Lecture 29

Readings

http://umdberg.pbworks.com/w/page/125132021/Workout%3A%20Superposition%20of%20waves

http://umdberg.pbworks.com/w/page/65139402/Sinusoidal%20waves%20%282013%29

http://umdberg.pbworks.com/w/page/107013348/Beats

Sinusoidal Waves

• Waves can have any "shape" , but we often use sinusoids

  • Nature often does this
  • Any wave, even strange shapes, can be broken down into sum of sinusoids, each with a different frequency ( superposition )

• We've been sloppy with units for waves

• We've written stuff like $y(t)=3\ast m\ast cos(2\ast \pi \ast (x-(5\ast t)))$$y(t) = 3*m*cos(2*\pi*(x-(5*t)))$

  • But argument of sine or cosine always must be dimensionless
  • So in above expression, $x$$x$ must be multiplied by something with units of $\frac{1}{length}$$\frac{1}{length}$
  • And $t$$t$ must be multiplied by something with units of $\frac{1}{time}$$\frac{1}{time}$

• So let's write it as $y(t)=A\ast cos((k\ast x)-(\omega \ast t))$$y(t) = A*cos((k*x)-(\omega*t))$

  • $\omega$$\omega$ is our familiar angular frequency with dimensions of $\frac{radians}{second}$$\frac{radians}{second}$ or just $\frac{1}{second}$$\frac{1}{second}$

• This $k$$k$ , is completely new

  • It's not a spring constant or any other $k$$k$ we've used
  • It's called "wave number"
  • Dimensions = $\frac{1}{length}$$\frac{1}{length}$

• Example, earlier expression: $y(t)=3\ast m\ast cos(2\ast \pi \ast (x-(5\ast t)))$$y(t) = 3*m*cos(2*\pi*(x-(5*t)))$

  • $k$$k$ = coefficient in front of $x\to \frac{1}{2\ast \pi meters}$$x \rightarrow \frac{1}{2*\pi\ meters}$

    • $k=\frac{2\ast \pi }{Wavelength(\lambda )}$$k = \frac{2*\pi}{Wavelength\ (\lambda)}$

  • $\omega$$\omega$ = coefficient in front of $t\to \frac{1}{2\ast \pi \ast 5}=\frac{1}{10\ast \pi seconds}$$t \rightarrow \frac{1}{2*\pi*5} = \frac{1}{10*\pi\ seconds}$

  • Don't worry about signs.

    • Signs determine direction wave is moving, but you're pretty safe letting $k$$k$ and $\omega$$\omega$ always be positive

• Wave speed = speed that the crest is moving

  • Remember, not transverse speed of mass
  • $WaveVelocity(v)=\frac{AngularFrequency(\omega )}{WaveNumber(k)}$$Wave\ Velocity\ (v) = \frac{Angular\ Frequency\ (\omega)}{Wave\ Number\ (k)}$

• So if you have equation, you can find wave speed by looking at coefficients of $x$$x$ and $t$$t$

  • Example, our wave: $y(t)=3\ast m\ast cos(2\ast \pi \ast (x-(5\ast t)))$$y(t) = 3*m*cos(2*\pi*(x-(5*t)))$

    • $\omega =10\ast \pi$$\omega = 10*\pi$
    • $k=2\ast \pi$$k = 2*\pi$
    • $v=\frac{10\ast \pi }{2\ast \pi }=\frac{5meters}{1second}$$v = \frac{10*\pi}{2*\pi} = \frac{5\ meters}{1\ second}$

   

Beats

• 2 waves, close in frequency

• When they are "in sync"

  • the amplitude is doubled
  • Crest adds to Crest

• When they are "out of sync"

  • Crests adds to Trough

  • You get zero amplitude

    • Aka, no sound

• The frequency that they go "in and out of sync" depends on difference in 2 original frequencies

Wavelength

• $\lambda$$\lambda$ = distance between crests

  • or length ( in distance ) of 1 full wave

• also, $k=\frac{2\ast \pi }{\lambda }$$k = \frac{2*\pi}{\lambda}$

  • or $\lambda =\frac{2\ast \pi }{k}$$\lambda = \frac{2*\pi}{k}$