Wave Propagation in a String of Varying Density
Consider a string of total length , made up of three segments of equal length. The mass
per unit length of the first segment is , that of the second is
, and that of the third
.
The third segment is tied to a wall, and the string is stretched by a force of magnitude
applied to the first segment;
is much greater than the total weight of the string.
Part A
How long will it take a transverse wave to propagate from one end of the string to the
other?
Express the time in terms of , , and Ts.
= ________________
Answer:
The tension in the string is much greater then the weight of the string means we
need not to consider the sagging effects of gravity on the string. The velocity of
transverse wave in a stretched string is given by
v
T

Where T is the tension in the string and  is the mass per unit length (linear mass
density of the string).
As the three segments are joined together and stretched by the same force, the
tension in the three segments will be the same. As the mass per unit length of the
different segments is different, the velocities of the wave in the three segments will
be different. Hence the velocities in the three segments are respectively given by
v1 
Ts
;

v2 
Ts
2
and
v3 
Ts

( / 4)
4Ts

Now the lengths of the each segment is equal to L/3, the time required by the wave
to pass through the segments given by
Time = displacement/velocity hence
The time required for the wave to travel through the first segment
t1 
L 
3 Ts
The time required for the wave to travel through the second segment
t2 
L 2
;
3 Ts
t3 
L 
3 4Ts
and
The time required for the wave to travel through the third segment
Hence the total time required for the wave to travel through whole string is
t = t 1 + t2 + t 3
Or
t=


L 
1 L 

.
3  2 2  0.9714 * L
1  2   
3 Ts 
2  6 Ts
Ts
---------------------------------------------