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### Standing Waves on a String

• 1. Standing Waves on a String By: Aysha Allard Brown http://pixshark.com/standing-waves-on-a-string.htm
• 2. 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/
• 3. 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?
• 4. 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)
• 5. Harmonics 1st→ Nodes: 2 Antinodes: 1 2nd→Nodes: 3 Antinodes: 2 3rd→Nodes: 4 Antinodes: 3 •=node • • • • • • • • • ○=antinode ○ ○ ○ ○ ○ ○ 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
• 6. What is the difference between a standing wave and a travelling wave? Travelling Wave Standing Wave The 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
• 7. 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
• 8. 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
• 9. 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.
• 10. 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 Hz 2) 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
• 11. 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
• 12. 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.
• 13. Solution Answer: 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
• 14. YouTube Videos Standing Waves: Demo https://www.youtube.com/watch?v=-gr7KmTOrx0 Standing Waves: Calculations https://www.youtube.com/watch?v=QcoQvzNQp6Q
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