The thirteenth homework of problem 6.12

kongdefeng 2014301020081

1.Background

problem

6.12 Gaussian initial string displacements are convenient for the calculations of this section,but are not very realistic.When a real string,such as a guitar string,is plucked,the initial string displacement is more accurately described by two straight lines that start at the ends of the string (we assume fixed end ）and end at the excrition point,as illustrated in figure 6.4. compare the power spectrum for a string excited in this manner with the results found above for a Gaussian initial wavepacket.

waves

This article is about waves in the scientific sense. For waves on the surface of the ocean or lakes, see Wind wave. For other uses, see Wave (disambiguation).

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.

Wave equation

Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling.
This wave can then be described by the two-dimensional functions
${\displaystyle u(x,t)=F(x-v\ t)}$ (waveform F F traveling to the right)
${\displaystyle u(x,t)=G(x+v\ t)}$ (waveform G G traveling to the left)
or, more generally, by d'Alembert's formula:
${\displaystyle u(x,t)=F(x-vt)+G(x+vt).\,}$
representing two component waveforms F and G traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation

${\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}$,

Standing wave

In physics, a standing wave – also known as a stationary wave – is a wave in a medium in which each point on the axis of the wave has an associated constant amplitude. The locations at which the amplitude is minimum are called nodes, and the locations where the amplitude is maximum are called antinodes.

2.The body and conclusion

First, we consider a kind of simple situation. While the initial displacement has a Gaussian profile, it quickly splits into two separate wavepackets, or pluses, which propagate in oppose directions along the string.

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Second：we can consider two Gaussian profile, semicircle profile, square profile and triangular profile.

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Some situations including one Guassian profile, semicircle profile, square profile and triangular profile

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3.Thanks

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