Dr. Jie Zou PHY 1151G Department of Physics1 Chapter 14 Waves and Sound (Cont.)

Dr. Jie Zou PHY 1151G Department of Physics 1

Chapter 14

Waves and Sound (Cont.)

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Outline Standing waves

Standing waves on a string Vibrating columns of air

Sound intensity Doppler effect

Moving observer Moving source

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)

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Standing waves on a string

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

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L

vn2

Node (N)

Antinode (A)

An application – guitar

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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

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L

vn4

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 440.0 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)

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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/(4r2). SI unit: W/m2.

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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.

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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.

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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.

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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.

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Example: Doppler effect 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?

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Homework See online homework assignment

on www.masteringphysics.com