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