1 / 27

waves_03

waves_03. TRANSVERSE WAVES ON STRINGS. Animations courtesy of Dr. Dan Russell, Kettering University. waves_03: MINDMAP SUMMARY - TRANSVERSE WAVES ON STRINGS.

mahlah
Download Presentation

waves_03

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. waves_03 TRANSVERSE WAVES ON STRINGS Animations courtesy of Dr. Dan Russell, Kettering University

  2. waves_03: MINDMAP SUMMARY - TRANSVERSE WAVES ON STRINGS Travelling transverse waves, speed of propagation, wave function, string tension, linear density, reflection (fixed and free ends), interference, boundary conditions, standing waves, stationary waves, SHM, string musical instruments, amplitude, nodes, antinodes, period, frequency, wavelength, propagation constant (wave number), angular frequency, normal modes of vibrations, natural frequencies of vibration, fundamental, harmonics, overtones, harmonic series, frequency spectrum, radian, phase, sinusoidal functions

  3. TRANSVERSE WAVES ON STRINGS Wave speed v (speed of propagation) • The string (linear density ) must be under tension FTfor wave to propagate • increases with increasing tension FT • decreases with increasing mass per unit length  • independent of amplitude or frequency linear density

  4. Problem 1 A string has a mass per unit length of 2.50 g.m-1 and is put under a tension of 25.0 N as it is stretched taut along the x-axis. The free end is attached to a tuning fork that vibrates at 50.0 Hz, setting up a transverse wave on the string having an amplitude of 5.00 mm. Determine the speed, angular frequency, period, and wavelength of the disturbance. [Ans: 100 m.s-1, 3.14x102 rad.s-1, 2.00x10-2 s, 2.00 m] use the ISEE method

  5. Solution 1 F = 25.0 N  = 2.50 g.m-1 = 2.50×10-3 kg.m-1f = 50.0 Hz A = 5.00 mm = 5×10-3 m v = ? m.s-1 = ? rad.s-1T = ? s  = ? m Speed of a transverse wave on a string speed of a wave

  6. Pulse on a rope • When pulse reaches the attachment point at the wall the pulse is reflected • If attachment is fixed the pulse inverts on reflection • If attachment point can slide freely of a rod, the pulse reflects without inversion • If wave encounters a discontinuity, there will be some reflection and some transmission • Example: two joined strings, different . What changes across the discontinuity - frequency, wavelength, wave speed?

  7. Reflection of waves at a fixed end Reflection of waves at a free end Reflected wave is inverted  / 2 PHASE CHANGE Reflected wave is not inverted ZERO PHASE CHANGE

  8. Refection of a pulse - string with boundary condition at junction like a fixed end CP 510

  9. Refection of a pulse - string with boundary condition at junction like a free end CP 510

  10. Refection of a pulse - string with boundary condition at junction like a fixed end Refection of a pulse - string with boundary condition at junction like a free end

  11. STANDING WAVES • If we try to produce a traveling harmonic wave on a rope, repeated reflections from the end produces a wave traveling in the opposite direction - with subsequent reflections we have waves travelling in both directions • The result is the superposition (sum) of two waves traveling in opposite directions • The superposition of two waves of the same amplitude travelling in opposite directions is called a standing wave • Examples: transverse standing waves on a string with both ends fixed (e.g. stringed musical instruments); longitudinal standing waves in an air column (e.g. organ pipes and wind instruments)

  12. Standing waves on strings Two waves travelling in opposite directions with equal displacement amplitudes and with identical periods and wavelengths interfere with each other to give a standing (stationary) wave (not a travelling wave - positions of nodes and antinodes are fixed with time) amplitude oscillation each point oscillates with SHM, period T = 2 /  CP 511

  13. A string is fixed at one end and driven by a small amplitude sinusoidal driving force fdat the other end. The natural frequencies of vibration of the string (nodes at each end) are fo = 150 Hz, 300 Hz, 450 Hz, 600 Hz, ... The string vibrates at the frequency of the driving force. When the string is excited at one of its natural frequencies, large amplitude standing waves are set up on the string (resonance). fd= 150 Hz fd= 450 Hz fd= 200 Hz fundamental 3rd harmonic Click the next image after an animation stops

