University Physics I - Classical Mechanics, 2019

300 CHAPTER 12. WAVES IN ONE DIMENSION
Now let us try to see if we can get a sinusoidal solution to this system of differential equations. By
analogy with Eq. (12.3) let
[ ( xn
)]
ξ n (t) =A sin 2π
λ − ft
where x n = nd is the equilibrium position of the n-th mass. Then for each of the three masses
considered, we have
ξ n−1 (t) =A sin[2π((n − 1)d/λ − ft)] = A sin[2π(nd/λ − ft) − 2πd/λ]
ξ n (t) =A sin[2π(nd/λ − ft)]
ξ n+1 (t) =A sin[2π((n +1)d/λ − ft)] = A sin[2π(nd/λ − ft)+2πd/λ] (12.22)
We want to substitute all this in Eq. (12.21). We can use the trigonometric identity sin(a − b)+
sin(a + b) =2sina cos b to simplify ξ n−1 + ξ n+1 :
[ ( )] ( )
nd
2πd
ξ n−1 + ξ n+1 =2A sin 2π
λ − ft cos
(12.23)
λ
then use 1 − cos x =2sin 2 (x/2) to yield
kξ n−1 − 2kξ n + kξ n+1 = −4kA sin 2 ( πd
λ
) [ ( )]
( )
nd
πd
sin 2π
λ − ft = −4k sin 2 ξ n (12.24)
λ
It is clear now that Eq. (12.21) will be satisfied provided the following condition holds:
( ) πd
m(2πf) 2 =4k sin 2 λ
Or, taking the square root and simplifying,
f = 1 π
√ ( )
k πd
m sin λ
(12.25)
(12.26)
This is clearly a more complicated relation between f and λ than just Eq. (12.4). However, since
we can argue that Eq. (12.4) must always hold for a sinusoidal wave, what we have actually found
is that the chain of masses and springs in Fig. (12.8) will support a sinusoidal wave provided the
wave velocity depends on the wavelength as required by Eqs. (12.4) and(12.26):
c = λf =
√
k
m
( )
λ πd
π sin λ
(12.27)
This is an instance of the phenomenon called dispersion: sinusoidal waves of different frequencies
(or wavelengths) have different velocities. One thing that happens in the presence of dispersion is
that, although a single (infinite), sinusoidal wave can travel without changing its shape (provided
f and λ satisfy Eq. (12.26)), a general pulse will be distorted as it propagates through the medium,
often severely so.