Progressive wave with initial conditions f(x,0)=0, f'(x,0)=g(x)

This allows us to simplify our solution to:y(x,t) = 1/2v[H(x-vt) - H(x+vt)] + CNow, let's use the initial condition y(x,0) = 0 to solve for C:0 = 1/2v[H(x) - H(x)] + C0 = CTherefore, our final solution is:y(x,t) = 1/2v[H(x+vt) - H(x-vt)]This solution represents a wave traveling in the positive x direction with velocity v. I hope this helps clarify the problem for you. Let me know if you have
  • #1
wakko101
68
0
Consider a long straight string that is given an initial impulse. The transverse displacement of the string y(x,t) satisfies the initial condition:

y(x,0) = 0 and y'(x,0) = G(x)

Show that the solution to the wave eq'n satisfying the intitial condition is

y(x,t) = 1/2v[H(x+vt)-H(x-vt)]

where H'(u) = G(u)

Solve for an impulse about x=0 with a step profile of the form:

0 x less than/equal to -a
-b -a < x less than/equal to 0
b 0 < x less than/equal to a
0 x > a

where a and b are positive constants.

I'm not entirely sure where to start with this. In class, we derived a solution for initial conditions of f(x,0) = f(x) and f'(x,0) = 0 by using the standing wave eq'n. Physically, that represented a long string with an initial displacement, released from rest. It's easier to picture that physical situation. This is the opposite, whereby the string has no initial displacement, but an initial velocity. Should I start with the standing wave eq'n again? Or should I go back to the wave equation (d^2f/dx^2=v^2(d^2f/dt^2)?

Any suggestions or clarifications would be appreciated.

Cheers,
W. =)
 
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  • #2


Dear W.,

Thank you for your post. It seems like you are on the right track with using the wave equation to solve this problem. Let's start with the general solution to the wave equation:

y(x,t) = f(x-vt) + g(x+vt)

where f and g are arbitrary functions. This solution represents a wave traveling in the positive x direction with velocity v. Now, let's plug in the initial conditions given in the problem:

y(x,0) = 0 and y'(x,0) = G(x)

This gives us the following equations:

f(x) + g(x) = 0
-vf'(x) + vg'(x) = G(x)

From the first equation, we can solve for g(x) as g(x) = -f(x). Substituting this into the second equation, we get:

-vf'(x) - vf'(x) = G(x)
-2vf'(x) = G(x)
f'(x) = -G(x)/(2v)

Now, we can integrate both sides to solve for f(x):

f(x) = -1/2v * ∫G(x) dx

Substituting this back into our solution for y(x,t), we get:

y(x,t) = -1/2v * ∫G(x-vt) dx + g(x+vt)

Now, let's use the fact that H'(u) = G(u). This means that we can rewrite the integral as:

y(x,t) = -1/2v * ∫H(x-vt) dx + g(x+vt)

Now, we need to find the function g(x+vt) that satisfies the initial condition y'(x,0) = G(x). We can do this by taking the derivative of g(x+vt) and setting it equal to G(x):

g'(x+vt) = G(x)

Integrating both sides with respect to x, we get:

g(x+vt) = ∫G(x) dx + C

where C is a constant of integration. Finally, we can substitute this back into our solution for y(x,t):

y(x,t) = -1/2v * ∫H(x-vt) dx + ∫G(x) dx + C

Now, let's use the fact that H(u) is a step function, which means that it is equal to
 

1. What is a progressive wave?

A progressive wave is a type of wave that moves through a medium, transporting energy from one point to another without displacing the medium itself. Examples of progressive waves include sound waves and water waves.

2. What are initial conditions?

Initial conditions refer to the values of a system at the beginning of a given time period. In the case of a progressive wave with initial conditions of f(x,0)=0 and f'(x,0)=g(x), it means that the displacement (f) and velocity (f') of the wave at time t=0 are both determined by the function g(x).

3. How are initial conditions important in understanding progressive waves?

Initial conditions are important because they provide the starting point for the wave's behavior. The values of f(x,0) and f'(x,0) determine how the wave will propagate through the medium and how it will interact with any obstacles or boundaries.

4. How does the function g(x) affect the behavior of the progressive wave?

The function g(x) is the initial displacement of the wave at time t=0. It determines the amplitude and shape of the wave at the beginning of its motion. Additionally, the derivative of g(x) (represented by f'(x,0)) determines the initial velocity of the wave, which can also affect its behavior.

5. What other factors can affect the motion of a progressive wave?

In addition to initial conditions, factors such as the properties of the medium (such as density and elasticity) and the presence of obstacles or boundaries can also affect the motion of a progressive wave. The type of wave (such as transverse or longitudinal) and the frequency or wavelength can also impact its behavior.

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