Oscillations and Waves 3

Question 1

Superposition and Interference

Description

Answer 1

The superposition of two harmonic waves of equal amplitude, wave number and frequency, but phase difference 𝛿, results in a harmonic wave of the same wave number and frequency, but differing in phase and amplitude from each of the two waves.

Question 2

Superposition and Interference

Equation

Answer 2

y = y1 + y2
= y0 sin(kx - ωt) + y0 sin(kx - ωt + 𝛿)
=2y0 cos(𝛿/2) sin(kx - ωt + 𝛿/2)

Question 3

Constructive Interference

Answer 3

If waves are in phase or differ in phase by an integer times 2π, then the amplitude of the waves add, and the interference is constructive.

Question 4

Destructive Interference

Answer 4

If waves differ in phase by π or by an odd integer times π, then the amplitudes subtract and the interference is destructive.

Question 5

Beats

Answer 5

Beats are the result of the interference of two waves of slightly different frequencies.
The beat frequency equals the difference in the frequencies of the two waves.
f beat = Δf

Question 6

Phase Difference Due to a Path Difference

Answer 6

𝛿 = kΔx = 2π Δx/λ

𝛿 = phase difference
Δx = path difference

Question 7

Standing Waves

Description

Answer 7

Standing waves occur for certain frequencies and wavelengths when waves are confined in space. If they occur, then each point of the system oscillates in simple harmonic motion and any two points not at nodes move either in phase or 180° out of phase.

Question 8

Standing Waves

Wavelength

Answer 8

The distance between a node and an adjacent antinode is a quarter of a wavelength

Question 9

Standing Waves

String Fixed at Both Ends

Answer 9

For a string fixed at both ends, there is a node at each end so that the integral number of half wavelengths must fit into the length of the string.
The standing wave condition in this case is:
L = nλn/2 n = 1, 2, 3, …
fn = nf1 n = 1, 2, 3, …

and f1 = v/2L is the lowest or fundamental frequency

Question 10

Standing Wave

Organ Pipe Open at Both Ends

Answer 10

Standing sound waves in the air in a pipe that is open at both ends have a pressure node (and a displacement antinode) near each end so that the standing-wave condition is the same as for a string fixed at both ends.

Question 11

Standing Wave

String Fixed at One End, Free at the Other

Answer 11

For a string with one end fixed and one end free, there is a node at the fixed end and an antinode at the free end. An integral number of quarter wavelengths must fit into the length of the string.
The standing wave condition in this case is:
L = nλn/4 n = 1, 3, 5, …
Only the odd harmonics are present, their frequencies are given by:
fn = nf1
where f1 = v/4L

Question 12

Standing Wave

Organ Pipe Open at One End and Stopped at the Other

Answer 12

Standing sound waves in a pipe that is open at one end and stopped at the other end have a displacement antinode at the open end and a displacement node at the stopped end. The standing wave equation is the same as for a string fixed at one end

Question 13

Wave Functions For Standing Waves

Answer 13

yn (x,t) = An sin(xkn) cos(tωn + 𝛿n)

where kn = 2π/λn and ωn = 2πfn

Question 14

Superposition of Standing Waves

Answer 14

A vibrating system typically does not vibrate in a single harmonic mode, but in a superposition of the allowed harmonic modes.

Question 15

Harmonic Analysis and Synthesis

Answer 15

Sounds of different tone quality contain different mixtures of harmonics. The analysis of a particular tone in terms of its harmonic content is called harmonic analysis. Harmonic synthesis is the construction of a tone by the addition of harmonics.

Question 16

Wave Packets

Answer 16

A wave pulse can be represented by a continuous distribution of harmonic waves. The range of frequencies Δω is related to the width in time Δt, and the range of wave numbers Δk is related to the width in space Δx.

Question 17

Wave Packets

Frequency and Time Ranges

Answer 17

ΔωΔt ~ 1

Question 18

Wave Packets

Wave Number and Space Ranges

Answer 18

ΔkΔx ~ 1

Question 19

Dispersion

Answer 19

In a nondispersive medium, the phase velocity is independent of frequency, and a pulse (wave packet) travels without change in shape. In a dispersive medium, the phase velocity varies with frequency, and the pulse changes shape as it moves. The pulse moves with a velocity called the group velocity of the packet.