Standing Waves on a StringBy: Aysha Allard Brown

http://pixshark.com/standing-waves-on-a-string.htm

Definitions

Node: a point on a standing wave along a string that does not move

ex. the end points of a string

Antinode: the region of maximum amplitude between two adjacent nodes in a standing wave along a string

Incident Wave: a wave that strikes a boundary, where it is then reflected/flipped

Reflected Wave: the reflected/flipped incident wave (180°)

λ: wavelength (m)

L: length of the string (m)

http://www.clemson.edu/ces/phoenix/labs/224/standwave/

In a standing wave, the string is held fixed at the end points

Some specific points do not move (nodes) and the points between them vibrate (antinodes)

The maximum amplitude of the wave corresponds to the antinode

The minimum amplitude of the wave corresponds to the node

A standing wave is a result of two similar waves travelling in opposite directions

What is a standing wave?

What is a standing wave?

Different frequencies are associated with different wave patterns for standing waves

These frequencies along with their corresponding patterns are referred to as harmonics

A harmonic is an integer which is a multiple of the fundamental frequency (the lowest frequency→ when the number of nodes=2)

Harmonics

1st→ Nodes: 2 Antinodes: 1

2nd→Nodes: 3 Antinodes: 2

3rd→Nodes: 4 Antinodes: 3

•=node

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○=antinode

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○ ○http://cnx.org/contents/07970e19-2e42-4b8e-9a7d-2749bf5d8529@15/Standing_Waves_and_Musical_Ins

http://hep.physics.indiana.edu/~rickv/Standing_Waves_on_String.html

What is the difference between a standing wave and a travelling wave?

Travelling Wave Standing WaveThe wave is not confined to a given space

The wave is confined to a given space (fixed ends)

Transports energy from one point to another

Does not transport energy from one point to another

The waves interfere The waves interfere

Can have any value for frequency

Frequency is quantized (only certain values are allowed) http://www.chegg.com/homework-help/questions-and-answers/standing-waves-

guitar-string-form-whenwaves-traveling-string-reflect-point-thestring-tied--q445454

Equations

1) T- tension force (N)

m- mass of string (kg)

L- length of string (m)

f- frequency (Hz)http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

Equations

2) n- harmonic number

(# of antinodes)

λ- wavelength (m)

L- length of string (m)

http://www.physicsclassroom.com/class/waves/Lesson-4/Mathematics-of-Standing-Waves

Question #1

A string is 5.0 meters long and is vibrating at the 3rd harmonic. The string vibrates up and down with 48 complete vibrational cycles in 20 seconds. Determine the frequency, period, wavelength and speed for this wave.

Solution (pt. 1)1) Determine the frequency

The frequency refers to how often a point on the string goes back-and-forth (hence the number of cycles per unit time).

Therefore, f = (48 cycles) / (20 seconds) = 2.4 Hz2) Determine the period

The period is referring to the time needed for one complete cycle of vibration to pass a given point. The period and frequency share a reciprocal relationship so therefore,

T= 1 / f

T = 1 / (2.4 Hz) = 0.417 seconds

Solution (pt. 2)3) Determine the wavelength

The wavelength for the 3rd harmonic is represented by λ=2/3*L. The length of the string is given in the question, 5.0m.

Therefore, λ = 2/3 * (5.0m) = 10/3m = 3.3m

4) Determine the wave speed

Since we calculated the frequency and wavelength above, we can now find the wave speed by using the following formula:

v = λf = (3.3m)(2.4Hz) = 7.9 m/s

Question #2

Which statement is CORRECT about the amplitude of a standing wave created from the interference of two waves, each with amplitude ‘A'?

A. The amplitude reaches its maximum value of 2A at the anti-nodes.

B. The amplitude reaches its maximum value of A at the nodes.

C. The amplitude reaches its maximum value of A at the anti-nodes.

D. The amplitude reaches its maximum value of 2A at the nodes.

SolutionAnswer: A) The amplitude reaches its maximum value of 2A at the anti-nodes.

Both interfering waves have the same amplitude “A” in the same direction. Hence, both waves have a positive/upward amplitude. As the two waves meet the medium’s shape will become the net of the two interfering waves. This is known as constructive interference, where the resultant wave is bigger than the two original interfering waves. The maximum amplitude occurs at the antinodes. It cannot occur at the nodes since these points represent the minimum amplitude and do not move.

http://www.physicsclassroom.com/class/waves/Lesson-3/Interference-of-Waves

YouTube Videos

Standing Waves: Demo

https://www.youtube.com/watch?v=-gr7KmTOrx0

Standing Waves: Calculations

https://www.youtube.com/watch?v=QcoQvzNQp6Q

Thank you.

Bibliography

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

http://www.physicsclassroom.com/class/sound/Lesson-4/Standing-Wave-Patterns

http://hep.physics.indiana.edu/~rickv/Standing_Waves_on_String.html

http://www.physicsclassroom.com/class/waves/Lesson-4/Harmonics-and-Patterns

http://astarmathsandphysics.com/a-level-physics-notes/waves-and-oscillations/a-level-physics-notes-the-difference-between-standing-waves-and-travelling-waves.html