# Differential Equation Help - Vibrating String

• bunkergirl198
In summary, the conversation discusses deriving a partial differential equation for a tightly stretched vibrating string with variable density, taking into consideration gravity and frictional force. The suggested approach involves finding the general wave equation and considering the motion of a point mass using Newton's equation and the relevant forces. The conversation also mentions solving the equation with initial and boundary conditions and assuming wave propagation from left to right.
bunkergirl198
Differential Equation Help -- Vibrating String

## Homework Statement

Suppose a tightly stretched vibrating string has a variable density $$\rho$$(x). Assume that the vibration is small and is only in the vertical direction (transverse-vibration). Derive teh PDE taking into consideration the gravity and the frictional force.

## Homework Equations

Gravity = -mg
Friction force on an object moving with velocity v = -$$\beta$$v.

## The Attempt at a Solution

Well. That's the hard part :)

utt + $$\frac{mg+Bv+T}{\rho(x)}$$ uxx=f(x,t)

Initial conditons
u(x,0)=f(x)
ut(x,0)=g(x)

Boundary conditions
u(0,t)=0
u(l,t)=0

I don't really think my differential equation is right, now do I know how to derive it. It was just my spin on an equation I came across a while ago (if I knew what it was I'd cite it.)

Thanks for ANY help :)

There are more than one way to tackle this problem.
First of all, you should find wave equation in string (without point mass). This equation will give you a general wave equation.
Second, consider motion of the point mass by using Newton's equation. There are three force relevant to the mass: frictional force, gravitational force, and string tension (on the left and right of the mass). For string tension, you may make approximation:
$$sin\theta \approx tan\theta \approx \frac{dy}{dx}$$
Solve these these equation with initial condition, their boundary conditions, and continuity of string.
You may further assume that wave is propagating from the left to right. Therefore there are two waves on the left of the mass (propagating to right and left) and only one wave on the right (propagating to the right, or transmission)

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It involves the use of derivatives and is used to describe many scientific phenomena, such as the motion of objects and the change in populations over time.

## 2. How is a vibrating string described by a differential equation?

A vibrating string can be described by the wave equation, which is a type of differential equation that models the behavior of waves. This equation takes into account the tension, density, and length of the string, as well as the speed of the wave.

## 3. What is the general solution to the differential equation for a vibrating string?

The general solution to the wave equation for a vibrating string involves a combination of sine and cosine functions. It also includes constants that depend on the initial conditions of the string, such as its position and velocity at a given time.

## 4. How can differential equations help us understand the vibrations of a string?

Differential equations allow us to mathematically model and analyze the behavior of a vibrating string. By solving the equations, we can determine the displacement, velocity, and acceleration of the string at any point in time, which helps us understand the motion and behavior of the string.

## 5. Are there any real-world applications of differential equations for vibrating strings?

Yes, differential equations for vibrating strings have many real-world applications. They are used in fields such as acoustics, engineering, and music to study and design stringed instruments, as well as in the study of earthquake waves and other types of vibrations.

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