Simple Nature - Light and Matter

Our final result for the speed of the pulses is

v=w/t

=

√

2 T

μ

.

The remarkable feature of this result is that the velocity of the

pulses does not depend at all onworh, i.e., any triangular pulse

has the same speed. It is an experimental fact (and we will also

prove rigorously below) that any pulse of any kind, triangular or

otherwise, travels along the string at the same speed. Of course,

after so many approximations we cannot expect to have gotten all

the numerical factors right. The correct result for the speed of the

pulses is

v=

√

T

μ

.

The importance of the above derivation lies in the insight it

brings —that all pulses move with the same speed — rather than in

the details of the numerical result. The reason for our too-high value

for the velocity is not hard to guess. It comes from the assumption

that the acceleration was constant, when actually the total force on

the segment would diminish as it flattened out.

Treatment using calculus

After expending considerable effort for an approximate solution,

we now display the power of calculus with a rigorous and completely

general treatment that is nevertheless much shorter and easier. Let

the flat position of the string define thexaxis, so thatymeasures

how far a point on the string is from equilibrium. The motion of

the string is characterized byy(x,t), a function of two variables.

Knowing that the force on any small segment of string depends

on the curvature of the string in that area, and that the second

derivative is a measure of curvature, it is not surprising to find that

the infinitesimal force dFacting on an infinitesimal segment dxis

given by

dF=T

∂^2 y

∂x^2

dx.

(This can be proved by vector addition of the two infinitesimal forces

acting on either side.) The symbol∂stands for a partial derivative,

e.g., ∂/∂xmeans a derivative with respect toxthat is evaluated

while treatingtas a constant. The acceleration is thena= dF/dm,

or, substituting dm=μdx,

∂^2 y

∂t^2

=

T

μ

∂^2 y

∂x^2

.

The second derivative with respect to time is related to the second

derivative with respect to position. This is no more than a fancy

362 Chapter 6 Waves