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Chapter 18

Superposition and Standing Waves (Cont.)

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Outline

Standing waves Standing waves in a string fixed at

both ends Standing waves in air columns Resonance

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Standing waves Standing wave: An oscillation

pattern with a stationary outline that results from the superposition of two identical waves traveling in opposite directions.

y1 = A sin(kx - t), y2 = A sin(kx + t)

y = y1 + y2 = (2A sinkx) cos t Nodes: The points of zero

displacement are called nodes; Antinodes: The positions in the medium at which the maximum displacement of 2A occurs.

The distance between adjacent antinodes = /2; The distance between adjacent nodes = /2; The distance between a node and an adjacent antinode = /4.

Important features for a standing wave:• It is not a traveling wave.

• Every particle of the medium oscillates in simple harmonic motion with the same frequency .• The amplitude of a given particle depends on the location x of the particle.

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Problem #14 Two waves in a long string are given by

where y1, y2, and x are in meters and t is in seconds.

(a) Determine the positions of the nodes of the resulting standing wave.

(b) What is the maximum transverse position of an element of the string at the position x = 0.400 m?

y1 0.0150m cos

x2

40t

y2 0.0150m cos

x2

40t

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Standing waves in a string fixed at both ends

Normal modes: Natural patterns of oscillation, each of which has a characteristic frequency.

Wavelengths of the normal modes: n = 2L/n, n=1,2,3…

Index n: Refers to the nth normal mode of oscillation.

Natural frequencies of normal modes:

Fundamental (frequency): fn = nf1- harmonic series and the

normal modes are called harmonics.

T

Lf

2

11

3,... 2, 1, n ,22

T

L

n

L

vn

vf

nn

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Example 18.4 The high E string on a

guitar measures 64.0cm in length and has a fundamental frequency of 330Hz. By pressing down on it at the first fret (Fug.18.8), the string is shortened so that it plays an F note that has a frequency of 350 Hz. Haw far is the fret from the neck end of the string?

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Standing waves in air columns

Standing waves can be set up in a tube of air.(1) In a pipe open at

both ends, natural frequencies of oscillation: fn=nv/(2L), n=1,2,3…

(2) In a pipe closed at one end and open at the other, natural frequencies of oscillation: fn = nv/(4L), n=1,3,5…

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Resonance Resonance: If a periodic force is

applied to a system, the amplitude of the resulting motion is greater than normal when the frequency of the applied force is equal or nearly equal to one of the natural frequencies (resonant frequencies) of the system.

An example of resonance

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Example 18.6 A vertical pipe open at both ends

is partially submerged in water, and a tuning fork vibrating at an unknown frequency is placed near the top of the pipe. The length L of the air column can be adjusted by moving the pipe vertically. The sound waves generated by the fork are reinforced when L corresponds to one of the resonance frequencies of the pipe. For a certain tube, the smallest value of L for which a peak occurs in the sound intensity is 9.00 cm. What are

(a) the frequency of the tuning fork?

(b) the value of L for the next two resonance frequencies?

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Homework

Ch. 18, Problems: #14, 25, 42, 46. Hints on #42:

The change in volume V and the change in the height h of a cylinder are related by: V = (r2) h or h = V/(r2).

Since the volume flow rate is R, i.e. V/t = R, the height flow rate is: h/t = R/(r2).