1. Oscillations and Waves
Unit 7

2. Opening Questions What is an oscillation (or vibration)?
What is a wave?

3. Simple Harmonic Motion

4. Simple Harmonic Motion
The most basic type of oscillation is when an object moves back and forth over the same path with each motion taking the same amount of time.
This motion is periodic and is often referred to as simple harmonic motion (SHM).
The most commonly used example of SHM is a mass on a spring.

5. Review of Springs
A spring is a coil of metal that can be compressed or stretched.
When the spring is displaced, it exerts a restoring force given by Hooke’s Law:
Recall that k is the spring constant and is a measure of how much force the spring exerts for every meter it is displaced.

6. Review of Springs
Recall that a compressed or stretched spring also has and associated potential energy:
For both force and energy, x is the distance the spring has been displaced from its equilibrium position, the point where the spring exerts no force.

7. Simple Harmonic Motion
Consider a mass on a horizontal, frictionless table. The mass is connected to a spring shown.
The mass is initially at rest at its equilibrium position.

8. Simple Harmonic Motion
If we displace the mass to the right, the spring exerts a force pulling the mass back to the left.
Similarly, if we displace the mass to the left, the spring pushes it back to the right.

9. Simple Harmonic Motion
This is why the spring force is called a restoring force.
The spring always exerts a force to restore the mass to its equilibrium position.

10. Cart Demo

11. Simple Harmonic Motion
What happened when the cart was released from rest?
Why did this happen?
Let’s examine this system in greater detail.

12. Simple Harmonic Motion
Suppose we take the mass and push it to the left, compressing the spring.
The mass is then released from rest.

13. Simple Harmonic Motion
When the mass is released, the spring exerts a force pushing it back towards the equilibrium position.
This force causes the mass to accelerate.

14. Simple Harmonic Motion
Once at the equilibrium position, the spring exerts no force on the mass.
However, the mass has now been accelerated to a v > 0.
So the mass continues moving past the equilibrium position.

15. Simple Harmonic Motion
But once the mass has passed the equilibrium position on the right, the spring exerts a force pulling to the left.
This causes the mass to decelerate.
The mass continues decelerating until it is at rest.

16. Simple Harmonic Motion
The mass stays at rest for only an instant because the spring is still exerting a force to the left.
So the mass accelerates back to the left.
When it reaches the equilibrium position again, there is no force exerted by the spring.

17. Simple Harmonic Motion
As before, the block’s velocity causes it to continue moving past the equilibrium position.
The spring exerts a force to the right, causing the block to decelerate.
The block comes to rest at its starting position, and the cycle repeats.

18. Conceptual Example
For the mass/spring system, which of the following statements are true at some point in the motion? a) v = 0, a ≠ 0 b) v = 0, a = 0 c) v ≠ 0, a = 0 d) v ≠ 0, a ≠ 0

19. Vocabulary of SHM

20. Vocabulary of SHM
Displacement: How far the mass is from the equilibrium position. WARNING: here displacement is represented by x instead of Δx.
Amplitude: The maximum displacement the mass has over the course of its motion.
Cycle: One complete motion from an initial point back to that same point in an oscillating system. If the system involves a rotation, this is sometimes called a revolution.

21. Vocabulary of SHM
Period (T): The amount of time it takes the object to complete one cycle. Period is measured in seconds.
Frequency (f): The number of cycles the system completes in 1 second. Frequency is the inverse of period and is measured in hertz (Hz).
The period and frequency of an oscillator do not depend on the initial amplitude.

22. Vocabulary of SHM
Any system where the restoring force is directly proportional to the displacement (e.g. F = -kx) exhibits simple harmonic motion.
A system exhibiting SHM is often called a simple harmonic oscillator (SHO).
Other examples of SHOs include pendulums, atoms within molecules, and molecules within solids.

23. Homework
Read section 11-3.
Do problems 1 and 2 on page 317.

24. Energy in the Simple Harmonic Oscillator

25. Energy in the SHO
Another way to look at the SHO is using conservation of energy.
This is especially useful here since the spring force is not constant (it changes as the mass moves).
As with the forces example, we will assume there is no friction here.

