7. Adding Two Linear Waves (Superposition).
The waves we have been
discussing so far and the ones that are most often seen in everyday
life, such as light and sound, are for the most part linear waves.
Linear waves have the
property that their amplitudes add linearly if they arrive at the same
point at the same time. This gives rise to several interesting
phenomena in nature.
The simulation shows the function f(x, t) in the top panel, g(x,t) in the middle panel and f(x,t)+g(x,t) in the bottom panel.
Note: Animations may take a few seconds to load.
Enter the functions you want to see added and click the 'set' button to
activate the function:
Questions:
7.1. Holding the left mouse button down in the panels gives the x and y coordinates in a yellow box at the lower left. Click the
'Stop'
button and click and measure the amplitude (half the height from highest point to lowest point)of f(x,t) or
g(x,t) (they are the same) and the amplitude of the sum, f(x,t)+g(x,t). How does the amplitude of the sum compare with the amplitude of f(x,t) or
g(x,t)?
7.2. How does the wavelength, frequency and speed of f(x,t) or
g(x,t) compare with the wavelength, frequency and speed of f(x,t) or
g(x,t)? This is an example of constructive interference.
7.3. Change f(x,t) to have a phase of p (type 2.5*sin(x-t+pi) for f(x,t) ). What happens to the amplitude of the sum of the two waves? This is an example of destructive interference.
7.4. Experiment with cases in between total destructive and total constructive interference by changing the phase of f(x,t) to be 1/2 p, 1/3 p, and 1/4 p. Stop the simulation each time and record the amplitude of the sum compared to the amplitude of f(x,t) or
g(x,t) .
7.4. Click on 'reset to original case' and then change the amplitude of f(x,t) from 2.5 to 4 and the amplitude
of g(x,t) from 2.5 to 1 clicking 'set' for both. What is the amplitude of f(x,t)+g(x,t) in this case? How does this amplitude
compare to the original case?
7.5. Go back to the original amplitudes but change one of the minus signs to a plus 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 (an example would be the waves on a guitar string). Describe what you see. How does the period and wavelength
of the combined wave compare to the period and wavelength of two
components? How is the maximum amplitude of the sum related to the
amplitudes of the two components? What can you say about the speed of
the sum?
7.6. Now enter the following functions: f(x,t) = 2.5*sin(x-t) and g(x,t) =
2.5*sin(1.1*x-1.1*t) (you and cut and paste instead of typing). Watch the sum f(x,t)+g(x,t) for a while and
describe what happens. Two waves with slightly different frequencies
added together give rise to the phenomena of beats. Notice that the waves are still traveling at the same speed. Find the
beat frequency the
following way: Stop the simulation
when the two source waves exactly cancel (f(x,t)+g(x,t) is a straight
line)
and record the time (use the step buttons if you
overshoot). 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)?
Credits.