Waves

Review Objectives from Physics 1

Waves
1. Describe a wave pulse and a continuous travelling wave.
· Students should be able to distinguish between the oscillations and the wave motion.
· Individual particles (molecules) within the medium vibrate (oscillate) about an equilibrium position.
· The pulse (one packet of energy) or wave (continuous energy) translates through the medium
2. State that waves transfer energy.
· Students should understand that there is no net motion of the medium through which the wave travels. Particles oscillate, wave energy travels. That's it.
3. Describe and give examples of transverse and longitudinal waves.
· Students should know that sound is a longitudinal wave and that light is a transverse wave.
· Transverse waves are those for which particle vibration is perpendicular to the direction of energy travel (wave propagation)
·
Longitudinal (compression) waves are those for which particle vibration is parallel to the direction of energy travel (wave propagation)
4. Describe waves in two dimensions, including the concepts of wave fronts and rays.
· When describing a traveling wave, we can represent the wave as either a wave front (think waves at the beach or ripples in a pond after dropping in a rock) or a series of rays that show the direction of energy propagation.

Wave characteristics
5. Define displacement, amplitude, period, frequency, wavelength and wave speed.
· Displacement refers to the amount of motion individual points or particles within the medium.
· Amplitude (A) refers to the maximum displacement of a particle from its equilibrium position.
· Period (T) refers to the amount of time necessary for a particle to repeat its motion within the medium.
· Frequency (f) refers to the number of times that a particle within the medium repeats its motion in a given time interval.
· Wavelength (l) refers to the distance within the medium over which the waveform repeats itself.
· Wave speed (v) refers to the speed with which the wave form travels within the medium. v = fl
6. Describe the terms crest, trough, compression and rarefaction.
· Crest refers to the part of the wave form in which the particles experience positive displacement.
· Trough refers to the part of the wave form in which the particles experience negative displacement.
· Compression refers to the portion of a longitudinal wave where the relative separation of particles is decreased
· Rarefaction refers to the portion of a longitudinal wave where the relative separation of particles is increased.
7. Draw and explain displacement–time and displacement–position graphs for transverse and longitudinal waves.
· Displacement-time graphs show the displacement of a single particle or molecule as a function of time. These graphs are useful for determining the period and frequency of motion, as well as the amplitude of the motion.
· Displacement-position graphs show all the particles in the medium at a single instant in time (think of a freeze-frame image). These graphs can also be used to determine the amplitude of the wave motion, but they are useful for determining the wavelength of the wave
· The diagram shown above is an example of a displacement-position graph.
· The difficulty is that both graphs tend to look the same -- pay attention to the units on the horizontal axis to determine which one you have
8. Derive and apply the relationship between wave speed, wavelength and frequency.
· Wave speed, like any other speed, is the rate at which the wave covers distance. The best distance to use to describe the motion of a wave is the wavelength, l. This distance corresponds to one complete oscillation of a particle in the medium, so the time that corresponds to this distance is T, the period. So v = l / T. Since the period and the frequency are related according to f = 1 / T, we can write speed as v = f l


New Material

Reflection, refraction and transmission of waves – Giancoli, - ch 11, sections 11 & 13, Ch 24, sections 1 & 2
1. Describe the reflection and transmission of one-dimensional waves at a boundary between two media.
· This should include the sketching of incident, reflected and transmitted waves, and the cases of reflection at free and fixed ends.
·
The fundamental property that describes a wave is the frequency of the wave and this property does not change upon reflection or transmission – wave speed and wavelength may change during transmission at a boundary
· When reflection occurs at a fixed end (a rope attached to the wall) the reflected wave is inverted
· When reflection occurs at a free end the reflected wave is not inverted

2. State Huygens' principle.
·
Every point on a wavefront emits a spherical wavelet of the same velocity and wavelength as the original wave. The wavelet is assumed to be emitted in the forward direction only. The amplitude of the wavelet is maximum in the forward direction and decreases rapidly away from that direction. The new wavefront is the surface that is tangent to all of the wavelets.
· An example:
Each point on the original wavefront creates a spherical wavefront in the forward direction. The new wavefront is the surface tangent to all of these spherical wavefronts. In a time, Dt, the new wavefront is a distance vDt from the original wavefront.