26. Energy in the SHO
Since work is required to stretch or compress a spring, we know there is potential energy stored in the spring given by:
We also know that the total energy of the system at any point is given by the sum of the kinetic and potential energies.

27. Energy in the SHO
When the spring is compressed a distance A (so x = -A), the system is at rest.
This means all the energy in the system is spring stored energy.

28. Energy in the SHO
The system is then released from rest, and the spring is allowed to expand.
At the equilibrium position, the displacement is zero so there is no spring PE.
All the energy is now KE.

29. Energy in the SHO
The spring then stretches beyond equilibrium position until it has stretched a distance A to the right.
The mass is now at rest.
So, the energy in the system is once again spring PE.

30. Energy in the SHO
At points in between the equilibrium position and the max stretch or compression, the system has both kinetic and potential energy.

31. Energy in the SHO
This analysis gives us a new way to think about SHOs: as devices that convert PE into KE and then back into PE.
Also, at the extremes (x = A and x = -A), the total energy of the system is potential energy.
Since the potential energy is related to the displacement, this tells us something important.

32. Energy in the SHO
The total mechanical energy in a simple harmonic oscillator is proportional to the square of the amplitude.
As a corollary, this also means that the TME of the oscillator is constant as long as there are no dissipative forces (like friction) present.
We can use this fact, along with the principle of conservation of energy to find an equation for the velocity of the mass at any point along its motion.

33. Energy in the SHO
Consider two points in the motion: at the extreme and a point between the extreme and the equilibrium position.
We know the total mechanical energy at the two points must be equal. So,

34. Energy in the SHO
Solving for v2, we get
Factoring out an A2

35. Energy in the SHO
If we look at the energy at the equilibrium point, we can see. So

36. Energy in the SHO So
This function lets us find the velocity of the block at any position x.

37. Example A spring stretches 0.15 m when a 0.3 kg mass is hung from it.
a) What is the spring constant?
The spring is then set up horizontally, so that the mass is resting on a frictionless table. The mass is pulled so that the spring stretches 0.1 m from the equilibrium point.
b) What is the amplitude of the oscillation?
c) What is the max velocity of the mass?
d) What is the velocity of the mass when it is at x = m?

38. Homework
Do problems 13 and 14 on page 317.

39. Period, Frequency and Equations of Motion

40. Opening Question
We have seen that a mass in an SHO oscillates between two points: x = A and x = -A.
What type of mathematical function do you know that oscillates between two points?
Conclusion: SHM is sinusoidal (that is, it can be described using a sine or cosine function).

41. Period and Frequency
We know that any sine function has an associated period and frequency.
We also know the oscillator exhibits periodic motion.
The period of oscillation has been observed to increase with the mass of the object, and to decrease with the spring constant, k.

42. Period and Frequency
Mathematically, this relationship is expressed in the following equation:
Notice that this is not a direct relationship. The period varies as the square root of m/k.

43. Period and Frequency
If we recall the definition of frequency (f = 1/T), we can also write a formula for the frequency:
Note that the formulas for period and frequency are valid for any SHO (not just for a mass on a spring).

44. Angular Frequency
Another way to express the frequency of oscillation is with the angular frequency (aka angular velocity). This is defined as:
For the oscillator, the angular frequency is

45. Angular Frequency
The angular frequency shows up in the equations of motion (it’s actually more common than the linear frequency).
It also has applications to other types of systems.
The angular frequency has units of radians/sec.
WARNING: All angles in this unit must be measured in radians.

46. Example: Spider Web
A kg spider waits in its web (which has negligible mass). A slight movement causes the web to exhibit SHM at a frequency of 15 Hz. a) What is the “spring constant” of the web? b) What would the frequency of oscillation be if a kg insect were caught in the web?

47. Equations of Motion

48. Equations of Motion Consider again the mass on horizontal spring.
The spring is stretched a distance A and the mass is released from rest.
We know that the mass will oscillate between A and – A.

