Chapter
10 & 11
Elasticity,
Oscillations,
Waves,
and Sound
**Standing
Waves in Air Columns**
Chamber
- both ends closed (23)
Chamber
- one end open
(24)
(25)
Where
N = 1, 3, 5,...
Chamber
- both ends open
(26)
(27)
Where
N = 1, 2, 3, ....
Example
25
An
empty pop bottle is to be used as a musical instrument in a band.
In order to be tuned properly, the fundamental frequency of the bottle
must be 440.0 Hz. If the bottle is 26.0 cm tall, how high should
it be filled with water to produce the desired frequency?
Solution:
Solve
for the length (L) of the pipe:
Find
the height of the water:
Example
26
An
experimental way to tune the pop bottle from the above example is
to compare its frequency with that of a 440.0 Hz tuning fork. Initially,
a beat frequency of 4 Hz is heard. As a small amount of water is
added to that already present, the beat frequency increases steadily
to 5 Hz. What were the initial and final frequencies of the bottle?
Solution:
444
Hz and 445 Hz
**The
Doppler Effect**
**The
Doppler Effect** is the change in frequency due to relative motion
between a source and a receiver.
The
source moving:
(28)
Use
the minus sign when the source is moving toward the observer
and the positive sign when the source is moving away from the
observer.
The
observer moving:
(29)
Use
the plus sign when the observer is moving toward the source and
the negative sign when the observer is moving away from the source.
When
both the source and the observer are moving:
(30)
In
the numerator, the + sign corresponds to the case in which the
oberver moves in the direction of the source, whereas the –
sign indicates motion in the opposite
direction.
In the denominator, the – sign corresponds to the case in which
the source moves in the direction of the
observer, whereas the
+ sign indicates motion in the opposite
direction.
Example
27
A
street musician sounds the A string of his violin, producing
a tone of 440 Hz. What frequency does a bicyclist hear as he
**(a)** approaches and **(b)** recedes from the musician with a speed
of 11.0 m/s?
Solution:
Use
equation (29) above with a positive sign:
Use
equation (29) above with a minus sign:
Example
28
A
train sounds its whistle as it approaches a tunnel in a cliff.
The whistle produces a tone of 650.0 Hz, and the train travels
with a speed of 21.2 m/s. **(a)** Find the frequency heard by an
observer standing near the tunnel entrance. **(b)** The sound from
the whistle reflects from the cliff back to the engineer in the
train. What frequency does the engineer hear?
Solution:
(a)
Use equation (28) with a minus sign:
(b)
Use equation (29) with a plus sign:
Example
29 A
car moving at 18 m/s sounds its 550 Hz horn. A bicyclist on the
sidewalk, moving with a speed of 7.2 m/s, approaches the car.
What frequency is heard by the bicyclist?
Solution:
Using
equation 30 with the plus sign in the numerator and the minus
sign in the denominator:
The
End |