Gaussian Wave
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
Short
Video
\(\LaTeX\)
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Exercise:
A single wave crest can be described by a Gaussian function. In its simplest form this can be written as fxe^-x^. Find a physically correct Gaussian shape function and erpret the parameters. Derive the corresponding wave function and plot this for several different times.
Solution:
In order to turn the Gauss function o a physically sensible function we need to include two additional parameters: fx Ae^-fracleftfracxsigmaright^ The effect of the amplitude A should be clear. The parameter sigma describes the width of the Gaussian wave crest. For xsigma the height is equal to fsigma A e^-frac fracAsqrteapprox .times A The figure below displays two such Gaussian wave crests with different amplitudes and widths. center includegraphicswidthcm#image_path:gaussian-shape# center To turn this expression o an expression describing a wave we need to substitute x by x-v t: yx t fx-v t Ae^-fracleftfracx-v tsigmaright^ The figure below displays snapshots of a Gaussian wave propagating in the positive direction vvO t_taO t_tbO t_tcO. center includegraphicswidthcm#image_path:gaussian-wave# center An animation of the wave propagation is linked to this exercise.
A single wave crest can be described by a Gaussian function. In its simplest form this can be written as fxe^-x^. Find a physically correct Gaussian shape function and erpret the parameters. Derive the corresponding wave function and plot this for several different times.
Solution:
In order to turn the Gauss function o a physically sensible function we need to include two additional parameters: fx Ae^-fracleftfracxsigmaright^ The effect of the amplitude A should be clear. The parameter sigma describes the width of the Gaussian wave crest. For xsigma the height is equal to fsigma A e^-frac fracAsqrteapprox .times A The figure below displays two such Gaussian wave crests with different amplitudes and widths. center includegraphicswidthcm#image_path:gaussian-shape# center To turn this expression o an expression describing a wave we need to substitute x by x-v t: yx t fx-v t Ae^-fracleftfracx-v tsigmaright^ The figure below displays snapshots of a Gaussian wave propagating in the positive direction vvO t_taO t_tbO t_tcO. center includegraphicswidthcm#image_path:gaussian-wave# center An animation of the wave propagation is linked to this exercise.
Meta Information
Exercise:
A single wave crest can be described by a Gaussian function. In its simplest form this can be written as fxe^-x^. Find a physically correct Gaussian shape function and erpret the parameters. Derive the corresponding wave function and plot this for several different times.
Solution:
In order to turn the Gauss function o a physically sensible function we need to include two additional parameters: fx Ae^-fracleftfracxsigmaright^ The effect of the amplitude A should be clear. The parameter sigma describes the width of the Gaussian wave crest. For xsigma the height is equal to fsigma A e^-frac fracAsqrteapprox .times A The figure below displays two such Gaussian wave crests with different amplitudes and widths. center includegraphicswidthcm#image_path:gaussian-shape# center To turn this expression o an expression describing a wave we need to substitute x by x-v t: yx t fx-v t Ae^-fracleftfracx-v tsigmaright^ The figure below displays snapshots of a Gaussian wave propagating in the positive direction vvO t_taO t_tbO t_tcO. center includegraphicswidthcm#image_path:gaussian-wave# center An animation of the wave propagation is linked to this exercise.
A single wave crest can be described by a Gaussian function. In its simplest form this can be written as fxe^-x^. Find a physically correct Gaussian shape function and erpret the parameters. Derive the corresponding wave function and plot this for several different times.
Solution:
In order to turn the Gauss function o a physically sensible function we need to include two additional parameters: fx Ae^-fracleftfracxsigmaright^ The effect of the amplitude A should be clear. The parameter sigma describes the width of the Gaussian wave crest. For xsigma the height is equal to fsigma A e^-frac fracAsqrteapprox .times A The figure below displays two such Gaussian wave crests with different amplitudes and widths. center includegraphicswidthcm#image_path:gaussian-shape# center To turn this expression o an expression describing a wave we need to substitute x by x-v t: yx t fx-v t Ae^-fracleftfracx-v tsigmaright^ The figure below displays snapshots of a Gaussian wave propagating in the positive direction vvO t_taO t_tbO t_tcO. center includegraphicswidthcm#image_path:gaussian-wave# center An animation of the wave propagation is linked to this exercise.
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