49. Equations of Motion
Question: which sinusoidal function starts at its maximum value and oscillates from there?
This suggests that the position function is a cosine function.
However, cosine only goes between -1 and 1.

50. Equations of Motion
In order to make the position function go from –A to A, we need to scale the cosine.
This suggests the position function has the general form

51. Equations of Motion However, θ depends on time. So, we can let θ = ωt.
This gives us the equation of motion:

52. Equation of Motion
We can also express this equation more explicitly as a function of period or frequency:
WARNING: T is a constant (period), t is a variable (time).

53. Equations of Motion
Cool Video!

54. Example
The displacement of an oscillator is described by the equation: Determine the oscillator’s: a) Amplitude b) Frequency c) Period d) Max velocity e) Max acceleration

55. Homework
Do problems 4, 7, and 9 on page 317.

56. Velocity and Acceleration

57. Velocity
We saw yesterday that the position of a SHO is described by the function
We can use this to determine the object’s velocity and acceleration as a function of time.
Question: how do you determine the instantaneous velocity of an object graphically?

58. Velocity

59. Velocity
Based on this, we can conclude that the velocity function is also sinusoidal.
Which function describes this data?

60. Velocity

61. Velocity
Recall also, from conservation of energy, that vmax can be expressed

62. Velocity
So, we can also express the velocity as

63. Acceleration
We can repeat the same process to find acceleration:

64. Acceleration
Based on this, we can conclude that the acceleration function is also sinusoidal.
Which function describes this data?

65. Acceleration

66. Acceleration
Recall, from Newton’s second law, that amax can be expressed

67. Acceleration
So, we can also express the velocity as

68. Example: Loudspeaker
The cone of a loudspeaker exhibits SHM at a frequency of 262 Hz. The center of the cone has an amplitude of m and begins its oscillation at x = A. a) What is the angular frequency of the oscillator? b) What are the equations for the position, velocity, and acceleration of the cone? c) What is the position of the cone at t = 1 ms?

69. A few loose ends…
The equations we have derived assume that the mass starts from rest at the amplitude.
This is not always the case.

70. A few loose ends…
If the mass starts from the equilibrium position with a push, the function would be
Note that the sine can be converted back to a cosine with a simple phase shift

71. A few loose ends…
Therefore, a more general form of the position function is
Where δ is the phase shift angle needed to correct for the initial position of the oscillator.
You will not be required to know this for this class.

72. A few loose ends: Vertical Springs
So far we have only looked at horizontal springs.
However, vertical springs exhibit SHM as well.
However, we must take a slightly different approach.

73. A few loose ends: Vertical Springs
When the mass is hung on the spring, the spring stretches beyond its equilibrium position (even though it is a rest).
The trick is to measure x from this position for the oscillations instead of the natural equilibrium position.

74. Homework Do problems 12 and 23 on pages 317-318.
We will have a daily exercise quiz tomorrow.
AP review meets tomorrow morning.

75. Problem Day Do problems 5, 8, 21, and 22 on pages 317.
We will whiteboard at the end of class.

76. Homework
Reread 11-4.
Do problems 29 and 30 on page 318.

77. The Simple Pendulum

78. The Simple Pendulum
A pendulum consists of a mass attached to the end of a lightweight string.
When at equilibrium, the mass hangs vertically.
When displaced from equilibrium, the mass is observed to oscillate.

79. The Simple Pendulum Question: is this oscillation an example of SHM?
If we displace the mass from equilibrium, part of the weight of the bob goes into maintaining tension in the string.

80. The Simple Pendulum
The other component of the weight points back towards equilibrium.
Thus the restoring force is:

81. The Simple Pendulum
Notice that the force is proportional to sin(θ), not to θ itself.
This suggests that the motion is not SHM.
However, for small angles, sin(θ) is almost the same as θ.

