# Wave on string, c^2 = T/rho. Assume T proportional to rho => c^2 = constant.

Gold Member
We know the speed of a wave on a string of mass density rho and under tension T goes as c^2 = T/rho. Let us play with this fact.

Let there be an imaginary string of mass M between two points a distance D apart whose separation we may want to vary. Let us assume that the imaginary string is like a physical string in the sense that it is actually 3-dimensional.

Let us demand that our string maintains a constant volume, V, regardless of the separation of the two points. For large separations this will require that the cross sectional area gets smaller.

Assume that the tension in the string is proportional to the cross sectional area of the string, pi*r^2, this implies that the tension will vary as 1/D where D is the separation between the points. A long string will be under smaller tension and conversely a short string will be under greater tension.

Let us consider the mass/length of the string, it is:

M/D == rho.

We have required our string tension to be proportional to the cross sectional area of the string, which we can relate to the mass density rho.

V = (pi*r^2)*D --> (pi*r^2) = V/D the cross sectional area.

Tension = T = constant*(pi*r^2) = constant*V/D

speed of wave propagation squared is:

c^2 = T/rho = (constant*V/D)/(M/D) = constant*V/M = another constant.

Help me make some waves. I hold one end of the imaginary string and you hold the other end. Let us be separated by some distance D. You shake the string and so do I. Notice that waves propagate with constant velocity and carry energy and momentum.

Remember the mass density of our imaginary string, rho, is M/D. Think about what you feel as you shake the string as a function of our separation. When you and I have large separations rho gets smaller and so it gets easier to shake the string but the tension also gets smaller which also makes shaking the string easier. Conversely, when we are near each other it will be harder to give our imaginary string a shake.

Let us try to make contact with the virtual photons that are exchanged between charges. For large separations virtual photons must have small energy and momentum but for small separations the energy and momentum can become quite large. In a similar way the above string I can send you large energy and momentum waves only if we are close.

Thank you for any thoughts.

Gold Member
Let us also consider the imaginary string to have only a surface and require the area of the string to stay constant. Long string small circumference. If we also require the tension to be proportional to the circumference of the string we get the same results, c independent of separation.

Staff Emeritus