# Showing a general solution for a wave on a string fixed at one end

• catpants
In summary, the conversation discusses a problem involving a string at rest and an initial condition, and the wave equation is mentioned as a potential solution. The use of separation of variables is suggested to solve the equation.
catpants

## Homework Statement

http://img811.imageshack.us/img811/1989/problem1.png

## Homework Equations

All shown in the above link, AFAIK

## The Attempt at a Solution

For part b, when they say "assume the string is initially at rest" I took that to mean:
$$\frac{\delta\Psi(x,0)}{\delta t}=0$$
But I don't know if that is right. It would be used as some sort of initial conditions for solving some diff eq. But I don't understand what that diff eq would be, or how to set it up. I would imagine I solve this diff eq using sep of vars?

Last edited by a moderator:
The wave equation, which I presume you have already seen, since it is not mentioned in the problem is
$$\frac{\partial^2 \psi}{\partial x^2}= \frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2}$$.

Yes, an initial condition is
[tex]\frac{\partial \psi(x, 0)}{\partial t}= 0[/math]

(By the way, use "\partial" to get the $\partial$ in LaTex.)

I think the wave equation is what I was missing. Do I try and solve it using separation of variables? Also, the end of your post got mangled, what were you trying to say?

Thanks!

## 1. What is a general solution for a wave on a string fixed at one end?

The general solution for a wave on a string fixed at one end is a mathematical expression that describes the displacement of the string at any point in time and space. It takes into account the initial conditions, such as the shape and velocity of the string at t=0, and the properties of the string, such as its tension and mass per unit length.

## 2. How is the general solution derived?

The general solution is derived using the wave equation, which is a second-order partial differential equation that describes the motion of a wave. By applying boundary and initial conditions, the general solution can be obtained through mathematical techniques such as separation of variables or the method of characteristics.

## 3. What are some common forms of the general solution for a wave on a string fixed at one end?

The most common forms of the general solution for a wave on a string fixed at one end include the d'Alembert solution, the Fourier series solution, and the Fourier integral solution. Each form has its own advantages and is used in different situations depending on the properties of the string and the initial conditions.

## 4. How does the general solution change if the string is not fixed at one end?

If the string is not fixed at one end, the general solution will include additional terms that take into account the boundary conditions at the other end of the string. These terms will depend on the type of boundary conditions, such as a free end or a fixed and free end combination, and will affect the overall shape and behavior of the wave on the string.

## 5. Can the general solution be applied to real-life situations?

Yes, the general solution for a wave on a string fixed at one end can be applied to real-life situations, such as musical instruments, earthquake vibrations, and communication systems. However, in most cases, the general solution will need to be modified to account for additional factors, such as damping, non-uniform strings, and external forces.

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