1. Superposition and Standing Waves EXAMPLES
Chapter 18
Superposition and Standing Waves EXAMPLES

2. Chapter 18 Superposition and Standing Waves: EXAMPLES

3. Example 18-1: Formation of a Standing Wave
Assume two waves are traveling in opposite direction to produce a standing wave
y = (2A sinkx) cosωt = ymax cos ωt
y1 = 4 sin(3x – 2t) and y2 = 4 sin(3x + 2t )
x and y in centimeters and t in seconds
(A) Find the amplitude of the SHM (2A sinkx) of the element located at x = 2.3 cm
A = 4 cm, k = 3 rad/cm,  = 2 rad/s
From the resultant wave will have:
ymax = 2Asin(kx) = 8sin(kx) 
= 8 sin(3 · 2.3) = 4.6 cm

4. Example 18-1: Formation of a Standing Wave, final
(B) Find the position of the nodes and antinodes if one end of the string is at x = 0
k = 3 rad/cm = 2/   = 2/3 cm 
Nodes: Antinodes:
(C) Find ymax for an element located at an antinode
An antinode occurs at a point of maximum displacement
ymax = ±2A = ± 8.0 cm

5. Example 18-2: Harmonics
A middle “C” on a piano has a fundamental frequency of 262 Hz. What are the next two harmonics of this string?
ƒ1 = 262 Hz
ƒ2 = 2ƒ1 = 524 Hz
ƒ3 = 3ƒ1 = 786 Hz

6. Example 18-3: More About Harmonics
The first A above middle C has a fundamental frequency of 440 Hz. If both strings have the same linear mass density  and length L, find the ratio between the two tensions.

7. Example 18-4: Resonance
If pendulum A is set into motion, the other pendulums begin to oscillate due to waves transmitted through the beam
Pendulum C has a greater amplitude than B or D
C’s length is closest to A’s and so C’s natural frequency is closest to the driving frequency from A

8. Standing Wave on a String, Example Set-Up
One end of the string is attached to a vibrating blade.
The other end passes over a pulley with a hanging mass attached to the end.
This produces the tension in the string.
The string is vibrating in its second harmonic.
Section 18.3

9. Material from the book to Study!!!
Material for the Midterm
Material from the book to Study!!!
Objective Questions: 1-6
Conceptual Questions: 1-2
Problems: