# What values of a and b satisfy the following wave equation?

• s_gunn
In summary, the values of the constants a and b that satisfy the wave equation for u(x,t) = sin(ax)sin(bt) are b = ca and a = b/c. Rearranging these equations will give you the two solutions needed.
s_gunn

## Homework Statement

For which values of the constants a and b does u(x,t) = sin(ax)sin(bt) satisfy the wave equation Utt = C2uxx?

## The Attempt at a Solution

I've taken the partial differentials:
Ux = acos(ax)sin(bt)
Uxx = -a2sin(ax)sin(bt)
Ut = bcos(bt)sin(ax)
Utt = -b2sin(bt)sin(ax)

and subbed them into the equation to get:
-b2sin(bt)sin(ax)+c2a2sin(ax)sin(bt)=0
so:
(-b2+c2a2)(sin(bt)sin(ax)) = 0

But then I'm lost!

I have a feeling that I may have gone in completely the wrong direction so any help would be greatly appreciated!

You have not gone in the wrong direction.

If p*q=0, what can you say about p and q?

I know that each of them must equal zero making
b = ca and a = b/c
but is rearranging them to make a the subject really all their asking for?! Am i missing something!?

s_gunn said:
I know that each of them must equal zero making
b = ca
This is correct, but you skipped a step, and in the process lost one of the solutions.

But once have the other solution, that's basically all you are asked to do.

Thanks Gokul43201, sometimes you just need a nudge in the right direction!

## What values of a and b satisfy the following wave equation?

The wave equation is a second-order partial differential equation that describes the behavior of waves in a variety of physical systems. It is given by the equation a2u/∂t2 = b2u/∂x2. To find the values of a and b that satisfy this equation, we need to consider the properties of the physical system and the type of wave being modeled.

## What is the physical interpretation of a and b in the wave equation?

The value of a represents the wave speed, while b is related to the stiffness or tension of the medium through which the wave is propagating. These values are often determined experimentally or through theoretical analysis.

## Can a and b be negative in the wave equation?

Yes, both a and b can be negative in the wave equation. Negative values of a represent waves that travel in the opposite direction of the positive direction of x, while negative values of b can indicate a change in the direction or shape of the wave.

## What happens if a or b is equal to zero?

If a is equal to zero, it means that the wave is not propagating or moving at all. If b is equal to zero, the wave equation simplifies to the one-dimensional heat equation, which describes the diffusion of heat over time.

## What are the boundary conditions for the wave equation?

The boundary conditions for the wave equation are the initial conditions, which specify the initial position and velocity of the wave, and the boundary conditions at the edges of the system, which describe how the wave behaves at the boundaries. These conditions are necessary to fully determine the solution to the wave equation.

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