How Do You Solve the Wave Equation Using Coefficient Equations?

[Lavace]

Homework Statement

[PLAIN]http://img33.imageshack.us/img33/8236/waveeq.jpg

The Attempt at a Solution

We calculate second differential with respect to x, and t, substitute into the wave equation.

We then equate the coefficients: [A''(x) + (w/v)^2A(x)]sin(wt)=0

We know from SHM equation that: A''(x) = -(w/v)^2A(x), and hence A''(x) = -k^2 A(x)

But where do we go from here? Any hints?

Also, what about part b?

 

[Lavace]

From A''(x) = -k^2 A(x), we seek a solution of the form A(x) = Csin(kx + psi)

Apply our boundary conditions of y(0,t) and y(L,t) both = 0.

We end up with sin(kL) = 0, where kL varies from 0 to 2PI, this implies that kL=nPI where n=1,2,3...

Because it's quantised, we can say k(n) = nPI/L, where n=1,2,3...

Since k = w/v, w(n) =nPI/L . vWhere w(n) are the normal mode frequencies.

Could someone verify this is correct?

 

[Lavace]

Also, any clues for b)?

 

[Redbelly98]

Looks good for part (a).
For (b), I'm not quite sure what they are getting at. In a sense, you already showed this in your derivation for part (a). Maybe they want you to think in terms of the wavelength λ and how it relates to the string length L.

 

[paranormal]

As a scientist, solving the wave equation involves understanding the physical principles behind the equation and using mathematical techniques to solve for the unknown variables. In this case, we are dealing with a second-order differential equation that describes the behavior of a wave in space and time. To solve this equation, we can use methods such as separation of variables, Fourier series, or Laplace transforms.

In this particular attempt at solving the wave equation, the first step is to calculate the second derivative of A(x) with respect to both x and t. This is done by using the chain rule and the product rule. Then, by substituting these derivatives into the wave equation and equating the coefficients, we can obtain a differential equation for A(x) that can be solved using known techniques.

In the case of the SHM equation, we know that the second derivative of A(x) is equal to -(w/v)^2 A(x). By equating this to the coefficient in the wave equation, we can determine that A''(x) = -k^2 A(x), where k is the wave number and is equal to w/v. This simplifies the differential equation and makes it easier to solve.

As for part b, it is important to understand the boundary conditions of the problem in order to fully solve the wave equation. These conditions can include the initial position and velocity of the wave, as well as any constraints on the medium in which the wave is propagating. By incorporating these conditions into the solution, we can obtain a complete solution to the wave equation.

In summary, solving the wave equation involves using mathematical techniques and understanding the physical principles behind the equation. By following the steps outlined in this attempt at solving the equation, we can obtain a solution that describes the behavior of the wave in space and time.