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.

Wave - III

Sound Resonances

Consider a pipe of length L, open at one end, closed at the other end.

L

vvfL

4or

4

1

111

L

vnfn 4

12

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

1311

2 54

5 and 3

4

3f

L

vff

L

vf

Harmonics:

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:

12nf

L

vnfn

The same expression as in string with both ends fixed.

Beats

Two sound waves with different but close frequencies give rise to BEATS

s1 x,t sm cos1tConsider

s2 x,t sm cos2t1 2

s s1 s2 2sm cos t cost

cos cos 2cos1

2 cos

1

2

1

21 2 Very

small

1

21 2 ≈1≈2

s 2sm cos t cost

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

1

21 2

Beat frequency fbeat: fbeat f1 f2

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

f vt

t

v

The detector stationary:

Distance the sound

travels in time t

Divided by to get the number of periods in time t

Periods in unit time: frequency

The detector moving toward the source: more periods reaches detector. Equivalently:

f vt vDt

t

v vD

f v vD

vf

vD is the SPEED, always positive

The detector moving toward the source:

In general:

f v vD

vf

+ : toward S-: away from S

vf

Dv v

f

Source Moving, Detector Stationary

vT

The source stationary:

Distance between two wavefronts period T apart

f v

The source moving toward the detector : waves are squeezed. Equivalently:

vT vST f v

f

f

vT vT vST

f v

v vS f vS is the SPEED, always positive

The source moving toward the detector :

In general:

f

v

v vS f-: toward D +: away fromD

f

f

In General

f

v vD v vS f

+: away from D -: toward D

+ : toward S -: away from S

All speeds are measured with respect to the medium of propagation: the air

At Low Speed

f f 1u

v

u vS vDRelative speed:

+ : toward each other

-: away from each other

Supersonic Speed

f v

v vS f

The source moving toward the detector :

When vS>v, the equation no longer applicable: Supersonic speed

A Shock Wave is generated: abrupt change of air pressure

The wavefronts form a Mach Cone

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:

f

vnL

L

vnf nn 4

12or

412

mnLn 8

7,

8

5,

8

3,

8

1

6864

34312

mLn 125.0,375.0,625.0,875.00.1Lwater

L

vnfn 4

12

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

Mass increases f1 decreases

Therefore, f1 = 387 Hz

fbeat f1 f2

f n

2L

Resonant frequency

fbeat decreases f1 becomes closer to 384 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?

f

v vD v vS f

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.

f v vD v vS

f 343 m/s + 250 m/s

343 m/s + 200 m/s17,500 Hz

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.

Hz 467=Hz) 440(m/s 20.0 - m/s 343

m/s 343

f

vv

vf

S

(a) The source moving toward the detector :

(b) The person (detector) moves toward the source at the wall withf’ = 467 Hz:

Hz 494=Hz) 467( m/s 343

m/s 20.0+m/s 343

f

v

vvf D

r