(b)

zero speed

Less

Greater

(c)

(f)

Less

Need more information

Example: The Doppler Effect

A toy rocket flies with a velocity of 242 m/s toward a mast while emitting a roaring sound with frequency 1250 Hz. The sound velocity is 343 m/s.

(a) What is the frequency heard by an observer who is standing at the mast?

(b) A fraction of the soundwaves is reflected by the mast and propagates back to the rocket. What is the frequency detected by a detector mounted on the head of the rocket?

Homework 5: Ambulance Siren

An ambulance with a siren emitting a whine at 1600 Hz overtakes and passes a cyclist pedaling a bike at 8 m/s. After being passed, the cyclist hears a frequency of 1590 Hz.

(a) How fast is the ambulance moving?

(b) What frequency did the cyclist hear before being overtaken by the ambulance?

Illustration only

- Concorde, the supersonic turbojet-powered supersonic passenger airliner
- Average cruise speed Mach 2.02 or about 2495 kmh

Homework 5A: Bat and Insect

1. An iron bar produces sound with a frequency of 335 Hz when struck. When the iron bar is struck together with a steel bar, beats with a frequency of 5.5 Hz will be heard. A piece of thread is then tied to the iron bar and its frequency is lowered slightly. When struck at the same time again, both the iron and steel bars now produce a beat with frequency of 8.2 Hz.

(a) What is the frequency of the iron bar after the thread is tied to it?

(b) What is the frequency of the steel bar?

2. (a) A stationary observer hears a frequency of 560 Hz from an approaching car. After the car passes, the observed frequency is 460 Hz. What is the speed of the car? (speed of sound is air is 343 m/s.)

(b) Abat, moving at 5 m/s, is chasing a flying insect. If the bat emits a 40 kHz chirp and receives back an echo at 40.4 kHz, at what speed is the insect moving toward or away from the bat?

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