3. Apply Huygens' principle to two-dimensional plane waves to show that the angle of incidence is equal to the angle of reflection.
·
A plane wavefront, I, is incident on a mirror at some angle, , as shown in the picture.
· Points A and E both lie on the incident wavefront and they will emit wavelets at the same time. The two circles represent the two wavelets emitted.
· In a time, Dt, the wavelet from point E will just reach the mirror at point B. The wavelet from point A will travel an equal distance.
· Therefore, the new wavefront must contain both point B and a point that lies on the wavelet formed by point A so that it is tangent to that wavelet. This means it must pass through point D.
· We know that the distances EB = AD
· The triangles = because they are both right triangles and have an equal side.
· This means that
· But, and
· So,
4. Explain refraction using Huygens' principle.
· As a plane wavefront passes through a boundary from one medium into another one at an angle, the wavefront will change speed and direction as it moves into the second medium
· Because wave speed changes and frequency doesn't (it's the fundamental property of a wave), the wavelength has to change
· Because the wavelength changes, the wavelets in the second medium will travel a different distance than the wavelets in the first medium.
· This causes the wave to change direction.






5. Derive, using Huygens' principle, Snell's law for refraction.
·
Use the diagram to the right
· The plane wavefront CD comes upon the boundary and makes the angle qI with the normal
· All points on the wavefront below point D will make wavelets like the one shown, creating a new plane wave AB.
· Point C will make a wavelet like the one shown, which will travel a shorter distance (smaller wavelength) in the second medium because it has a slower speed there.
· To find the wavefront in the second medium, we must connect point A and the smaller wavelet with a tangent line (to point E) as in the diagram.
· This wavefront makes a different angle, qR, with the normal line
· We can show that sin qI = DA / AC
· We can also show that sin qR = CE / AC
· This means that AC = DA / Sin qI = CE / sin qR
· So, sin qR / sin qI = CE / DA = lR / lI = vR / vI

6. State and apply Snell's law.
· This is Snell's Law:
Wave diffraction and interference – Giancoli, ch 11, sections 11 & 13
7. Explain and discuss qualitatively, using Huygens' principle, the diffraction of waves by apertures and obstacles.
· The effect of wavelength compared to obstacle size or aperture size should be discussed.
· Diffraction is the spreading of waves as they pass through an aperture (hole) or around an obstacle
· The size of the aperture/obstacle in comparison with the wavelength of the wave is the critical factor in determining the degree of diffraction that takes place
· If the wavelength of a plane wavefront is small compared to the size of the aperture or obstacle then no diffraction takes place - based on Huygen's principle you can say that if the aperture or obstacle appears to be large compared to the wave then many points on the wavefront will act as sources of wavelets and the new wave will also be a planar wavefront
· If the wavelength of a plane wavefront is comparable to the size of the aperture or obstacle then diffraction will take place - based on Huygen's principle you can say that if the aperture is small it is like only one point on the plane wave will produce a wavelet and these will spread out as circles through the aperture or beyond the boundary.
8. Describe examples of diffraction.
· You can hear around a door opening but you can't see around the corner. This is because sound waves are long wavelength so as they pass through the doorway they spread out and fill all of the space beyond the doorway. Light waves, however, have very small wavelengths compared to the size of the doorway and so they pass straight through and you can only see directly through the doorway.
9. State the principle of superposition and explain what is meant by constructive and destructive interference.
· The principle of superposition allows you to determine the resulting displacement of interfering waves. To find the displacement of the medium at a single point, simply add the displacements from each of the interfering waves.
· Constructive interference occurs when the resulting displacement is larger than the displacement of the interfering waves.
·
Destructive interference occurs when the resulting displacement is smaller than the displacement of the interfering waves.
10. Apply the principle of superposition to find the resultant of two waves.
· Only one-dimensional situations need to be considered.




Nature and production of standing waves – Giancoli, Ch 11, section 12
11. Describe the nature of standing waves.
· Standing waves are called “standing” because the crests of the wave do not appear to move.
· No energy is transferred by a standing wave
· There are a number of points where there is never any disturbance – these are called nodes
· The points for which disturbance is maximum are called antinodes.
12. Explain the formation of standing waves in one dimension.
· Standing waves are formed through the constructive interference of two identical waves traveling in opposite directions as follows








The left end of the string is lifted up and down, creating a pulse that travels to the right as shown.








The left end is pushed down and lifted up, creating pulse #2 as shown. Both pulses are traveling to the right.