82. The Simple Pendulum
For angles less than 15°, we can make the small angle approximation.
Now, notice that

83. The Simple Pendulum Thus, the force is now
This is very similar to Hooke’s law if we let k = (mg)/L.
Thus the motion is SHM (for small angles).

84. The Simple Pendulum
To determine the angular frequency of oscillation, recall
If we plug in our new value of “k”

85. The Simple Pendulum
This means the angular frequency of the oscillator is
The linear frequency is then

86. The Simple Pendulum Taking the inverse, we get the period
Notice that none of these formulas depend on the mass of the bob.

87. The Simple Pendulum
Notice also that none of these formulas depend on the initial amplitude, θ0.
Lastly, these formulas all break down if the amplitude gets larger than 15°.

88. Example: Measuring g
A pendulum can be used to get a precise measure of the acceleration due to gravity (g). A scientist determines his pendulum has a period of Hz. If the length of the string is 37.1 cm, what is the value of g at his location?

89. Homework
Read 11-5 and 11-6.
Do problems 32 and 35 on page 318.

90. Damped Oscillations

91. Opening Questions
In the lab last week, what happened to the amplitude of the oscillator when we taped a paper plate to the bottom?
Why did this happen?

92. Damped Oscillations
The system in the lab was an example of damped harmonic motion.
In a damped system, the amplitude of the oscillation decreases over time as a result of a resistive force like friction or air resistance.
A damping force does alter the frequency of the oscillator. This effect is usually small unless the damping force is large.

93. Three Types of Damping
The system we saw in the lab was an example of underdamped motion.
In an underdamped situation, the goes through several cycles before coming to rest.
This is represented by graph A to the right.

94. Three Types of Damping
The second type of damping is called overdamping.
In this situation, the system goes back to equilibrium without oscillating. This takes a significant amount of time.
Overdamping is represented by curve C.

95. Three Types of Damping
The last type of damping is called critical damping.
In this situation, the system also goes directly back to equilibrium. However, this happens in the shortest possible amount of time.
This corresponds to curve B.

96. Why is Damping Good?
In many systems, periodic motion is desired. Damping effects need to be minimized in these cases.
However, in many other systems oscillations are a problem.
In these cases, the system must be critically damped in order to return to equilibrium as quickly as possible.

97. Examples
Car shocks
Building supports

98. Resonance

99. Natural Frequency
When you release an oscillator from rest, it vibrates at a specific frequency given by:
If the system is not changed or affected by an outside force, it will always vibrate at this frequency. The frequency is known as the natural frequency of the oscillator.

100. Driven Oscillations
However, suppose we apply a periodic force that has its own frequency.
In that case, the oscillator would vibrate at the frequency of the applied force instead of its natural frequency.
This is known as a driven (forced) oscillation.

101. Driven Oscillations
From experiments, we know that the amplitude of the driven oscillation depends on the difference drive frequency and the natural frequency.
If f << f0 the amplitude will be fairly small.
Likewise, if f>>f0 the amplitude will also be fairly small.

102. Resonance
However, if the drive frequency is close to f0, the amplitude of the oscillations.
If f = f0, this increase is greatest.
This effect is known as resonance and the natural frequency is also the resonant frequency.

103. Examples Pushing a swing Breaking a wine glass
Tacoma Narrows Bridge collapse

104. Homework
Do problem 26 on page 318

105. Waves
Unit 7-b

106. Waves
What is a wave?
In simplest terms, a wave is a traveling oscillation.
There are many types of waves: mechanical, sound, and electromagnetic.

107. Wave Motion
From our experiences, we know that waves travel forward through a material (called a medium).
However, the wave does not move particles forward through the medium.

108. Wave Motion
Instead, the wave is the result of individual particles oscillating vertically about an equilibrium point.
The wave propagates through internal forces between the particles of the medium.

109. Wave Motion
The simplest type of wave is a single wave bump. This is known as a pulse.
A pulse can be made by a simple up and down motion on the end of a rope.
As a result of the applied force, the first particle begins to move upward.

