Chapter 6 Problem 16: Realistic String

计算物理

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Name: 陈锋
Student Number: 2013301020145
May 29, 2016

Abstract

This article is about using 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.

Introduction

In this case, the equation of vertical motion for a point in the string is:

$$\begin{equation} \rho\Delta x\frac{d^2y}{dt^2} = Tsin\theta_{i+1} - Tsin\theta_{i} \end{equation}$$
When $\Delta x$ is sufficient small, we have:
$$\begin{align} \Delta x = dx \\ sin\theta \approx tan\theta = \frac{\partial y}{\partial x} \end{align}$$
Then, the equation (1) becomes:
$$\begin{equation} \frac{\partial ^2 y}{\partial t^2} = \frac{T}{\rho}\frac{\partial ^2 y}{\partial x^2} = c^2\frac{\partial ^2 y}{\partial x^2} \end{equation}$$
Also, if we consider the friction, the equation of vertical motion becomes:
$$\begin{equation} \frac{\partial ^2 y}{\partial t^2} = c^2(\frac{\partial ^2 y}{\partial x^2} - \epsilon L^2\frac{\partial ^4 y}{\partial x^4}) \end{equation}$$
In the numerical approach, we rewrite the equation into this form:
$$\begin{equation} \frac{y(i,n+1)+y(i,n-1) - 2y(i,n)}{(\Delta t)^2} \approx c^2[\frac{y(i+1,n)+y(i-1,n) - 2y(i,n)}{(\Delta x)^2}] \end{equation}$$
Rearranging the former expression we can express $y(i, n+1)$ in terms of y at pervious time steps, with the result:
$$\begin{equation} y(i,n+1) = 2[1-r^2]y(i,n) - y(i,n-1) +r^2[y(i+1,n)+y(i-1,n)] \end{equation}$$
where $r = c\Delta t/\Delta x$. Generally, we let $r = 1$ to keep the stability for the solution.
Also, in the case of realistic string, the numerical equation becomes:
$$\begin{equation} y(i,n+1) = [2-2r^2 - 6\epsilon r^2M^2]y(i,n) - y(i,n-1) +r^2[1 + 4\epsilon M^2][y(i+1,n)+y(i-1,n)] - \epsilon r^2 M^2[y(i+2,n)-y(i-2,n)] \end{equation}$$
where $M = L/\Delta x$ is the number of spatial units along the string.

Solving the equation in different initial condition

The velocity of wave is $c = 300m/s$, the space speration is $\Delta x = 0.01m$ and $\Delta t = \Delta x /c$.

Figure 1: The initial condition is a sine wave. Obviously, in this situation, the wave we solved is a standing wave.

Figure 2: The initial condition is a "Gaussian pluck". That is, we have taken the initial string profile to be $y_0(x) = exp[-k(x-x_0)^2]$. Where the displacement is centerd at $x_0$, and $k$ is a factor that determines the width of the Guassian envolope. In this case, I used $x_0 = 0.3m$ and $k = 0.1m^{-2}$

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.

String signal and power spectra for ideal strings

The recording point is $x = 0.2m$.

Figure 4: In this standing wave case, every point on the string does harmonic motion, which means the string signal must be a sine or cosine wave.

Figure 5: This string was excited with a Guassian initial pluck centered at $x = 0.3m$. Either the odd frequencies or the even frequencies were excited in this situation.

Figure 6: This string was excited with a realistic initial pluck centered at $x = 0.125m$. Only a few low frequencies could be excited in this case.

Power spectra for "realistic" strings

In this section, I will consider the influence of the friction of the string. In all cases data of the point at $x = 0.2m$ was analyzed, and I used $\Delta t = \Delta x/4c$ to ensure stability for all values of $\epsilon$ employed.

Figure 7: The power spectra for "realistic" string in the case of Guassian pluck.

Figure 8: The power spectra for "realistic" string in the case of realistic pluck.
As the figure showed above, there are large frequency shifts in the high frequency. I construct a high frequency standing wave in order to verify it.

Figure 9: This is what we expect.

Conclusion

The friction of the string will cause a frequency shift especially for the high frequency. Also, the frequencies which excited in a guitar are always low frequencies.

Reference

1.Giodano, N.J., Nakanishi, H. Computational Physics. Tsinghua University Press, December 2007.

Programme Code

• Chapter 6 Problem 6.16