Exercise 13: Waves

python homework

Introduction and background

In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium (space or mass). Frequency refers to the addition of time. Wave motion transfers energy from one point to another, which displace particles of the transmission medium–that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations (of a physical quantity), around almost fixed locations.
There are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed. Restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When the molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave.The second main type, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields originally generated by charged particles, and can therefore travel through a vacuum. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.

Abstruct

In this exercise we consider wave motion. Use numerical method to solve wave function, especially for a wave spreading in a string. Further more, it will discuss the realistic case, which means considering friction.

Content

• problem 6.16
  Perform the calculations described in this section. One interesting possibility is to compare the size of the octave stretching, that is, the magnitude of the deviations from a purely harmonic spectrum, for short (treble) and long (bass) strings. The relevant string parameters for a good grand piano are given in Table 6.1
  Table 6.1

  $Note$	$f(Hz)$	$L(m)$	$c(m/s)$	$ε(unitless)$	$b(s^{-1})$
  $C2$	65.4	1.9	250	$7.5×10^{-6}$	0.5
  $C4$	262	0.62	330	$3.8×10^{-5}$	0.5
  $C6$	2093	0.09	380	$8.7×10^{-4}$	0.5

• Numerial approach
  The central equation of wave motion is
  

  $$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}$$
  In the numerical approach, we rewrite the equation into this form:
  $$\frac{y(i,n+1)+y(i,n-1)-2y(i,n)}{\Delta t^2}≈c^2[\frac{y(i+1,n)+y(i-1,n)-2y(i,n)}{(\Delta x)^2}]$$
  Rearranging the former expression we can express $y(i,n+1)$ in terms of y at pervious time steps, with the result:
  $$y(i,n+1)=x[1-r^2]y(i,n)-y(i,n-1)+r^2[y(i+1,n)+y(i-1,n)]$$
  where $r=c\Delta t/\Delta x$. Generally, we let $r=1$ to keep the stability for the solution.
  In the case of realistic string, the numerical equation becomes:
  $$y(i,n+1)=[2-2r^2-6\varepsilon r^2M^2]y(i,n)-y(i,n-1)+r^2[1+4\varepsilon M^2][y(i+1,n)+y(i-1,n)]-\varepsilon r^2M^2[y(i+2,n)+y(i-2,n)]$$

• Figures and Codes
  Code
  The velocity of wave is $c=300m/s$, the space speration is $\Delta x=0.01m$.
  Figure 1: The initial condition is a sine wave.
  
  
  Figure 2: The initial condition is a "Gaussian pluck".
  
  
  Figure 3: The initial condition is a realistic excitation for a guitar string that is plucked, as shown in the FIGURE 6.4 of the textbook. The excited amplitude is 0.1m.
  
  

Thanks to

Chen Feng