phy1020.DVI

are under the same tension, these coefficients depend only on the densities of the two strings. Writing the
string density (mass per unit length) as Dm=L, the coefficientsRandTturn out to be (Ref. [8])

RD

p

1 

p

(^2)
p
1 C
p
(^2)

 2

(9.5)

TD

4

p

(^1 2)
p
1 C
p
2 /^2

(9.6)

where the subscripts 1 and 2 refer to the two strings. Note the following about these equations:

1. RCTD 1. (This is due to the conservation of energy.)


2. If 1 D 2 , thenRD 0 andTD 1 : if both strings have the same density, then all of the incident wave
   is transmitted, and none is reflected.


3. If 2 D 0 , thenRD 1 andTD 0 : for a “free” end, all the wave is reflected and none is transmitted.


4. Similarly, if 2 !1, thenRD 1 andTD 0 : for a “fixed” end, all the wave is reflected and none is
   transmitted.

The coefficientsRandTshow how the initial waveenergyis divided among the reflected and transmittted
waves. Theamplitudesof the reflected and transmitted waves (ArandAt, respectively) are related to the
incident wave amplitudeAiby (Ref. [8])

Ar

Ai

D

p

1 

p

(^2)
p
1 C
p
(^2)

(9.7)

At

Ai

D

2

p

(^1)
p
1 C
p
(^2)

(9.8)

9.5 Superposition

What happens when two waves collide? It turns out that while they overlap (a situation calledsuperposition),
their displacementsywill add algebraically. Given two wavesy 1 .x;t/andy 2 .x;t/, the total wavey.x;t/
will be the sum of the two:yDy 1 Cy 2.
An example is shown in Fig. 9.3, where two wave pulses are shown colliding with each other. During the
time that the wave pulses overlap, they add algebraically. Afterwards, the two pulses continue, as if they just
passed right through each other.
This ability of waves to pass through each other is fortunate, and we observe it in everyday life. For
example, you can talk with someone directly across from you, at the same time people to your left and right
can talk to each other. The sound waves pass right through each other, and each person is able to hear and
understand his partner without difficulty. The same is true of light waves: each person is able to see the other
three, because the light waves are able to pass through each other.

9.6 Interference

Closely related to the idea of superposition is the concept of waveinterference. When two waves overlap and
their displacementsyare in the same direction, the two waves will, by superposition, add together to make