Open Graphing Calculator
A sine wave has the equation ...
The constant k is written as 2p/l where lambda is the wavelength. When k is one the wavelength l equals 2p.
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Enter the wave ... Experiment with different values of the constants. What are the amplitudes and what is the wavelengths of these waves? |
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Enter the wave ... What is the amplitude and what is the wavelength of this special wave? |
The expression ...
... where k and w are constants is a standing wave.
In graphing calculator the expression wt is represented by a constant times n (eg. 4n) where n can be set to change from 0 to 10 in 100 steps.
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Enter the wave ... Note: the dot in the equation is a multiply sign. Hold down shift and type 8. The machine displays a dot. |
Instructions:
Click the n on the panel below ....
[The panel is at the bottom of the display in Graphing Calculator.]
i Set the number of steps to 100 and click OK.
ii Watch the standing wave.
iii Experiment with the constants.
The traveling wave equation has the form ...
The variable x is replaced by the new variable (x - vt). As time passes the wave moves to the right.
Note: replacing x with (x-a) in any equation moves the graph to the right. Try it with the straight lines; y = x and y = (x - 1).
In graphing calculator the expression vt is represented by a constant times n (eg. 4n) where n can be set to change from 0 to 10 in 100 steps. (See above if you need instructions)
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Enter the traveling wave... |
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Enter the new traveling wave... |
You cannot enter two waves at once. Put these in one at a time.
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Enter the traveling wave ... ... and the traveling wave ... |
The two traveling waves have the same amplitude and velocity but the have different wavelengths.
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Enter the sum ... Experiment by typing in different amplitudes in Graphing Calculator. i Describe the effect of changing one of the amplitudes. ii What would you hear if you were listening to a sound with a wave pattern like that in the bottom diagram at right? |
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Note the beat-length and the wavelength here and then double the frequency difference by replacing 2.2 in the second wave equation with 2.4. i Describe the effect of increasing the frequency difference. ii What would you hear if you were listening to a sound with a wave pattern like that in the diagram at right? |
It can be shown that the intensity of a wave - (the energy per second that is carried by the wave) - is proportional to amplitude squared.
Adding waves of different frequency
If a modulated wave is the sum of two waves of amplitude 2A, the sum of the two intensities is proportional to....
The average intensity of the modulated wave is proportional to....
Adding waves of different amplitude and frequency
If a modulated wave is the sum of two waves of amplitudes A and 4A, the sum of the two intensities is proportional to....
The average intensity of the modulated wave is proportional to....
Ring modulation is achieved when two signals are multiplied together.
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Enter the wave ... ... and then the wave ... |
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Enter the product ... |
Explanation
The process of multiplying the two signals has effectively removed the single frequency and replaced it with two frequencies - one above and one below the original. Effectively, the two signals beat, producing amplitude modulation.
* This type of modulation is called Ring Modulation because it was originally achieved with a ring of transformers. The effect is now more commonly produced digitally.
The figure shows two waves of different wavelength, amplitude and velocity.
The two waves are now multiplied together to give a ring modulated signal.
The beats as a whole move slowly to the right with a speed known as the group velocity.
Any information on a modulated signal of any kind travels at the group velocity. For electromagnetic waves the group velocity is less than the velocity of light in the medium.
If you watch the figure carefully you will see a second effect running to the right at a higher velocity. The effect is known as the phase velocity and may, for electromagnetic waves be greater than the speed of light.
Audio frequency information can be transmitted by modulating a carrier wave of high frequency (typically 100 Kilohertz). The modulation can be of two types. Amplitude modulation (AM) or frequency modulation (FM). We consider only amplitude modulation in this article.
The modulated carrier wave can be considered as the sum of three waves. The strong carrier and two weaker secondary signals with higher and lower frequencies called side bands.
If the carrier is modulated by many frequencies the side band structure is more complex . (Lower diagram)
The presence of side bands leads to a characteristic band width for all modulated signals. For this reason carrier frequencies are licensed to broadcasters to avoid multiple reception. Only a few broad band signals (carrying much information) can be fitted into a given frequency range.
Any modulated signal can be resolved into a carrier and side bands. Side bands can be found in frequency spectra of sounds made by the human voice and by musical instruments at high intensities because instabilities often lead to modulation.
An amplitude modulated signal can be produced by adding a single side band to a carrier frequency. Single side band modulation is not mathematically identical to double side band modulation but the information carrying capacity of the modulated signal is the same for AM radio transmission and reception. Single side band transmission is commonly used on crowded amateur bands to reduce band width.
*Topics not required for IB examination.