Waves: Types, Properties, and Interference

1. Chapter 17 Waves

2. Wave Motion • Fundamental to physics (as important as particles) • A wave is the motion of a disturbance • All waves carry energy and momentum • Mechanical waves require • Some source of disturbance • A medium that can be disturbed • Some physical connection between or mechanism though which adjacent portions of the medium influence each other

3. Types of Waves – Traveling Waves • Flip one end of a long rope that is under tension and fixed at one end • The pulse travels to the right with a definite speed • A disturbance of this type is called a traveling wave

4. Types of Waves – Transverse • In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion

5. Types of Waves – Longitudinal • In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave • A longitudinal wave is also called a compression wave

6. Other Types of Waves • Waves may be a combination of transverse and longitudinal • Mainly consider periodic sinusoidal waves

7. Waveform – A Picture of a Wave • The brown curve is a “snapshot” of the wave at some instant in time • The blue curve is later in time • The high points are crests of the wave • The low points are troughs of the wave

8. Longitudinal Wave Represented as a Sine Curve • A longitudinal wave can also be represented as a sine curve • Compressions correspond to crests and stretches correspond to troughs • Also called density waves or pressure waves

9. Amplitude and Wavelength • Amplitude is the maximum displacement of string above the equilibrium position • Wavelength, λ, is the distance between two successive points that behave identically

10. Speed of a Wave • v = ƒ λ • Is derived from the basic speed equation of distance/time • This is a general equation that can be applied to many types of waves

11. Speed of a Wave on a String • The speed of wave on a stretched rope under some tension, F • m is called the linear density • The speed depends only upon the properties of the medium through which the disturbance travels

12. Example String vibrates at 10 hz and a snapshot. Determine wavelength, period, amplitude, speed.

13. Example Mass and length of the string are 0.9 kg and 8 m. What is the speed of wave on the string?

14. Wave fronts & rays • Wave fronts – locate crests of waves • Ripples from a pebble dropping in a pond • concentric arcs • The distance between successive wave fronts is the wavelength • Rays are the radial lines pointing out from the source and perpendicular to the wave fronts

15. Plane Wave • Far away from the source, the wave fronts are nearly parallel planes • The rays are nearly parallel lines • A small segment of the wave front is approximately a plane wave

16. Reflection of Waves • Waves reflect when they hit boundaries • Fixed end: wave inverts upon reflection • Free end: no inversion

17. Superposition Principle • Two traveling waves can meet and pass through each other without being destroyed or even altered • Waves obey the Superposition Principle • If two or more traveling waves are moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point • Actually only true for waves with small amplitudes

18. Constructive Interference • Two waves, a and b, have the same frequency and amplitude • Are in phase • The combined wave, c, has the same frequency and a greater amplitude

19. Constructive Interference in a String • Two pulses are traveling in opposite directions • The net displacement when they overlap is the sum of the displacements of the pulses • Note that the pulses are unchanged after the interference

20. Destructive Interference • Two waves, a and b, have the same amplitude and frequency • They are 180° out of phase • When they combine, the waveforms cancel

21. Destructive Interference in a String • Two pulses are traveling in opposite directions • The net displacement when they overlap is decreased since the displacements of the pulses subtract • Note that the pulses are unchanged after the interference

22. Standing Waves • When a traveling wave reflects back on itself, it creates traveling waves in both directions • The wave and its reflection interfere according to the superposition principle • With exactly the right frequency, the wave will appear to stand still • This is called a standing wave

23. Standing Waves, cont • A node occurs where the two traveling waves have the same magnitude of displacement, but the displacements are in opposite directions • Net displacement is zero at that point • The distance between two nodes is ½λ • An antinode occurs where the standing wave vibrates at maximum amplitude • The distance between two antinodes is ½λ • Distance between node and antinode λ/4

24. Standing Waves on a String • Nodes must occur at the ends of the string because these points are fixed

25. Standing Waves, cont. • The pink arrows indicate the direction of motion of the parts of the string • All points on the string oscillate together vertically with the same frequency, but different points have different amplitudes of motion

26. Resonance • Can have resonance in strings (these are actually standing waves) • Amplitude increases • How to determine resonance frequencies?

27. Standing Waves on a String, final • The lowest frequency of vibration (b) is called the fundamental frequency

28. Standing Waves on a String – Frequencies • ƒ1, ƒ2, ƒ3 form a harmonic series • ƒ1 is the fundamental and also the first harmonic • ƒ2 is the second harmonic (1st overtone) • Waves in the string that are not in the harmonic series are quickly damped out • In effect, when the string is disturbed, it “selects” the standing wave frequencies

29. Example A guitar has 0.6 m long string. Wave speed on the string is 420 m/s. What are the frequencies of the first few harmonics?

30. Example String 80 cm long is driven with frequency of 120 Hz when both ends fixed. There are 4 nodes in the middle of the string. Find speed of wave on string?

31. Producing a Sound Wave • Sound waves are longitudinal waves traveling through a medium • A tuning fork can be used as an example of producing a sound wave

32. Using a Tuning Fork to Produce a Sound Wave • A tuning fork will produce a pure musical note • As the tines vibrate, they disturb the air near them • As the tine swings to the right, it forces the air molecules near it closer together • This produces a high density area in the air • This is an area of compression

33. Using a Tuning Fork, cont. • As the tine moves toward the left, the air molecules to the right of the tine spread out • This produces an area of low density • This area is called a rarefaction

34. Using a Tuning Fork, final • As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the fork • A sinusoidal curve can be used to represent the longitudinal wave • Crests correspond to compressions and troughs to rarefactions

35. Categories of Sound Waves • Audible waves • Lay within the normal range of hearing of the human ear • Normally between 20 Hz to 20,000 Hz • Infrasonic waves • Frequencies are below the audible range • Earthquakes are an example • Ultrasonic waves • Frequencies are above the audible range • Dog whistles are an example

36. Applications of Ultrasound • Can be used to produce images of small objects • Widely used as a diagnostic and treatment tool in medicine • Ultrasounds to observe babies in the womb • Cavitron Ultrasonic Surgical Aspirator (CUSA) used to surgically remove brain tumors • Ultrasonic ranging unit for cameras

37. Speed of Sound, General • The speed of sound is higher in solids than in gases • The speed is slower in liquids than in solids

38. Speed of Sound in Air • 331 m/s is the speed of sound at 0°C and 1 atm • Changes with temperature • T in °C • At 20 °C, 343 m/s • In other substances in He: 1000 m/s in Water: 1500 m/s in Al: 5000 m/s

39. Standing Waves in Air Columns • If one end of the air column is closed, a node must exist at this end since the movement of the air is restricted • If the end is open, the elements of the air have complete freedom of movement and an antinode exists

40. Tube Open at Both Ends

41. Resonance in Air Column Open at Both Ends • In a pipe open at both ends, the natural frequency of vibration forms a series whose harmonics are equal to integral multiples of the fundamental frequency

42. Tube Closed at One End Closed pipe

43. Resonance in an Air Column Closed at One End • The closed end must be a node • The open end is an antinode • There are no even multiples of the fundamental harmonic

44. Example An open organ pipe has a fundamental frequency of 660 Hz at 0 C and 1 atm. • Frequency of 2nd overtone? • Fundamental at 20 C? • Replacing air with He?