  14. Standing waves on strings • String fixed at both ends • A steady pattern of vibration will result if the length corresponds to an integer number of half wavelengths • In this case the wave reflected at an end will be exactly in phase with the incoming wave • This situations occurs for a discrete set of frequencies Boundary conditions  Speed transverse wave along string Natural frequencies of vibration CP 511

  15. Why do musicians have to tune their string instruments before a concert? CP 518

  16. Modes of vibrations of a vibrating string fixed at both ends Natural frequencies of vibration Fundamental node antinode CP 518

  17. Harmonic series Nth harmonic or (N-1)th overtone N = 2L / N = 1 / NfN = N f1 N = 3 3nd harmonic (1st overtone) 3 = L = 3 / 2 f3 = 2 f1 N = 2 2nd harmonic (1st overtone) 2 = L = 1 / 2 f2 = 2 f1 N = 1 fundamental or first harmonic 1 = 2L f1 = (1/2L).(FT / ) Resonance (“large” amplitude oscillations) occurs when the string is excited or driven at one of its natural frequencies. CP 518 22 23

  18. violin – spectrum viola – spectrum CP 518

  19. Problem solving strategy: I S E E Identity: What is the question asking (target variables) ? What type of problem, relevant concepts, approach ? Set up: Diagrams Equations Data (units) Physical principals Execute: Answer question Rearrange equations then substitute numbers Evaluate: Check your answer – look at limiting cases sensible ? units ? significant figures ? PRACTICE ONLY MAKES PERMANENT

  20. Problem 2 A guitar string is 900 mm long and has a mass of 3.6 g. The distance from the bridge to the support post is 600 mm and the string is under a tension of 520 N. 1 Sketch the shape of the wave for the fundamental mode of vibration 2 Calculate the frequency of the fundamental. 3 Sketch the shape of the string for the sixth harmonic and calculate its frequency. 4 Sketch the shape of the string for the third overtone and calculate its frequency.

  21. Solution 2 L1 = 900 mm = 0.900 m m = 3.6 g = 3.610-3 kg L = 600 mm = 0.600 m FT = 520 N  = m / L1 = (3.610-3 / 0.9) kg.m-1 = 0.004 kg.m-1 v = (FT / ) = (520 / 0.004) m.s-1= 360.6 m.s-1 1 = 2L = (2)(0.600) m = 1.200 m Fundamental frequency f1 = v / 1 = (360.6 / 1.2) Hz = 300 Hz fN = N f1 sixth harmonic N = 6 f6 = (6)(300) Hz = 1800 Hz = 1.8 kHz third overtone = 4th harmonic N = 4 f4 = (4)(300) Hz = 1200 Hz = 1.2 kHz

  22. Problem 3 A particular violin string plays at a frequency of 440 Hz. If the tension is increased by 8.0%, what is the new frequency?

  23. Solution 3 fA = 440 Hz fB = ? Hz FTB = 1.08 FTA A = BA = BLA = LBNA = NB v = fv = (FT / ) string fixed at both ends L = N /2  = 2L / N natural frequencies fN = Nv / 2L = (N / 2L).(FT / ) fB / fA = (FTB / FTA) fB = (440)(1.08) Hz = 457 Hz

  24. Problem 4 Why does a tree howl? The branches of trees vibrate because of the wind. The vibrations produce the howling sound. Length of limb L = 2.0 m Transverse wave speed in wood v = 4.0103 m.s-1 Fundamental L =  / 4  = 4 L v = f  f = v /  = (4.0 103) / {(4)(2)} Hz f = 500 Hz N A Fundamental mode of vibration

  25. Standing waves in membranes NB the positions of the nodes and antinodes http://www.isvr.soton.ac.uk/spcg/tutorial/tutorial/Tutorial_files/Web-standing-membrane.htm

  26. CHLADNI PLATES

  27. Some of the animations are from the web site http://paws.kettering.edu/~drussell/demos.html

More Related