• N-Mass Systems vs. Distributed Systems:
• The Wave Equation (for a stretched string):
• Traveling Waves:
• Reflections at Boundaries:
• Sound Waves
• The Doppler Effect

Waves

N-Mass Systems vs. Distributed Systems:

• An N-mass system has N modes per degree of freedom.
• As N gets very large, it becomes convenient to view the system as a continuous string with a uniform mass density and tension.

The Wave Equation (for a stretched string):

• ``Equation of motion for a wave''.

  Figure 10: Short string section.

  

• The mass of the short string section (length $\Delta x$) is $\mu \Delta x$, where $\mu$ is the mass per unit length of the string.
• The net vertical force on the section is $T \sin \theta_{2} - T \sin \theta_{1}$.
• For small angles, $\sin \theta \approx \tan \theta$ and $\tan \theta = \frac{\partial y}{\partial x} = m$, the slope of the string.
• The net vertical force can thus be rewritten: $T \Delta m$.
• By Newton's Second Law: $T \Delta m = (\mu \Delta x) \frac{\partial^{2}y}{\partial t^{2}} \hspace{0.1in}... ...e{0.1in} T \frac{\Delta m}{\Delta x} = \mu \frac{\partial^{2}y}{\partial t^{2}}$.
• As $\Delta x \rightarrow 0$,
  
  $$\lim_{\Delta x \rightarrow 0} \frac{\Delta m}{\Delta x} = \f... ...artial m}{\partial x} = \frac{\partial^{2} y}{\partial x^{2}}.$$
  
  

• The expression based on Newton's Second Law then becomes
  
  $$\frac{\partial^{2} y}{\partial t^{2}} = c^{2} \frac{\partial^{2} y}{\partial x^{2}},$$
  
  
  where $c = \sqrt{T/\mu}$ is the speed of wave motion on the string. This is the one-dimensional wave equation which describes small amplitude transverse waves on a stretched string.

Traveling Waves:

• Solution to the wave equation of the form $y = f_{1}(ct - x) + f_{2}(ct + x)$.
• $f_{1}(ct - x)$ represents a wave traveling to the right with a velocity $c$. $f_{2}(ct + x)$ represents a wave traveling to the left with the same velocity.
• The functions $f_{1}$ and $f_{2}$ are arbitrary.

Reflections at Boundaries:

• At a fixed end, $y=0$. If the string is fixed at $x=0$, then $y=0$ and $f_{1}(ct) = -f_{2}(ct)$.
• At a free end, $\partial y/\partial x = 0$ because no transverse force is possible. In this case, $f_{1}(ct) = f_{2}(ct)$.

Sound Waves

• Longitudinal waves that travel in a solid, liquid, or gas.
• The speed of sound in air is approximately given by $c = 331.3 + 0.6 t$ where $t$ is the temperature of the air in degrees Celsius.

The Doppler Effect

• Observer moving toward source:
  
  $$f' = f_{s} \frac{v + v_{o}}{v},$$
  
  
  where $f_{s}$ is the frequency of the source, $v_{o}$ is the speed of the observer, and $v$ is the speed of sound.

• Source moving toward observer:
  
  $$f = f_{s} \frac{v}{v-v_{s}},$$
  
  
  where $v_{s}$ is the speed of the source.