Working Content > Oscillations and Waves > Waves in 1D > Waves on an elastic string
Prerequisites:
What we saw when we watched the video of a pulse moving on a taut spring was that the pulse seemed to move with a constant velocity. What could that velocity depend on? In the beads & massless springs model, the velocity of a pulse is about the springs pulling on the beads due to the springs not being perfectly aligned, as in the figure below.
How quickly the bead responds depends on how hard its pulled  the tension, T  and the inertia of the beads  the mass of a bead, m. We expect a bigger tension would make the beads respond faster and a bigger mass would make the beads respond slower.
Now to actually derive the wave speed is not TOO bad  it just requires applying Newton's second law to a small bit of string. But the problem is that since we are describing the motion of the shape of the string, we are writing a differential equation for a function of two variables  y(x,t). The result is a partial differential equation  one that relates not just the position and its time derivative like when we had one mass and Newton's second law  but one that relates the time and space derivatives. This is definitely "beyond our pay grade" here. So we will not go through this. (See The Wave Equation on a String (technical).) Instead, we will see what we can do with dimensional analysis alone.
The wave speed is going to be determined by the force parameter, T (the tension), and the mass parameter. For a long spring what should we choose? In the beaded model it might depend on the mass of each bead and the space between them. But we don't really have a beaded model. What might we use for a continuous elastic string? In this case, we expect the speed to depend on the local properties of the string. If we doubled the length of the string (and therefore doubled the mass) we wouldn't expect the moving pulse to know  at least until it got to the end of the first string. So it wouldn't make sense for the speed to depend on the total mass. So if it is going to depend on the mass, it probably should depend not on the total mass, but on the mass per unit length. If m is the mass of the whole string, and it has length L, we would expect it to depend not on m itself, but on the mass density, which we define to be μ (mu):
μ = M/L.
So the question for dimensional analysis is: can we make a speed out of μ and T?
Be careful here! Remember that when we do a dimensional analysis we label what kind of quantity everything is: mass (M), length (L), time (T), or charge (Q). And we use these symbols M, L, T, and Q. These can be confusing since we also use these for variables and for units. But when you see a quantity surrounded by a square bracket "[ ...]" it means that we are asking its dimensionality so what goes on the right are to be interpreted as the symbols MLTQ and NOT as variables or units.
So what are the dimensionalities of the quantities we are trying to relate? These are μ, T, and v_{0} (the wave speed). Since μ is a mass per unit length, T is a force, and v_{0} is a velocity, we have dimensionalities
[μ] = M/L
[T] = [ma] = ML/T^{2}
[v_{0}] = L/T
We want to use T to get L/T but is has a mass in it. So it suggests we divide it by μ. This gives dimensions:
[T/μ] = (ML/T^{2})/(M/L) = L^{2}/T^{2}
This sort of looks like a velocity, or at least like the square of one. If we take the square root of T/μ it will at least have the right dimensionality (and therefore the right units). This suggests that we propose the hypothesis that the wave speed is given by
where T = tension of the string, μ = mass density of the string = M (mass of the string) / L (length of the string).
It turns out that when we do the full math (with the partial differential equations!) this turns out to be exactly correct. There are no extra factors of 2 or π or whatever  that would be invisible to our dimensional analysis argument.
Remember! This is the speed that displacements move along the string; it is NOT the speed with which the bits of the string are moving transversely. Those can be anything and depend on how the original motion moving the first bit of string up and down were moved. 

Workout: Wave speed
Joe Redish 3/28/12
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