__Summary of Chapter 17__

**1.
**Wave is the motion of a disturbance.

**2.
****There are two (2) types of
waves:**

**Transverse Waves**: *The particles of the disturbed medium
move perpendicular to the wave velocity *

*(example: waves on a
string).*

**Longitudinal Waves**: *The particles of the disturbed medium
move parallel to the wave velocity *

*(example: sound waves).*

**3.
**For a ** sinusoidal
wave** the transverse displacement

_{}

This equation represents the displacement of the particles of the
medium as a function of *x* and *t*.

where *y _{m}* is the amplitude
of the wave in (m),

_{}

The expression (_{}) is called the phase angle and its units is ** radians**.

The plus (+)
sign means the wave is traveling to the left.

The minus (-) sign means the wave is traveling to the right.

A
wave is said to be stationary if is ** time
independent**, i.e., y (x) = f (x).

A
travelling wave can be longitudinal or transverse.

Ø **The transverse velovity ( the velocity of
the particles) is given by ;**

_{}

The maximum transverse
velocity is u_{max} = y_{m} w (m/s)

** **

Ø **The transverse acceleration ( the
acceleration of the particles) is given by**;

_{}

The maximum transverse acceleration is a_{max} = y_{m} w^{2}
(m/s^{2})

**4.
**The
** speed of the wave** is constant
and given by

The wavelength *l* is the **distance** between **two**
successive maxima (crests) or minima (throughs) or any two identical points on
the wave.

The frequency of the wave is
also related to the period T in (s) by _{}

**5.
**In
the case of a ** stretched string**
as in the figure, the wave speed is given by:

where *T*
is the tension in the string in (N), *m* is the
linear mass density (kg/m) and v is the speed
of the wave (m/s).

**6.
**The
power transmitted by a harmonic wave moving in a stretched string is given by; _{} with
units of *Watt*

**7.
****Superposition of Waves**

Suppose two waves *y _{1}
(x,t)* and

_{}

*y(x,t)*
is called** ***the resultant
wave***.**

**8.
****Interference of Waves**

The net displacement of two equal sinusoidal waves
(the resultant) traveling in the ** same**
direction equals the algebraic sum of the displacement of the two waves.

*y _{1} = y_{m} sin(k x *

*y _{2}
= y_{m} sin(k x *

The
resultant wave function is *y = y _{1} + y_{2} = [2 y_{m}
cos *

**Amplitude of the resultant
wave**

3 Cases:

a)
The amplitude of the
resultant wave is maximum (constructive interference) when f = 0, 2p, 4p, 6p,

b)
The amplitude of the
resultant wave is zero (destructive interference) when f = p, 3p, 5p,

c) Intermediate
interference when 0 < f
< p etc, ..

**9.
****Standing Waves**

If
two equal sinusoidal waves travel in ** opposite**
directions along a stretched string, their interference with each other produce
a standing wave.

* y _{1} = y_{m}
sin(k x - *

* *

* y = y _{1} + y_{2}
=[ 2 y_{m} sin(k x)] cos(*

* *

** **

** **

**2
Cases**:

a)
position of the ** nodes **or

*kx = n**p*, for
*n = 0, 1, 2, 3, *

* *

but *k
= 2**p**/**l*

* *

Þ *x = **l**/2, **l**, 3**l**/2, = n**l**/2***
for n = 0, 1, 2, 3, **

b)
position of the ** anti-nodes
**or

*k x = **p**/2, 3**p**/2, 5**p**/2, = n**p**/2 *for *n = 0, 1, 2, 3, *

but *k
= 2**p**/**l*

Þ *x = **l**/4, 3**l**/4, 5**l**/4, = n**l**/4***
for n = 1, 3,**

** **

**A standing Wave **

** **

** **

** **

** **

** **

** **

**10.
****Standing Waves and Resonance
in a stretched string**

In
the case of a stretched string resonance occur and standing wave patterns are
formed at certain freqencies called **resonant
frequencies**.

The
standing wave patterns in the case of a stretched string are as follows:

** Fundamental mode**
or

## Second harmonic

The relation between the wavelength (l) and the length of the string (L) is:

_{} or
_{} for n = 1, 2, 3,

As you can see from the figures, n represents the number of
loops or segments.

_{} for n = 1, 2, 3,

In this case we can either **fix
the tension** and varie the frequency to obtain the various harmonics
or **fix the frequency**

and change the tension to obtain the various
harmonics.

__This latter case was done in the Phys. 102 lab.
(Standing waves in a string). __