The left end is raised and lowered again. This creates pulse #3 going to the right. Pulse #2 is still moving to the right, but pulse #1 is reflected upside down and is now moving to the left. Pulse #1 and pulse #2 interfere making the dotted line.








The left end is lowered and raised one more time, creating pulse #4 moving to the right. It interferes with pulse #1 moving to the left (making the dotted line). Pulse #3 moved to the right and interferes with pulse #2 (which is reflected to the top) that is moving to the left, making the dotted line.

· Once the process described above has been completed, the four pulses will continue to move back and forth, reflecting and flipping at each end so that the dotted line pattern just keeps flipping vertically. There is no net travel and no net transfer of energy from the left end to the right end. As you watch this you do not detect any horizontal motion.
13. Compare standing waves and travelling waves.
Boundary conditions and resonance – Giancoli - Ch 11, section 12 & Ch 12, section 5
14. Explain the concept of resonance and state the conditions necessary for resonance to occur.
· Resonance occurs when two waves of nearly the same frequency interfere. The result is a dramatic increase in amplitude.
· To make an object resonate it must be forced to vibrate. If the frequency of the forced vibration is very nearly the same as the “natural frequency” at which the object would like to vibrate, then a standing wave situation is created and resonance occurs.
· The natural frequency of vibration of an object is determined by its geometry (size, shape, etc.) and the material out of which it is made.
·
15. Describe the fundamental and higher resonant modes in strings and open and closed pipes.
· A whole series of standing waves can be set up in an object. This series is called a harmonic series. The series is determined by the amount of a wave that will create a standing wave (resonance) in the object.
· The fundamental mode (also called the first harmonic) describes the smallest part of a wave that can create a standing wave.
· Subsequent modes (harmonics) can then be determined based on geometry and the location of nodes and antinodes.
· We will limit ourselves to three possibilities:
1. A string that is fixed at both ends (this means there must always be a node at each end)
2. A pipe that is closed at one end (node) and open at the other end (antinode)
3. A pipe that is open at both ends (antinode at both ends)



















String fixed at both ends – must have a node at each end
General rule: L = nl / 2 where n is the number of the harmonic
Pipe open at one end, closed at the other – must have node at closed end and antinode at open end
General rule: L = nl / 4 where n is the number of the harmonic and you can only have odd harmonics
Pipe open at both ends – must have an antinode at each end
General rule: L= nl / 2 where n is the number of the harmonic (this one looks just like the string with nodes and antinodes switched)

16. Solve problems involving the fundamental and higher harmonic modes for stretched strings and open and closed pipes. End correction is not required.
· A typical problem might go something like this: A 0.4 m long fixed string is vibrated in its fundamental at a frequency of 60 Hz. What would be the frequency of the 3rd harmonic for this string?
· Solution: We know that the fundamental (1st harmonic) mode has L = l/2, so this means that the fundamental wavelength is 2L = 0.8 m. This allows us to calculate the wave speed: v = fl = (60 Hz)(0.8 m) = 48 m/s. As long as we don’t change the string, the wave speed will not change. So for the third harmonic, v = fl3 = 48 m/s. We know that the third harmonic means L = 3l / 2 and l3 = 2L / 3 = 2(0.4)/3 = 0.267 m. This allows us to calculate the frequency of the third harmonic: 48 m/s = f (0.267), so f = 180 Hz.




Problems: Chapter 11 #35, 50, 51, 55, 63, 64, 67 Chapter 12 #29, 33, 39

Change of Approach

We will now be guided by the conceptual level of understanding rather than the HL/SL grouping.
To be efficient we shall be covering the core and the Extended concurently. This is to make sure that we link the concepts together.

We hope to improve in our level of understanding.

List of Practicals

25/09/07 Motion Investigation-Cantilever
02/10/07 Measuring Acceleration
02/10/07 Measuring gravity
09/10/07 Motion in a horizontal circle
23/10/07 Mass on a elastic string
30/10/07 The simple Pendulum
13/11/07 Determining the Specific Heat Capacity of Water
29/01/08 Rate of Cooling

List of Practicals and Dates

02/10/07 Measuring Acceleration
13/11/07 Determining the Specific Heat Capacity of Water
02/10/07 Measuring gravity
09/10/07 Motion in a horizontal circle
25/09/07 Motion Investigation-Cantilever
23/10/07 Mass on a elastic string
30/10/07 The simple Pendulum
29/01/08 Rate of Cooling