Superposition

Adding Two Waves Simulation

Description:

Be sure the simulation has finished loading before you begin.

The generic equation for a traveling sine wave is A*sin(k*x-w*t) where A is the amplitude, k is the wave vector (k = 2*pi/wavelength) and w is the angular frequency (w =2*pi*f where f is the frequency in Hertz).This simulation adds the function f(x,t) (shown in the top graph) to the function g(x,t) (middle graph) to get the graph on the bottom.

Click and drag inside the graphs to read the coordinates to obtain numerical values for the amplitudes and wavelengths.

Enter the functions you want to see added and click the 'Set' button to activate the function:

Ignore the sum f(x,t)+g(x,t) for questions 1 and 2. Don't forget to click the 'Set' button to make the changes take place.

Question 1:

For the initial f(x,t), click the 'Stop' button and click and drag the mouse to measure:

The amplitude (half the height from top to bottom) of f(x,t).
The wavelength (peak to peak distance).
The period (start the simulation, hold the mouse in one location and time how long it takes for an entire wave to pass that location).
The speed of the wave (v = wavelength/period).

Question 2:

Double the wavelength by changing f(x,t) to be f(x,t) = 2.5*sin(0.5*x-t). Measure the new wavelength of f(x,t) on the simulation. How does it compare with the wavelength of g(x,t)?
Go back the original wavelenght and double the frequency: f(x,t) = 2.5*sin(x-2*t). Now what are the differences between f(x,t) and g(x,t)?
Double the wavelenght and double the frequency: f(x,t) = 2.5*sin(0.5*x-2*t). Now what are the differences between f(x,t) and g(x,t)?

Question 3:

The intial simulation adds two identical sine waves. Click the 'Stop' button and click and drag the mouse to measure:

The amplitude (half the height from top to bottom) of f(x,t) or g(x,t) (they are the same).
The amplitude of the sum, f(x,t)+g(x,t). How does it compare with the amplitudes of f(x,t) and g(x,t)?
The wavelength (peak to peak distance) of the sum. How does it compare with the wavelenghts of f(x,t) and g(x,t)?
The period of the sum (start the simulation, hold the mouse in one location and time how long it takes for an entire wave to pass that location). How does it compare with the period of f(x,t) and g(x,t)?
The speed of the sum (v = wavelength/period). How does it compare with the speed of f(x,t) and g(x,t)?

Question 4:

Change the amplitude of f(x,t) from 2.5 to 4 and the amplitude of g(x,t) from 2.5 to 1 and click the 'Set' buttons to make the changes take effect. What is the amplitude of f(x,t)+g(x,t)? How does this compare to the original case?

Question 5:

Go back to the original amplitudes but change the minus sign to a plus sign in both f(x,t) and g(x,t) (so now f(x,t) = 2.5*sin(x+t) and g(x,t) = 2.5*sin(x+t). Do the amplitudes of any of the waves change? How about the period or wavelength? What does change?

Question 6:

Change one of the plus signs back to a minus sign (so now f(x,t) = 2.5*sin(x+t) and g(x,t) = 2.5*sin(x-t). The sum f(x,t)+g(x,t) is called a standing wave in this case. Do the amplitudes of any of the waves change? How about the period or wavelength? What does change (describe the added waves)?

Question 7:

Now enter the following functions: f(x,t) = 2.5*sin(x-t) and g(x,t) = 2.5*sin(x-1.2*t). Watch the sum f(x,t)+g(x,t) for a few seconds and describe what happens. Two waves with slightly different frequencies added to gether give rise to the phenomena of beats. Find the beat frequency the following way: Stop the simulation (use the step buttons if you overshoot) when the two source waves exactly cancel (f(x,t)+g(x,t) is a straight line) and record the time. Start the simulation and stop it again the next time the waves cancel. Record the new time and subtract to get the elapsed time. The beat frequency is 1/(time elapsed). How does this compare to the frequency of f(x,t) subtracted from the frequency of g(x,t)? (Hint: Use f = w/(2*pi) to get the frequencies of f(x,t) and g(x,t).)

Question 8:

Not all waves are sinusoidal. Click the following button to see two Gaussian pulses (note that you can use the 'Reset' button under the graph to restart the pulses). Click the 'Stop' button and click and drag the mouse to measure the amplitudes before collision. How do the amplitudes before collision compare with the amplitude of the sum f(x,t) + g(x,t):

When they are exactly on top of each other?
When they are overlapping by about half (use the step button)?
In general for all cases (sine and Gaussian), what can you say about the amplitude of f(x,t)+g(x,t) at any point on the graph compared to the amplitude of f(x,t) and g(x,t) at the same point?

Applet by Wolfgang Christian, Davidson University. Web page modifications by J. Sullivan (1/99). Questions by Kyle Forinash (6/03).