1. An Introduction to Waves and Wave Properties
Waves and Sound
An Introduction to Waves and Wave Properties

2. Animation courtesy of Dr. Dan Russell, Kettering University
Mechanical Wave
A mechanical wave is a disturbance which propagates through a medium with little or no net displacement of the particles of the medium.
Water Waves
Wave “Pulse”
People Wave
Animation courtesy of Dr. Dan Russell, Kettering University

3. Parts of a Wave crest : wavelength 3 equilibrium A: amplitude 2 4 6
x(m) -3
trough
y(m)

4. Speed of a wave
The speed of a wave is the distance traveled by a given point on the wave (such as a crest) in a given interval of time.
v = d/t
d: distance (m)
t: time (s)
v = ƒ
v : speed (m /s)
 : wavelength (m)
ƒ : frequency (s–1, Hz)

5. Period of a wave
T = 1/ƒ
T : period (s)
ƒ : frequency (s-1, Hz)

6. Problem: Sound travels at approximately 340 m/s, and light travels at 3.0 x 108 m/s. How far away is a lightning strike if the sound of the thunder arrives at a location 2.0 seconds after the lightning is seen?

7. Problem: The frequency of an oboe’s A is 440 Hz
Problem: The frequency of an oboe’s A is 440 Hz. What is the period of this note? What is the wavelength? Assume a speed of sound in air of 340 m/s.

8. Types of Waves Refraction and Reflection

9. Wave Types
A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction which the wave moves.
Example: Waves on a String
A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction which the wave moves. These are also called compression waves.
Example: sound

10. Wave types: transverse

11. Wave types: longitudinal

12. Longitudinal vs Transverse

13. Other Wave Types Earthquakes: combination Ocean waves: surface
Light: electromagnetic

14. Reflection of waves
Occurs when a wave strikes a medium boundary and “bounces back” into original medium.
Completely reflected waves have the same energy and speed as original wave.

15. Reflection Types
Fixed-end reflection: The wave reflects with inverted phase.
Open-end reflection: The wave reflects with the same phase
Animation courtesy of Dr. Dan Russell, Kettering University

16. Animation courtesy of Dr. Dan Russell, Kettering University
Refraction of waves
Transmission of wave from one medium to another.
Refracted waves may change speed and wavelength.
Refraction is almost always accompanied by some reflection.
Refracted waves do not change frequency.
Animation courtesy of Dr. Dan Russell, Kettering University

17. Sound is a longitudinal wave
Sound travels through the air at approximately 340 m/s.
It travels through other media as well, often much faster than that!
Sound waves are started by vibration of some other material, which starts the air moving.
Animation courtesy of Dr. Dan Russell, Kettering University

18. Hearing Sounds
We hear a sound as “high” or “low” depending on its frequency or wavelength. Sounds with short wavelengths and high frequencies sound high-pitched to our ears, and sounds with long wavelengths and low frequencies sound low-pitched. The range of human hearing is from about 20 Hz to about 20,000 Hz.
The amplitude of a sound’s vibration is interpreted as its loudness. We measure the loudness (also called sound intensity) on the decibel scale, which is logarithmic.
© Tom Henderson,

19. Doppler Effect
The Doppler Effect is the raising or lowering of the perceived pitch of a sound based on the relative motion of observer and source of the sound. When a car blowing its horn races toward you, the sound of its horn appears higher in pitch, since the wavelength has been effectively shortened by the motion of the car relative to you. The opposite happens when the car races away.

20. Doppler Effect Stationary source Moving source
Animations courtesy of Dr. Dan Russell, Kettering University
Supersonic source

21. Pure Sounds
Sounds are longitudinal waves, but if we graph them right, we can make them look like transverse waves.
When we graph the air motion involved in a pure sound tone versus position, we get what looks like a sine or cosine function.
A tuning fork produces a relatively pure tone. So does a human whistle.
Later in the period, we will sample various pure sounds and see what they “look” like.

22. Graphing a Sound Wave

23. Complex Sounds
Because of the phenomena of “superposition” and “interference” real world waveforms may not appear to be pure sine or cosine functions.
That is because most real world sounds are composed of multiple frequencies.
The human voice and most musical instruments produce complex sounds.

24. The Oscilloscope
With the Oscilloscope we can view waveforms in the “time domain”. Pure tones will resemble sine or cosine functions, and complex tones will show other repeating patterns that are formed from multiple sine and cosine functions added together.

25. The Fourier Transform
We will also view waveforms in the “frequency domain”. A mathematical technique called the Fourier Transform will separate a complex waveform into its component frequencies.

26. Superposition of Waves

27. Principle of Superposition
When two or more waves pass a particular point in a medium simultaneously, the resulting displacement at that point in the medium is the sum of the displacements due to each individual wave.
The waves interfere with each other.

28. Types of interference.
If the waves are “in phase”, that is crests and troughs are aligned, the amplitude is increased. This is called constructive interference.
If the waves are “out of phase”, that is crests and troughs are completely misaligned, the amplitude is decreased and can even be zero. This is called destructive interference.

29. Constructive Interference
crests aligned with crest
waves are “in phase”

30. Constructive Interference

31. Destructive Interference
crests aligned with troughs
waves are “out of phase”

32. Destructive Interference

33. Sample Problem: Draw the waveform from its two components.

34. Sample Problem: Draw the waveform from its two components.

35. Standing Waves

36. Standing Wave
A standing wave is a wave which is reflected back and forth between fixed ends (of a string or pipe, for example).
Reflection may be fixed or open-ended.
Superposition of the wave upon itself results in a pattern of constructive and destructive interference and an enhanced wave.
Let’s see a simulation.

