1. Standing Waves and Resonance Standing Wave: “Standing waves” are formed from two or more traveling waves that collide and are “in tune” with one another in such a way that their amplitudes add or subtract in repetitive ways. This is called “Resonance”. Wave moving right Wave moving left

2. Standing Waves and Resonance Given a string whose length is L: The longest standing wave possible is called the “Fundamental”, or “1 st harmonic”. L

3. Standing Waves and Resonance What fraction of a sine wave is this? L

4. Standing Waves and Resonance L What fraction of a sine wave is this? ½ wave Therefore, using “ L”, how long is one full sine wave ( )

5. Standing Waves and Resonance What fraction of a sine wave is this? ½ wave Therefore, using “ L”, how long is one full sine wave ( )  2L L

6. Standing Waves and Resonance 1 st harmonic 2 nd harmonic 3 rd harmonic 4 th harmonic

7. Standing Waves and Resonance 1 st Harmonic: 2 nd Harmonic: 3 rd Harmonic: 4 th Harmonic: L = 2L = ___?

8. Standing Waves and Resonance 1 st Harmonic: 2 nd Harmonic: 3 rd Harmonic: 4 th Harmonic: L = 2L = ___? = L

9. Standing Waves and Resonance 1 st Harmonic: 2 nd Harmonic: 3 rd Harmonic: 4 th Harmonic: L = 2L = ___? = L = 2L/3

10. Standing Waves and Resonance 1 st Harmonic: 2 nd Harmonic: 3 rd Harmonic: 4 th Harmonic: L = 2L = L = 2L/3 = L/2

11. Standing Waves and Resonance What’s the pattern here? In general, where: Resonant wavelength formula

12. Standing Waves and Resonance Nomenclature: Node Antinode Node: a region of zero amplitude Antinode: a region of maximum amplitude

13. Standing Waves and Resonance Resonant Frequencies Of Harmonic Standing Waves: We learned = 2L/n since v = f,

14. Standing Waves and Resonance Resonant Frequencies Of Harmonic Standing Waves: We learned = 2L/n since v = f,

15. Standing Waves and Resonance Resonant Frequencies Of Harmonic Standing Waves: We learned = 2L/n since v = f,

16. Standing Waves and Resonance where: Resonant frequency formula Resonant Frequencies Of Harmonic Standing Waves: We learned = 2L/n since v = f,

17. Standing Waves and Resonance Therefore the 1 st resonant frequency corresponds to the 1 st resonant wavelength… --- 1 st Harmonic --- 2 nd Harmonic --- 3 rd Harmonic --- 4 th Harmonic

18. Standing Waves and Resonance

19. What two physical characteristics play a role in determining the velocity on a string?

20. Standing Waves and Resonance What two physical characteristics play a role in determining the velocity on a string? 1.Tension 2.Mass, or inertia

21. Standing Waves and Resonance What two physical characteristics play a role in determining the velocity on a string? 1.Tension 2.Mass, or inertia What do we expect the relationship with velocity to be?

22. Standing Waves and Resonance What two physical characteristics play a role in determining the velocity on a string? 1.Tension 2.Mass, or inertia We expect the relationship to be as follows:

23. Standing Waves and Resonance The mass is applied by using density; specifically linear density: linear density surface density volume density

24. Standing Waves and Resonance Finally, the equation used to find velocity can be described: F = tension  = linear density Units = m/s

25. Standing Waves and Resonance Wave Velocity on string: where F = Force Problem: A force of 45 Newtons is applied to a string with a mass of 0.012 kg and a length of 1.5 meters. Find the velocity of a wave on that string:

26. Standing Waves and Resonance Sound Velocity on string: where F = Force, and Example: A force of 45 Newtons is applied to a string with a mass of 0.012 kg and a length of 1.5 meters. Find the velocity of a wave on that string: