1. Dr. Jie Zou PHY 1151G Department of Physics
Chapter 14
Waves and Sound (Cont.)
Dr. Jie Zou PHY 1151G Department of Physics

2. Dr. Jie Zou PHY 1151G Department of Physics
Outline
Standing waves
Standing waves on a string
Vibrating columns of air
Sound intensity
Doppler effect
Moving observer
Moving source
Dr. Jie Zou PHY 1151G Department of Physics

3. Dr. Jie Zou PHY 1151G Department of Physics
A standing wave
A standing wave is one that oscillates with time, but remains fixed in its location. It is in this sense that the wave is said to be “standing”.
Example:
(a) A string tied down at both ends;
(b) When plucked in the middle, a standing wave results-the fundamental mode of oscillation of the string.
1 = 2L (wavelength of the fundamental)
Dr. Jie Zou PHY 1151G Department of Physics

4. Standing waves on a string
Antinode (A)
Standing waves on a string
Node (N)
First harmonic (fundamental) frequency and wavelength: f1 = v/(2L); 1 = 2L
v = (F/)1/2: wave speed on the string; F: tension in the string; : mass per unit length of the string; L: length of the string
Frequency and wavelength of the nth harmonic, with n = 1, 2, 3,…: fn = nf1 = ; n = 1/n = 2L/n
Dr. Jie Zou PHY 1151G Department of Physics

5. An application – guitar
Dr. Jie Zou PHY 1151G Department of Physics

6. Vibrating columns of air
Standing waves in a column of air closed at one end:
f1 = v/(4L);
fn = nf1 = , n = 1, 3, 5,… odd harmonics only
n = 1/n = 4L/n
v: speed of sound in air
L: length of the air column
Dr. Jie Zou PHY 1151G Department of Physics

7. Dr. Jie Zou PHY 1151G Department of Physics
Example
An empty soda 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 Hz.
(a) If the bottle is 26.0 cm tall, how high should it be filled with water to produce the desired frequency? Treat the bottle as a pipe that is closed at one end (the surface of the water) and open at the other end. (6.5 cm)
(b) What is the frequency of the next higher harmonic for this bottle? (1320 Hz)
Dr. Jie Zou PHY 1151G Department of Physics

8. Dr. Jie Zou PHY 1151G Department of Physics
Sound intensity
The loudness of a sound is determined by its intensity.
Definition of intensity: If the energy E passes through the area A in the time t, the intensity, I, of the wave is
I = E/(At). Since the power P = E/t,
I = P/A. SI unit: W/m2.
The intensity falls off with the square of the distance:
I = P/(4r2). SI unit: W/m2.
Dr. Jie Zou PHY 1151G Department of Physics

9. Example: Intensity of sound
Two people relaxing on a deck listen to a songbird sing. One person, only 1.00 m from the bird, hears the sound with an intensity of 2.80 10-6 W/m2 .
What intensity is heard by the second person, who is 4.25 m from the bird? Assume that no reflected sound is heard by either person.
Dr. Jie Zou PHY 1151G Department of Physics

10. Dr. Jie Zou PHY 1151G Department of Physics
Doppler effect
Doppler effect: The change in pitch, due to the relative motion between a source of sound and the receiver, is called the Doppler effect.
The pitch increases when the observer and the source are moving closer together.
The pitch decreases when the observer and source are separating.
Dr. Jie Zou PHY 1151G Department of Physics

11. Doppler effect for a moving observer
For moving observer: f´= (1  u/v)f
f´= frequency to the observer.
f = frequency of the source of the sound.
u = speed of the observer.
v = speed of the sound.
“+”: when the observer moves toward the source.
“-”: when the observer moves away from the source.
Dr. Jie Zou PHY 1151G Department of Physics

12. Doppler effect for a moving source
For moving source: f’ = [1/(1  u/v)] f
u = speed of the source.
v = speed of the sound.
“-”: when the source moves toward the observer.
“+”: when the source moves away from the observer.
Dr. Jie Zou PHY 1151G Department of Physics

13. Example: Doppler effect
A train sounds its whistle as it approaches a tunnel in a cliff. The whistle produces a tone of 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?
Dr. Jie Zou PHY 1151G Department of Physics

14. Dr. Jie Zou PHY 1151G Department of Physics
Homework
See online homework assignment on
Dr. Jie Zou PHY 1151G Department of Physics