37. Fixed-end standing waves (violin string)
1st harmonic
2nd harmonic
Animation available at:
3rd harmonic

38. Fixed-end standing waves (violin string)
Fundamental
First harmonic
 = 2L
First Overtone
Second harmonic
 = L
Second Overtone
Third harmonic
 = 2L/3

39. Open-end standing waves (organ pipes)L
Fundamental
First harmonic
 = 2L
First Overtone
Second harmonic
 = L
Second Overtone
Third harmonic
 = 2L/3

40. Mixed standing waves (some organ pipes)L
First harmonic
 = 4L
Second harmonic
 = (4/3)L
Third harmonic
 = (4/5)L

41. Sample Problem
How long do you need to make an organ pipe that produces a fundamental frequency of middle C (256 Hz)? The speed of the sound in air is 340 m/s.
A) Draw the standing wave for the first harmonic
B) Calculate the pipe length.
C) What is the wavelength and frequency of the 2nd harmonic? Draw the standing wave

42. Sample Problem
How long do you need to make an organ pipe whose fundamental frequency is a middle C (256 Hz)? The pipe is closed on one end, and the speed of sound in air is 340 m/s.
A) Draw the situation.
B) Calculate the pipe length.
C) What is the wavelength and frequency of the 2nd harmonic?

43. Resonance and Beats

44. Resonance
Resonance occurs when a vibration from one oscillator occurs at a natural frequency for another oscillator.
The first oscillator will cause the second to vibrate.

45. Beats
“Beats is the word physicists use to describe the characteristic loud-soft pattern that characterizes two nearly (but not exactly) matched frequencies.
Musicians call this “being out of tune”.

46. What word best describes this to physicists?
Amplitude
Answer: beats

47. What word best describes this to musicians?
Amplitude
Answer: bad intonation
(being out of tune)

48. Periodic Motion
Motion that repeats itself over a fixed and reproducible period of time is called periodic motion.
The revolution of a planet about its sun is an example of periodic motion. The highly reproducible period (T) of a planet is also called its year.
Mechanical devices on earth can be designed to have periodic motion. These devices are useful timers. They are called oscillators.

49. Simple Harmonic Motion
You attach a weight to a spring, stretch the spring past its equilibrium point and release it. The weight bobs up and down with a reproducible period, T.
Plot position vs time to get a graph that resembles a sine or cosine function. The graph is “sinusoidal”, so the motion is referred to as simple harmonic motion.
Springs and pendulums undergo simple harmonic motion and are referred to as simple harmonic oscillators.

50. Analysis of graph 3 2 4 6 t(s) -3 x(m)
Equilibrium is where kinetic energy is maximum and potential energy is zero. 3
equilibrium 2 4 6
t(s) -3
x(m)

51. Analysis of graph 3 2 4 6 t(s) -3 x(m) Maximum and minimum positions
Maximum and minimum positions have maximum potential energy and zero kinetic energy.

52. Oscillator Definitions
Amplitude
Maximum displacement from equilibrium.
Related to energy.
Period
Length of time required for one oscillation.
Frequency
How fast the oscillator is oscillating.
f = 1/T
Unit: Hz or s-1

53. Springs Springs are a common type of simple harmonic oscillator.
Our springs are “ideal springs”, which means
They are massless.
They are both compressible and extensible.
They will follow Hooke’s Law.
F = -kx

54. Fs = -kx Fs mg Review of Hooke’s Law m
The force constant of a spring can be determined by attaching a weight and seeing how far it stretches. mg

55. Period of a spring
T: period (s)
m: mass (kg)
k: force constant (N/m)

56. Sample Problem
Calculate the period of a 300-g mass attached to an ideal spring with a force constant of 25 N/m.

57. Sample Problem
A 300-g mass attached to a spring undergoes simple harmonic motion with a frequency of 25 Hz. What is the force constant of the spring?

58. Sample Problem
An 80-g mass attached to a spring hung vertically causes it to stretch 30 cm from its unstretched position. If the mass is set into oscillation on the end of the spring, what will be the period?

59. Conservation of Energy
Springs and pendulums obey conservation of energy.
The equilibrium position has high kinetic energy and low potential energy.
The positions of maximum displacement have high potential energy and low kinetic energy.
Total energy of the oscillating system is constant.

60. Sample problem.
A spring of force constant k = 200 N/m is attached to a 700-g mass oscillating between x = 1.2 and x = 2.4 meters. Where is the mass moving fastest, and how fast is it moving at that location?

61. Sample problem.
A spring of force constant k = 200 N/m is attached to a 700-g mass oscillating between x = 1.2 and x = 2.4 meters. What is the speed of the mass when it is at the 1.5 meter point?

62. Pendulums
The pendulum can be thought of as a simple harmonic oscillator.
The displacement needs to be small for it to work properly.

63. Pendulum Forces  T
mg sin mg

64. Period of a pendulum T: period (s) l: length of string (m)
g: gravitational acceleration (m/s2)

65. Sample problem
Predict the period of a pendulum consisting of a 500 gram mass attached to a 2.5-m long string.

66. Sample problem
Suppose you notice that a 5-kg weight tied to a string swings back and forth 5 times in 20 seconds. How long is the string?

67. Sample problem
The period of a pendulum is observed to be T. Suppose you want to make the period 2T. What do you do to the pendulum?