standing waves on strings

Standing Waves on Strings

Physics 101

What is a Standing Wave?

A wave that does not appear to travel.

Created by two harmonic waves of equal amplitude, wavelength and frequency, moving in opposite directions.

Nodes and Antinodes Nodes occur where the amplitude is always zero.

Node Node NodeNode

Nodes and Antinodes Antinodes occur where the amplitude moves between -2A and 2A, the

maximum amplitude. (A – amplitude of constituent harmonic waves)

Antinode

Antinode

λ4λ2

3 λ4

λ 5λ4

λ2

λ2

Node to Node

Antinode to Antinode

Nodes and Antinodes Consecutive nodes and consecutive antinodes are (half a wavelength)

apart

The distance between a node and the nearest antinode is .

Nodes and Antinodes

λ4λ2

3 λ4

λ 5λ4

λ4

Antinode to Node

λ4

Node to Antinode

Frequency of Standing Waves on Strings m = mode number (1, 2, 3, …) (see next slide) v = wave speed = wavelength L = length of string T = tension of the string M = mass of string Recall that is the linear mass density of the string .

Normal Modes

Represented as n in the diagram

The mode number corresponds to the number of antinodes on the vibrating string

A mode of 2 is double the frequency of the fundamental frequency (n = 1)

The relationship between the first harmonic and other harmonics is represented by this equation:

m is a positive integer > 0

Harmonics/Resonant Frequencies

The allowed frequencies represented by the equation:

In lab 5, you found the fundamental frequency, the lowest frequency that results in a single antinode at its maximum frequency.

Knowing the fundamental frequency, we can find the harmonics/resonant frequencies.

Questions

#1 - The frequency of the fourth harmonic is…

A) Same as the frequency of the 2nd harmonic.

B) times greater than the frequency of the 3rd harmonic.

C) Triple the frequency of the fundamental frequency.

D) Double the frequency of the 3rd harmonic.

The frequency of the fourth harmonic is…

A) Same as the frequency of the 2nd harmonic.

B) times greater than the frequency of the 3rd harmonic.

C) Triple the frequency of the fundamental frequency.

D) Double the frequency of the 3rd harmonic.

#2 - A guitar string that is plucked produces a standing wave. Its angular velocity is 5 rad/m. The amplitude is 8mm. The length the string vibrating is 40cm. The mass of the string is 0.2g. The tension of the string is equal to the weight of a 36kg mass on the moon.

A) Imagine that the standing wave is produced by two harmonic waves of equal amplitude, wavelength, and frequency travelling opposite directions. What would be the amplitude of the two constituent waves?

B) Write an equation to represent the amplitude of the standing wave.

C) What is the distance between a node and the nearest antinode?

D) What is the wave speed in the string?

E) What is the fifth harmonic?

A guitar string that is plucked produces a standing wave. Its angular velocity is 5 rad/m. The amplitude is 8mm. The length the string vibrating is 40cm. The mass of the string is 0.2g. The tension of the string is equal to the weight of a 36kg mass on the moon.

A) Imagine that the standing wave is produced by two harmonic waves of equal amplitude, wavelength, and frequency travelling opposite directions. What would be the amplitude of the two constituent waves?

The amplitude of the standing wave is double the amplitude of the constituent wave.

2A = 8mm

A = 4mm

A guitar string that is plucked produces a standing wave. Its angular velocity is 5.0 rad/m. The amplitude is 8.0mm. The length the string vibrating is 40cm. The mass of the string is 0.2g. The tension of the string is equal to the weight of a 36kg mass on the moon.

B) Write an equation to represent the amplitude of the standing wave.

A(x) = A sin(ωx)

A(x) = (8.0 mm)sin(5.0x)

A guitar string that is plucked produces a standing wave. Its angular velocity is 5 rad/m. The amplitude is 8mm. The length the string vibrating is 40cm. The mass of the string is 0.2g. The tension of the string is equal to the weight of a 36kg mass on the moon.

C) What is the distance between a node and the nearest antinode?

Let us find the distance between nodes.

Recall from the text (eq 14-44): , where m =

We know that ω = 5rad/m and

5 = =

m = 0 x = 0

m = 1 x = = =

m = 2 x = =

(continued on next slide)

A guitar string that is plucked produces a standing wave. Its angular velocity is 5 rad/m. The amplitude is 8mm. The length the string vibrating is 40cm. The mass of the string is 0.2g. The tension of the string is equal to the weight of a 36kg mass on the moon.

C) What is the distance between a node and the nearest antinode?

The location of the nodes are at: x = 0, , , …

The location of the first antinode is at: = =

We can find the distance between any node the nearest antinode by finding the distance between the first node and antinode.

The first node is at x = 0

The first antinode is at x =

- 0 =

A guitar string that is plucked produces a standing wave. Its angular velocity is 5 rad/m. The amplitude is 8mm. The length the string vibrating is 40cm. The mass of the string is 0.2g. The tension of the string is equal to the weight of a 36kg mass on the moon.

D) What is the wave speed in the string?

=

A guitar string that is plucked produces a standing wave. Its angular velocity is 5 rad/m. The amplitude is 8mm. The length the string vibrating is 40cm. The mass of the string is 0.2g. The tension of the string is equal to the weight of a 36kg mass on the moon.

E) What is the fifth harmonic?

Find the fundamental frequency first.

Works Cited

Wave on a String PhET simulation (wave diagrams) https://phet.colorado.edu/en/simulation/wave-on-a-string

Physics 101 Textbook (definitions and equations) Physics for Scientists and Engineers: An Interactive

Approach, 1st Edition

Robert Hawkes, Javed Iqbal, Firas Mansour, Marina Milner-Bolotin, and Peter Williams