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Waves Learning Goals Fa13

August 22, 2013

Learning goals for the first unit in our calculus-based Physics 3 course…

Content Learning Goals

Upon completing this unit the student will be able to..

1. Explain what does and does not propagate as a wave travels from point A to point B and illustrate this idea by talking about a specific type of mechanical wave.

2. Explain what amplitude, frequency, period, wavelength, and wave speed mean physically and will be able to relate these quantities mathematically. The student should know what these quantities mean for both transverse and longitudinal waves.

3. Write the wave function for a wave traveling in a given direction, at a given speed, with a given amplitude. The student can also show that this wave function satisfies the wave equation (note that this requires knowing the wave equation).

4Write the wave function for a wave given its graph or vice versa.

5Derive mathematically, along with an accompanying explanation and diagram, one of the following (student’s choice): the linear wave equation, the speed of a transverse wave on a string, or the speed of a sound wave traveling through a gas.

6. Determine the speed of a wave on a string or the speed of sound through a gas based on properties of the medium. The student can also use these results to draw conclusions about the speeds of waves in different mediums or about the speeds of two different waves in a given medium.

7. Explain what intensity means and how it varies with distance from the wave source. The student can also use this idea to calculate the amplitude at different distances from the wave source for water waves, light waves, sound waves, or water waves.

8. Explain how decibels relate to sound intensity, and why we use decibels rather than just talking about sound intensity. The student can also explain how decibels and intensity relate to loudness and what change in decibels or change in intensity corresponds to a sound “twice as loud”.

9. Explain the idea of the Doppler effect and what changes (apparent wavelength or apparent speed) when the source is moving compared to when the observer is moving. The student can relate frequencies and speeds mathematically for when the source and/or the observer is moving.

10. Explain the idea of interference or superposition in words and diagrams and given examples of how this phenomenon can be observed in sound waves, water waves, light waves, and waves on a string.

11. Use the trigonometric identity  Asin(mx) + Asin(nx) = 2Acos[(m-n)x/2]cos[(m+n)x/2]  and the idea of superposition to mathematically combine two waves and determine the amplitude, frequency, wavelength, and energy of the resultant wave. Student can also discuss the meaning of two waves being “in phase” or “out of phase”.

12. Explain the role interference or superposition plays in creating a standing wave and describe the behavior of nodes and antinodes in a standing wave.

13. Sketch a string, fixed at both ends, exhibiting motion associated with its fundamental frequency or with higher harmonics. Student can also calculate the fundamental frequency, harmonic frequencies, and associated wavelengths for a string fixed at both ends and for a column of air open at one or both ends.

14. Apply (13) to musical instruments like guitar, flute, or blowing across a Coke bottle to explain how to produce sounds with higher and lower frequencies.

15. Describe the phenomenon of beats and their cause in words. Student can also calculate the resulting beat frequency and can explain the difference between the conditions that create standing waves and the conditions that create beats.

16. Explain in words the idea of a Fourier series and why this is useful. Student can also use this idea to explain the presence of overtones or harmonics in musical instruments.

17. Identify the two wave equations that define electromagnetic waves and use these equations to determine the speed of these waves. Student can also explain how electromagnetic waves differ from mechanical waves.

18. Explain how a dipole antennae produces electromagnetic waves and how linear and loop antennae detect electromagnetic waves.


General Skills Related to this Unit

Every unit in this course will have learning goals specific to the topics of that unit. In addition, every unit will also emphasize some general skills that are applicable across much of science and beyond. The following general skills will be emphasized in this unit.

1. Thinking about the ways in which derivatives enter into physical models. In this unit, we will see multiple examples of derivatives entering into models by definition (ex. acceleration) and through applying the concept of a limit.

2. Using boundary conditions to constrain possible solutions. There are many, many situations in which the solution to a problem must satisfy not only a differential equation but also a set of boundary conditions.

3. Using of a Fourier series to decompose a complex function into a sum of simpler functions. Fourier series arise frequently in science and engineering and are in fact just one of many ways that a complex function can be written as a sum of simpler functions.


Experiential Value of this Unit

Each unit should also impact the way you view the world. I want you to see and experience things differently as a result of taking this class. The following are just some ways in which this unit might impact the way you view the world.

1. This unit will change how you view waves. Where a wave may have previously been simply a shape, you will now naturally see water waves in terms of periods and wavelengths. You will see guitar strings in terms of tensions and harmonics. This will not cloud any of the natural beauty of the ocean or music, rather it will open you up to more details and more layers of beauty than you previously experienced.

2. This unit will change how you experience wind instruments. Where you may have previously heard beautiful sounds you will now be able visualize the music as well. You will be able to visualize the nodes and antinodes inside the instrument and will find enjoyment in watching how the player’s fingers or the instrument move as the frequency of the sound changes.

3. This unit will draw your attention to the medium and the space in which waves are occurring. You will take greater notice of the shapes of rooms, hallways, microwaves, and pools as well as the placement of speakers and other wave sources. You will naturally ponder the ways in which these shapes and placements might influence the resulting waves through reflections, interference, or beats.

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