Worksheet for Exploration 17.1: Superposition of Two Pulses

One of the most interesting phenomena we can explore is that of a superposition of waves. Each panel shows an individual wave that is traveling on a string.

 

If these two waves are traveling on the same string, draw the superposition of the two pulses between t = 0 and t = 20 seconds in 2-second intervals for each animation (position is given in meters and time is given in seconds).

 

When you have completed the exercise check your answers.

 

Animation 1 sketches:

 

Animation 2 sketches:

 

Animation 3 sketches:

 

Animation 4 sketches:

 

Discuss what is meant by superposition.

 

Worksheet for Exploration 17.2: Measure the Properties of a Wave

 

Shown in black is a traveling wave (position is given in centimeters and time is given in seconds). Measure the relevant properties of this wave and determine the wave function of the wave. Once you are finished, check your answer by importing a f(x, t) and look at the red wave to see if it matches.

 

i. The equation for the wave may be written in several ways that are equivalent. One way is:

 

]t T 2x2sin[A)t,x(f oφ+ π

 

− λ π

 

= . The velocity of the wave is T λ

 

. For the given wave

 

determine:

 

A=_______ λ=_______ T=_______

 

v=_______ f=_______

 

ii. What is the effect of changing the initial phase of your “check” function?

 

Worksheet for Exploration 17.3: Traveling Pulses and Barriers

 

A string can be approximated by many connected particles as shown in the animations (position is given in meters and time is given in seconds). Restart. This Exploration considers a pulse on a string and looks at the motion of the individual particles that make up such a string. Pulse 1 shows a Gaussian pulse incident from the left, while Pulse 2 shows a Gaussian pulse incident from the right. Notice how the particles never really move in the x direction, yet the creen. information in the pulse does travel across the s

 

In the other two animations the pulse is incident from the left and hits either a Hard or a Soft barrier. The hard barrier example is depicted by the hand that represents a string whose end is tied down; the soft barrier example represents a string with one end free.

 

a. During the hard barrier example, what is the direction of the force that is exerted on the hand?

 

Explain:

 

b. During the hard barrier example, what is the direction of the force that is exerted on the string?

 

Explain:

 

c. Describe the differences between the waves reflected at the two barriers (Hard or a Soft). Explain those differences.

 

Worksheet for Exploration 17.4: Superposition of Two Waves

 

The top two windows display waves that are traveling simultaneously in the same nondispersive medium: string, spring, air column, etc. (position is given in meters and time is given in seconds). The wave in the bottom window is the superposition (algebraic sum) of the two component waves in the upper windows. The superposition is what you would actually see. You wouldn’t see the component waves. Restart. You can adjust the amplitude, wavelength, and wave speed for g (x, t) (the middle window). For the waves described (traveling in the same medium), the two waves could have different amplitudes and wavelengths, but they must have the same speed (you will need to adjust the wave speed of g (x, t) appropriately).

 

a. Why must the two waves have the same speed? (Think in terms of what influences wave speed in the medium.) i. Note that in general different wavelengths can have different speeds. The question here

 

means that “in order to give a constant superposition waveform “ the speeds must be the same (and direction).

 

b. For each f (x, t) determine the amplitude, wavelength, frequency, and wave speed of the wave. Check your answer by making g (x, t) identical to f (x, t).

 

f(x,t) = 3*cos[2*pi(x/8-t/4)]

 

A=_______

 

λ=_______

 

f=_______

 

f(x,t) = 3*cos[2*pi(x/2-t/2)]

 

A=_______

 

λ=_______

 

f=_______

 

c. Determine the amplitude, wavelength, and wave speed of the wave, g (x, t), that will make f + g a standing wave. i. Do this for each of the waveforms f.

 

Worksheet for Exploration 17.5: Superposition of Two Waves

 

The top two windows display waves that are traveling simultaneously in the same nondispersive medium: string, spring, air column, etc. (position is given in meters and time is given in seconds). Restart. Note that the two waves are traveling at the same speed in opposite directions and that they have the same amplitude and wavelength. It is, of course, possible that the two waves could have different amplitudes and wavelengths. However, the waves that we are studying must have the same speed.

 

The wave in the bottom window is the superposition (algebraic sum) of the two component waves in the upper windows. The superposition is what you would actually see. You wouldn’t see the component waves.

 

a. Why must the two waves have the same speed? (Think in terms of what influences wave speed in the medium.)

 

b. Stop the top wave and measure its wavelength in units of divisions along the horizontal axis. Sketch the wave, showing the two points between which you measured the wavelength.

 

λ=_______

 

Sketch:

 

c. Now measure the period of the top wave in time units. Describe your method for doing this.

 

T=_______

 

d. Calculate the speed of the top wave. Show your work.

 

v=_______

 

e. Assume that the bottom wave shown represents the displacement of a string. What is the longitudinal speed of a point on the string?

 

f. Assume that the bottom wave shown represents the displacement of a string. Is there a time when the transverse speed of the string is zero?

 

g. What relationship, if any, do the speeds in (d), (e), and (f) have to one another?

 

Worksheet for Exploration 17.6: Make a Standing Wave

 

Find a wave function, g(x, t), that will produce a standing wave with a node at x = 0 m, i.e., at the center (position is given in meters and time is given in seconds). You may want to pause the animation before you click-drag the mouse to read position coordinates.

 

i. Write out the amplitude, wavelength, period, and wavespeed of the wave f(x,t).

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