110. Wave Motion The first particle is connected to the second particle.
As a result, the second particle moves upward.
As the hand comes to a stop and begins to move down, the same thing happens to the first particle.

111. Wave Motion
This results in a force on the second, which exerts a force on the third, etc.
Thus, the wave transmits the original oscillating motion down the length of the string.

112. Wave Motion
A second type of wave arises when the source is a continuous oscillation.
The result is a periodic wave. This is because the oscillating motion of each particle repeats over a set interval.

113. Wave Motion
If the vibration that causes the wave is sinusoidal (SHM), the wave will also have a sinusoidal shape in both space and time.
Space: if you take a snapshot of the wave (see below), it will have the shape of a sine or cosine wave as a function of position.

114. Wave Motion
Time: If you look at the motion of one point on the rope over a long period of time, you will see it oscillate in SHM.
Animations

115. Vocabulary of Waves

116. Vocabulary of Waves
Because waves are often sinusoidal, we will use many of the same terms to describe waves.
The high points of the wave are called crests while the low points are called troughs.

117. Vocabulary of Waves
The amplitude (A) of the wave is the maximum height of a crest.
The distance between two successive crests is known as a wavelength.

118. Vocabulary of Waves
A wavelength is also the distance between any two identical points on the wave.
Wavelength is represented by letter λ.

119. Vocabulary of Waves
The period (T) of the wave is then the time between two successive crests passing the same point in space.
The frequency (f) is the number of crests that pass a given point in space in a second.

120. Vocabulary of Waves
As with SHM, period and frequency are inversely related.

121. Vocabulary of Waves We can also define the wave velocity (v).
The wave velocity is the velocity at which crests move. This is not the same as the velocity at which individual particles are oscillating.

122. Vocabulary of Waves
A wave crest travels a distance of one wavelength in a time equal to one period.

123. Waves on a String
The specific formula for calculating the velocity of a traveling wave depends on the material it is moving through.
One common situation is a traveling wave on a string.
The speed depends directly on the tension in the string and inversely on the mass of the string per unit length.

124. Waves on a String Mathematically this is:
This should make some sense based on our knowledge of mechanics.
For waves on a string.

125. Example: Wave on a String
A wave has a wavelength of 0.3 m is traveling down a 300 m string whose total mass is 15 kg. If the string is under a tension of 1000 N, a) What is the speed of the wave? b) What is the frequency of the wave?

126. Types of Waves

127. Two Types of Waves
So far we have been talking only about transverse waves.
In transverse waves, the oscillations of individual particles are perpendicular to the direction the wave is traveling.

128. Two Types of Waves Another type of wave is a longitudinal wave.
In a longitudinal wave, the particles oscillate in the same direction the wave is moving.

129. Two Types of Waves This can easily be seen with a slinky (demo time).
Notice how the wave is characterized by a series of compressions and expansions.
These correspond to crests and troughs in the transverse wave.

130. Homework
Read 11-7 and 11-8.
Work on your paper proposals.

131. Energy Transported by Waves

132. Energy Transported by Waves
We mentioned yesterday that waves transport energy.
We have also seen waves move objects by giving them kinetic energy.
Recall also that for a sinusoidal wave, each particle vibrates in SHM.

133. Energy Transported by Waves
This means each particle has an energy
Based on this, we can infer an important result:
The energy transported by a wave is proportional to the square of the amplitude.

134. Intensity It is also useful to define a new term: intensity.
Intensity is the power (energy per unit time) delivered to across an area perpendicular to the direction the wave is moving.

135. Intensity
Intensity is represented by the variable I (don’t get this confused with current).
The SI unit for intensity is W/m2.
Since intensity is related to energy, we can conclude:

136. Spherical Waves Many waves flow out from a source in all directions.
Some common examples include earthquakes, sound, and light.
If the medium the wave is traveling through is uniform, the result will be a spherical wave.

