Lecture 27

Readings

http://umdberg.pbworks.com/w/page/125131262/Workout%3A%20Wave%20pulses

http://umdberg.pbworks.com/w/page/73086062/Wave%20pulses%20%282013%29

http://umdberg.pbworks.com/w/page/52293895/Propagating%20a%20wave%20pulse%20-%20the%20math

Waves - Continued

• Wave pulse activity = "wave velocity"

• Equation for wave

  • General Form = $f(x-(v_0\ast t)$$f(x - (v_0*t)$ or $f(x+(v_0\ast t))$$f(x + (v_0*t))$

  • When you see the $(x-(v_0\ast t))$$(x- (v_0*t))$ or $(x+(v_0\ast t))$$(x + (v_0*t))$ as arguments of a function, it means some quantity (like height, pressure, etc.) will move left or right (in space) as time goes on

  • Example: $cos(x-(v_0\ast t))$$cos(x-(v_0*t))$

    • At $t=0$$t=0$ , peak is at $x=0$$x=0$

      • $(x-(v_0\ast t))\to (0-(v_0\ast t))=0$$(x-(v_0*t)) \rightarrow (0 - (v_0*t)) = 0$
      • The argument of the cosine function is $0$$0$ , and $cos(0)=1$$cos(0) = 1$

• Now try $t=1$$t=1$

  • Where is the peak? Which peak are we talking about?
  • at $t=0$$t=0$ , the picture looked like:

  

  • Cosine has many peaks

  • I'm following the peak that was at $x=0$$x=0$ when $t=0$$t=0$ , which is equivalent to the peak where the argument $x-(v_0\ast t)=0$$x-(v_0*t)=\color{green}{0}$

  • Ok, at $t=1$$t=1$ , the argument becomes $x-(v_0\ast 1)$$x-(v_0*1)$

  • I want this to still equal 0 , so still get $cos(0)=1$$cos(0)=1$

  • $v_0$$v_0$ is fixed, so $x$$x$ is something new

  • it will be the new location for some peak:

    • $x-(v_0\ast 1)=0\to x=v_0\ast 1$$x-(v_0*1)=0 \rightarrow x = v_0*1$

    • call this $x_1$$x_1$

      • $x_1=v_0\ast 1$$x_1 = v_0*1$

      

    • As $t$$t$ increases, $x$$x$ must increase

      • This means "our" peak is moving to the right

  • What happens if we have equation $cos(x+(v_0\ast t))$$cos(x+(v_0*t))$

    • Notice the ${}^{\prime }{+}^{\prime }$$'+'$ sign

    • As $t$$t$ increases:

      • $x$$x$ must decrease in order to keep the argument $(x+(v_0\ast t))$$(x+(v_0*t))$ the same value
      • So this wave is moving to the left

  • $cos(x-(v_0\ast t))$$cos(x \color{red}{-} (v_0*t))$ means wave is moving to: $+x$$\color{red}{+}x$

  • $cos(x+(v_0\ast t))$$cos(x \color{red}{+} (v_0*t))$ means wave is moving to: $-x$$\color{red}{-}x$

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