Question

A) Is it length, time, length/time, or none of the above?

b) 1/time, length/time, length²/time, none of the above.

In the wave equation u_tt = c²u_sx, where u(x, t) is the displacement of the string from equilibrium at position x at time t, what are the dimensions of the constant c?

Answer 1

The constant c in the wave equation represents the wave speed with dimensions of length/time.

Step-by-step explanation:

In the wave equation u_tt = c²u_xx, the term c represents the wave speed, and it has a dimension of length/time. To find the dimensions of c, we can look at the equation provided, which shows a second derivative of displacement u with respect to time t on the left side, and a second derivative of u with respect to position x on the right side, multiplied by c². Displacement has a dimension of length [L], and time has a dimension of [T], thus the dimensions of c must be such that when squared, it has dimensions of [L]2/[T]2 to maintain dimensional consistency on both sides of the equation. We can confirm that the dimension of c is indeed [L]/[T], which simplifies to length over time, or simply wave speed.