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University Physics: Waves and Electricity. Ch1 7 . Longitudinal Waves. Lecture 5. Dr.-Ing. Erwin Sitompul. http://zitompul.wordpress.com. Homework 4 : Two Speakers. - PowerPoint PPT Presentation
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University Physics: Waves and Electricity
Ch17. Longitudinal WavesLecture 5
Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com
5/2Erwin Sitompul University Physics: Wave and Electricity
Homework 4: Two SpeakersTwo 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.
(a) What is the lowest frequency fmin,1 that gives minimum signal (destructive interference) at the listener’s ear?
(b) What is the second lowest frequency fmin,2 that gives minimum signal?
(c) What is the lowest frequency fmax,1 that gives maximum signal (constructive interference) at the listener’s ear?
(d) What is the highest frequency fmax,n that gives maximum signal?
5/3Erwin Sitompul University Physics: Wave and Electricity
2 2 23 1 2( ) ( ) ( )d d d
3d 2 23 (2) (3.75)d 4.25 m
3 2L d d 4.25 3.75 0.5 m
sound ,v f sound 343 m sv
Solution of Homework 4: Two Speakers
5/4Erwin Sitompul University Physics: Wave and Electricity
Solution of Homework 4: Two Speakers
(b) What is the second lowest frequency fmin,2 that gives minimum signal?
min 686 Hz 0.5,1.5,2.5,f
min,2 686 Hz 1.5f 1029 Hz
(a) What is the lowest frequency fmin,1 that gives minimum signal (destructive interference) at the listener’s ear? Fully destructive
interference
0.5,1.5, 2.5,L
sound min
0.5,1.5,2.5,Lv f
soundmin 0.5,1.5,2.5,vf
L
min,1 686 Hz 0.5f 343 Hz
343 0.5,1.5,2.5,0.5
686 Hz 0.5,1.5, 2.5,
5/5Erwin Sitompul University Physics: Wave and Electricity
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
0,1,2,L
sound max
0,1,2,Lv f
soundmax 0,1,2,vf
L
max,1 686 Hz 1f 686 Hz
343 0,1, 2,0.5
686 Hz 0,1,2,
(d) What is the lowest frequency fmax,1 that gives maximum signal (constructive interference) at the listener’s ear?max 686 Hz 0,1, 2,f
max, 686 Hz 29nf 19894 Hz Highest constructive frequency that still can be listened by human, < 20 kHz
5/6Erwin Sitompul University Physics: Wave and Electricity
Intensity and Sound Level There is more to sound than frequency, wavelength, and
speed. We are well with something called intensity. The intensity I of a sound wave at a surface is the average
rate per unit area at which enery is transferred by the wave through or onto the surface.
PIA
5/7Erwin Sitompul University Physics: Wave and Electricity
The Decibel Scale Human ear can bear the displacement amplitude that ranges
from about 10–5 m for the loudest torelable sound to about 10–11 m for the faintest detectable sound.
The ratio between the highest and the lowest amplitude is 106.
To deal with such an enormous range of values, people use logarithmic scale instead of linear scale.
0
(10 dB) log II
β is called the sound level. dB is the abbreviation for decibel, the unit of sound level. I0 is a standard reference intensity 10–12 W/m2, chosen
because it is near the lower limit of human range of hearing.
5/8Erwin Sitompul University Physics: Wave and Electricity
Intensity and Sound Level β increases by 3 dB every time
the sound intensity is doubled (increases by a factor of 2).
β increases by 10 dB every time the sound intensity increases by an order of magnitude (increases by a factor of 10).
5/9Erwin Sitompul University Physics: Wave and Electricity
Traveling Sound Waves Here we examine the displacements and pressure variations
associated with a sinusoidal sound wave traveling through air. The figure below displays such a wave traveling rightward
through a long air-filled tube. For a thin element of air
of thickness Δx, as the wave travels through this portion of the tube, the element of air oscillates left and right in a simple harmonic motion about its equilibrium position.
5/10Erwin Sitompul University Physics: Wave and Electricity
Traveling Sound Waves We choose to use a cosine
function to show the displacements s(x,t):
( , ) cos( )ms x t s kx t
5/11Erwin Sitompul University Physics: Wave and Electricity
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.
5/12Erwin Sitompul University Physics: Wave and Electricity
Beats Let the time-dependent variations of the displacements due to
two sound waves of equal amplitude sm be1 1 1( , ) cos( )ms x t s k x t
2 2 2( , ) cos( )ms x t s k x t
From superposition principle, the resultant displacement is:1 1 2 2( , ) cos( ) cos( )m ms x t s k x t s k x t
1 12 2cos cos 2cos ( )cos ( )
2 cos( ) cos( )2 2mks kx t x t
1 2k k k
11 22 ( )
11 22 ( )k k k k
1 2
2 cos( ) cos( )2 2mks x t kx t
Amplitude modulation, depends on
Δk/2 and Δω/2
Oscillating term, a traveling wave,
depends on k and ω
5/13Erwin Sitompul University Physics: Wave and Electricity
( , ) 2 cos( ) cos( )2 2mks x t s x t kx t
Beats
cos( )2 2k x t
1 2ampl 2 2
beat ampl 1 22f f f f
1 2ampl 2 2
f fff
In 1 amplitude cycle, we will hear 2 beats (maximum amplitude magnitude)
5/14Erwin Sitompul University Physics: Wave and Electricity
Example: BeatsThe 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?
beat 1 2f f f
beat string forkf f f
string4 440f string 436 Hz or 444 Hzf
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.
5/15Erwin Sitompul University Physics: Wave and Electricity
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 source S and a sound
detector D 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).
5/16Erwin Sitompul University Physics: Wave and Electricity
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.
5/17Erwin Sitompul University Physics: Wave and Electricity
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.
5/18Erwin Sitompul University Physics: Wave and Electricity
The Doppler Effect The emitted frequency f and the detected frequency f’ are
related by:D
S
v vf fv v
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.
D
S
v vf fv v
+ 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
5/19Erwin Sitompul University Physics: Wave and Electricity
Example: The Doppler EffectA 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?
242 m s, toward Sv
D
0Dv
D
S
v vf fv v
343 01250343 242
4245 Hz
0Sv
1250 Hzf
4245 Hzf
242 m s, toward Dv
S
D
S
v vf fv v 0
343 2424245343
7240 Hz
5/20Erwin Sitompul University Physics: Wave and Electricity
Supersonic Speeds
vsource = vsound(Mach 1 - sound barrier)
vsource > vsound(Mach 1.4 - supersonic)
5/21Erwin Sitompul University Physics: Wave and Electricity
Homework 5: Ambulance SirenAn 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
5/22Erwin Sitompul University Physics: Wave and Electricity
Homework 5: Ambulance Siren
(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?
(b) A bat, 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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