137. Spherical Waves
Since we know the shape of the spherical wave, we can write an equation for the intensity
Notice that the wave loses intensity the farther from the source you get.
Spherical wave only

138. Spherical Waves Spherical wave only
If the power generated is constant, we can also write a few proportions:
Spherical wave only

139. Example: Earthquake
The intensity of an earthquake wave traveling through this Earth is 106 w/m2 when measured 100 km from epicenter. What is the intensity of the wave 350 km from the source?

140. Homework
Do problems and 41 on page 318.
Finish topic proposals.

141. Reflection and Transmission

142. Slinky Demo Again

143. Reflection
When a wave strikes an obstacle or comes to the end of a medium, at least part of the wave is reflected back.
Depending on the type of obstacle, some of the wave may continue on in the same direction (it is transmitted).
We will discuss both effects today.

144. Reflection Consider a string that has fixed at one end.
A wave pulse is moving to the right.
As the wave approaches the peg, the rope is prevented from moving.

145. Reflection
This means the wave is forced to have zero amplitude at that point.
The front end of the wave pulse is pulled down at the end of the rope.
Now the front end of the pulse has a downward velocity.

146. Reflection This pulls the rest of the pulse down, creating a new wave.
However, the amplitude of this wave is inverted.
This results in a pulse being sent back (reflected) upside down.

147. Reflection What if the rope is free to move at that end?
In this case as the wave pulse arrives, the end of the rope is pulled upward.
As the rear end of the wave approaches, it creates a downward force that generates a pulse in the other direction.

148. Reflection So, we have learned two things here:
If the end of the rope is fixed, the reflected wave has an inverted amplitude.
If the end of the rope is free to move, the reflected wave has the same amplitude.

149. Reflection and Transmission
What happens if the boundary is a change in the medium?
Because the rope is still free to move on the other side of the boundary, at least part of the wave will be transmitted.

150. Reflection and Transmission
The heavier the second section is, the less the energy that is transmitted.
However, because energy must be conserved, the rest of wave is reflected.
The amount of transmission and reflection depends on the boundary type.

151. Reflection and Transmission
Lastly, when a wave is transmitted, its frequency does not change.
However, because the second section has more mass, the wave will have a different velocity.

152. Reflection As a result, the wavelength of the wave also changes.
This will be of great importance when we talk about sound and light.

153. Reflection for 2D and 3D Waves

154. 2D and 3D Reflection
For 2D and 3D waves, the oscillations can vary as you move through space.
We need a way to define the shape of the wave in space.
To do this, we use the concept of a wave front.

155. 2D and 3D Reflection
All the points that form a crest are considered a wave front.
When we refer to a wave in the ocean, we are really describing a wave front.

156. 2D and 3D Reflection
With wave fronts, we can then represent the direction the wave is moving through the use of rays.
Rays are always perpendicular to the wave front.

157. 2D and 3D Reflection
Lastly, if a circular or spherical wave has expanded to the point where the wave fronts have lost their curvature, it is called a plane wave.
We will approximate many waves as plane waves when looking at light and sound.

158. 2D and 3D Reflection Consider now a plane wave striking a surface.
The angle the incoming (or incident) wave makes with the surface is equal to the angle the reflected wave makes with the surface.

159. The angle of reflection is equal to the angle of incidence.
Law of Reflection
Law of Reflection
The angle of reflection is equal to the angle of incidence.

160. Interference

161. Interference
Now we will look at what happens when two different waves pass through the same region of space.
An example of this might be two pulses traveling in opposite directions along the same string.
What will happen when the two pulses meet?

162. Interference
Animation

163. Interference
When two waves meet at the same point in space, they continue past each other.
However, in the region where they overlap, the two wave interfere to produce a new amplitude.
The amplitude depends on the shape of the individual waves.

164. Principle of Superposition
In the region of overlap, the amplitude is found by using the principle of superposition.
Principle of Superposition
The amplitude of two interfering waves is the algebraic sum of their individual amplitudes.

165. Interference Let’s use this idea to consider two cases.
Suppose two wave pulses with positive amplitudes are traveling toward each other (animation).
What happened to the amplitude when they overlapped?

