University Physics: Waves and Electricity

University Physics: Waves and Electricity

Ch17. Longitudinal Waves

Lecture 5

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

5/2Erwin Sitompul University Physics: Wave and Electricity

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.

(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?


2 2 23 1 2( ) ( ) ( )d d d

3d2 2

3 (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



(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,L

v f

soundmin 0.5,1.5,2.5,

vf

L

min,1 686 Hz 0.5f 343 Hz

3430.5,1.5,2.5,

0.5

686 Hz 0.5,1.5,2.5,



(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,L

v f

soundmax 0,1,2,

vf

L

max,1 686 Hz 1f 686 Hz

3430,1,2,

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


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.

PI

A


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) logI

I

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.


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).


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.


Traveling Sound Waves

We choose to use a cosine function to show the displacements s(x,t):

( , ) cos( )ms x t s kx t


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

1 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 2m

ks kx t x t

1 2k k k

11 22

( )

11 22

( )k k k k

1 2

2 cos( ) cos( )2 2m

ks x t kx t

Amplitude modulation, depends on

Δk/2 and Δω/2

Oscillating term, a traveling wave,

depends on k and ω


( , ) 2 cos( ) cos( )2 2m

ks x t s x t kx t

Beats

cos( )2 2

kx 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)


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?

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.


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).


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:

D

S

v vf f

v 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 f

v 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


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?

242 m s, toward Sv

D

0Dv

D

S

v vf f

v v

343 01250

343 242

4245 Hz

0Sv

1250 Hzf

4245 Hzf

242 m s, toward Dv

S

D

S

v vf f

v v

0

343 2424245

343

7240 Hz


Supersonic Speeds

vsource = vsound(Mach 1 - sound barrier)

vsource > vsound(Mach 1.4 - supersonic)


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?


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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