phy1020.DVI

vD=T; and sinceTD1=f, we can write

vDf: (9.3)

This equation relates the temporal frequencyfof the wave to its “spatial frequency”, or wavelength,.

9.3 String Waves

Now let’s examine some properties of waves propagating in strings. Although string waves are occasionally
of interest (as in some musical instruments), the reason we’re interested in them here is that they form a
simple system that’s easy to visualize, yet illustrates many properties that we’ll find later in other kinds of
waves.
First, let’s look at a formula for the speedvof a wave in a string, in terms of the physical properties of
the string (its tension and density). We’ll skip the derivation and just present the result:

vD

s

FT

m=L

; (9.4)

wherevis the wave speed,FTis the tension in the string (in newtons), andm=Lis the mass density of the
string (mass per unit length, in kg/m). (We’ll see later that the speed of sound waves in a fluid follows a
similar formula:vD

p
B=, whereBis the bulk modulus and is the density of the medium. The speed of
sound waves in a solid isvD

p

Y=, whereYis the Young’s modulus.)

9.4 Reflection and Transmission

Next, let’s look at what happens when a wave pulse hits a boundary—for example, a boundary with a lighter
or heavier string. Generally at the boundary there will be areflected wavesthat returns in the opposite
direction as the incident wave, and there will be atransmitted wavethat continues into the new medium, in
the same direction as the incident wave. The various possibilities are shown in Fig. 9.2.
Note the following points:

• When the incident wave is incident on a “heavier” (denser) medium, the returning reflected wave will
  beinverted.


• When the incident wave is incident on a “lighter” medium, the returning reflected wave will be right-
  side up.


• The transmitted wave will always be right-side up.


• A fixed end may be regarded as an infinitely heavy medium, and may be thought of as an end that is
  attached to a heavy wall. In this case there is no transmitted wave.


• A free end may be regarded as a medium of zero density, and may be thought of as an end attached to
  a ring that is free to move up and down a vertical pole. In this case there is no transmitted wave.


• The transmitted wave will be largest when both media have the same density; in this case there is no
  reflected wave, and all of the incident wave is transmitted.

We might ask: in string waves, how much of the incident wave is reflected, and how much is transmitted?
We can define thecoefficient of reflectionRas the ratio of reflected to incident wave energy, and similarly
define acoefficient of transmissionTas the ratio of transmitted to incident wave energy. Since both strings