166. Interference
When two waves have the same sign amplitude (either both positive or both negative), they produce an amplitude that is greater than either individual amplitude when they overlap.
This is called constructive interference.

167. Interference
When two waves have amplitudes with different signs, they produce an amplitude that is less than either individual amplitude when they overlap.
This is called destructive interference.

168. Interference
If the two waves have identical amplitudes, they cancel out completely when they overlap.
This is known as perfectly destructive interference.

169. Interference in 2D and 3D
Imagine we throw two rocks into a pond at the same time.
What do we see?
Animation

170. Interference in 2D and 3D What we are seeing is 2D interference.
At some points, crests are repeatedly meeting and interfering constructively.
At these points, the amplitude is greater than the amplitude of either individual wave.

171. Interference in 2D and 3D
At other points, troughs are repeatedly meeting and interfering constructively.
At some points, the crests of one wave meet the troughs of the other, interfering destructively.

172. Describing Interference
When two waves are interfering, we use the term phase to describe the relative positions of the crests of the two waves.
When the crests and troughs of both waves are aligned, we say the two waves are in phase.

173. Describing Interference
When we add two waves that are in phase (principle of superposition), the result is a wave with a larger amplitude.
Thus, two waves that are in phase interfere constructively.

174. Describing Interference
If the crests and troughs are not perfectly aligned, we say the waves are out of phase.
If we add to waves that are out of phase, the result is a wave of smaller amplitude, indicating partially destructive interference.

175. Describing Interference
If the crests of one wave are aligned with the troughs of the other, the waves are out of phase by ½ a wavelength.
When added, the waves cancel out, leading to perfectly destructive interference.

176. Homework
Read and
Do problem 51 on page You might want to use the animation from class to help you. The link is on the blog.

177. Standing Waves

178. Announcements We will be having our next test on Wednesday.
The test will be mostly conceptual, but will include a few problems on waves like the ones we have done for HW.
There will be a daily exercise quiz on Friday.

179. Standing Waves
Imagine you have a string that is fixed at on end to a wall.
You send a periodic wave down the string.
At other end, the waves are inverted and reflected back toward you.

180. Standing Waves
Since you are still sending waves down the string, they will interfere with the reflected waves.
This usually results in a jumbled mess.
However, if you send waves down the string at the right frequency, a standing wave will be produced.

181. Slinky Demo Again

182. Standing Waves
In a standing wave, parts of the string appear to oscillate up and down, while other parts appear to remain still.
Why does this happen?
Animation

183. Vocabulary of Standing Waves
The frequencies that produce standing waves are called the natural or resonant frequencies of the string.
The points of destructive interference where the string does not vibrate are called the nodes.
The points of max amplitude are called the antinodes.

184. Standing Waves Standing waves can occur at more than one frequency.
The lowest resonant frequency produces a single antinode.
If you drive the wave a twice the lowest frequency you get two antinodes.
Three times produces three antinodes, etc.

185. Standing Waves
To quantify the resonant frequencies, notice that the wavelength for each wave is related to the length of the string.
The lowest frequency, called the fundamental frequency, produces a wave with wavelength equal to twice the length of the string.

186. Standing Waves
The second frequency produces a wave with wavelength equal to the length.
The third frequency produces

187. Standing Waves
In general, the wavelength is related to length by the formula:
Recalling the relationship between frequency and wavelength,

188. Standing Waves We get a formula for the resonant frequencies.
Keep in mind, the velocity can be found by

189. Standing Waves
The frequencies above the fundamental frequencies are called overtones.
You may have also heard the fundamental frequency referred to as the first harmonic.
The first overtone would then be the second harmonic.

190. Example: Piano String
A piano string is 1.1 m long and has a mass of 9.0 g. a) How much tension must the string have to vibrate with a fundamental frequency of 131 Hz? b) What are the frequencies of the first four harmonics?

191. Homework
Read section
Do problems all on page 319.

192. Whiteboarding Groups Group Members Problem 1 52 2 53 3 54 4 55 5 56 657 7 59 8 61 9 77