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# Section 5.3: Exploring the Properties of Waves

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Understanding wave properties includes developing a mathematical relationship for a traveling 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 entering a function, *f*(*x*, *t*), in the text box and looking at the red wave to see if it matches. Restart.

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With an understanding of the mathematical description of a traveling wave, we can now explore what happens when traveling waves *run into* each other: the superposition of waves.
Restart. A superposition of two traveling waves is nothing more than the arithmetic sum of the amplitudes of the two underlying waves. We represent the amplitude of a transverse wave by a wave function, *y*(*x*, *t*). Notice that the amplitude, the maximum value of *y*, is a function of position on the *x* axis and the time. Given two waves moving in the same medium, we call them *y*_{1 }= *f*(*x*, *t*)
and y_{2 }= *g*(*x*, *t*). Their superposition, arithmetic sum, is written as *f*(*x*, *t*) +
*g*(*x*, *t*). This may seem like a complicated process, but we simply need to add the waves together at each point on the string **(position is given in centimeters and time is given in seconds)**. Consider the two waves shown above. If these two wave packets are traveling on the same string, draw on a piece of paper the superposition of the two wave packets between *t* = 0 and *t* = 20 s in 2-s intervals for each animation.

When you have completed the exercise, check your answers with the animations provided below.

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