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Exploring the wave equation.

The following command adds a few of the ui's and vi's into a nice function:

JSFHQyQ+SSN1MUc2IiwoKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kieEdGJSIiIi1JJGNvc0dGKjYjSSJ0R0YlRi9GLyomLUYpNiMsJEYuIiIjRi8tRik2IywkRjNGOEYvISIlKiYtRik2IywkRi4iIiRGLy1GMTYjLCRGM0ZBRi9GQUYvLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Note that this does solve the wave equation (with c = 1 and L = Pi):

QyQsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiVJI3UxRzYiSSJ4R0YpRioiIiItRiU2JUYoSSJ0R0YpRi4hIiJGKw==

This is the plot of u1:

QyQtSSdwbG90M2RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JUkjdTFHRigvSSJ4R0YoOyIiIUkjUGlHRiYvSSJ0R0YoO0YuLCRGLyIiIyIiIg==

And the animation of the string (to animate it, click on the picture, and some picture controls will appear at the top of the worksheet):

QyYtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIiLUkoYW5pbWF0ZUdGKDYnSSVwbG90R0YlNyRJI3UxR0YoL0kieEdGKDsiIiFJI1BpR0YmL0kidEdGKDtGNSwkRjYiIiMvSSdmcmFtZXNHRigiI10vSSpwYXJhbWluZm9HRihJJmZhbHNlR0YmIiIi

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This command makes an alternating sum of ui's:

QyQ+SSN1Mkc2Ii1JJHN1bUdGJTYkKiopISIiSSJuR0YlIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJkYsRi1JInhHRiVGLUYtLUkkY29zR0YwNiMqJkYsRi1JInRHRiVGLUYtRixGKy9GLDtGLSIjNUYt

Then plot u2:

QyQtSSdwbG90M2RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2KEkjdTJHRigvSSJ4R0YoOyIiIUkjUGlHRiYvSSJ0R0YoO0YuLCRGLyIiIy9JJWdyaWRHRig3JCIkKyJGOC9JJnN0eWxlR0YoSShzdXJmYWNlR0YoL0krbGlnaHRtb2RlbEdGKEknbGlnaHQxR0YoIiIi

And then the curve animation:

QyQtSShhbmltYXRlRzYiNidJJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU3JEkjdTJHRiUvSSJ4R0YlOyIiIUkjUGlHRikvSSJ0R0YlO0YwLCRGMSIiIy9JJ2ZyYW1lc0dGJSIkKyIvSSpwYXJhbWluZm9HRiVJJmZhbHNlR0YpIiIi

Eh, one more. Here is u3:

QyQ+SSN1M0c2Ii1JJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiosJiEiIiIiIilGLkkibkdGJUYvRi9GMUYuLUkkc2luR0YoNiMqJkYxRi9JInhHRiVGL0YvLUkkY29zR0YoNiMqJkYxRi9JInRHRiVGL0YvL0YxO0YvIiM1Ri8=

It satisfies wave:

QyQsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiVJI3UzRzYiSSJ4R0YpRioiIiItRiU2JUYoSSJ0R0YpRi4hIiJGKw==

It's plot:

QyQtSSdwbG90M2RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2KEkjdTNHRigvSSJ4R0YoOyIiIUkjUGlHRiYvSSJ0R0YoO0YuLCRGLyIiIy9JJWdyaWRHRig3JCIkKyJGOC9JJnN0eWxlR0YoSShzdXJmYWNlR0YoL0krbGlnaHRtb2RlbEdGKEknbGlnaHQxR0YoIiIi

It's curve animation:

QyQtSShhbmltYXRlRzYiNidJJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU3JEkjdTNHRiUvSSJ4R0YlOyIiIUkjUGlHRikvSSJ0R0YlO0YwLCRGMSIiIy9JJ2ZyYW1lc0dGJSIkKyIvSSpwYXJhbWluZm9HRiVJJmZhbHNlR0YpIiIi

Your turn!!

1) In the sum defining u2 and u3, change the number 10 (the number of terms) to something else. See what happnes as the number of terms increases.

2) Create your own cool solution to the wave equation. The coolest you can create is by taking some interesting sum of the ui's and/or vi's, like we did for u2 and u3.

JSFH